# Asymptotic Methods for Solitary Solutions and Compactons

Authors: Ji-Huan He
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles  are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.


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## Body

## 1. Introduction

Soliton was first discovered in 1834 by Russell [1], who observed that a canal boat stopping suddenly gave rise to a solitary wave which traveled down the canal for several miles, without breaking up or losing strength. Russell named this phenomenon the “soliton.”In a highly informative as well as entertaining article [1], Russell gave an engaging historical account of the important scientific observation:I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.His ideas did not earn attention until 1965 when Zabusky and Kruskal [2] began to use a finite difference approach to the study of KdV equation, which was obtained by Korteweg and de Vires [3]. Various analytical methods also led to a complete understanding of solitons, especially the inverse scattering transform proposed by Gardner et al. [4] in 1967. The significance of Russell’s discovery was then fully appreciated. It was discovered that many phenomena in physics, electronics, and biology can be described by a mathematical and physical theory of “soliton.” For a historical account of the scientific development of solitons, the reader is referred to the “Encyclopedia of Complexity and Systems Science,” especially [5, 6]. Some analytical methods leading to our present state of the art are available in several review articles [7–9].


## 2. Basic Properties of Solitary Solutions and Compactons [5, 6]

A soliton is a special traveling wave that after a collision with another soliton eventually emerges unscathed. Solitons are solutions of partial differential equations that model phenomena like water waves or waves along a weakly anharmonic mass-spring chain. A soliton is a bell-like solution as illustrated in Figure1.Figure 1
Bell-like solitary wave.The soliton can be written in a standard form, which is(2.1)u(ξ)=psech2(qξ)=4pe2qξ+e-2qξ+2,
where u(x,t)=u(ξ),ξ=x-ct, and c is the wave velocity.It is obvious that(2.2)limξ→∞u(ξ)=0,limξ→-∞u(ξ)=0.
The soliton has exponential tails, which are the basic character of solitary waves. This property allows the exponential function to describe its solution, see Section 5.9 for detailed discussion.The soliton obeys a superposition-like principle: solitons passing through one another emerge unmodified, see Figure2.Collision of two solitary waves.
(a)
(b)
(c)
(d)
(e)A compacton is a special solitary traveling wave that, unlike a soliton, does not have exponential tails. A compacton-like solution is a special wave solution which can be expressed by the squares of sinusoidal or cosinoidal functions.


## 3. Explanation of Compacton Solutions [6]

Compactons are special solitons with finite wavelength. It was Rosenau and Hyman [10] who first found compactons in 1993.


### 3.1. Compacton: An Oscillatory Wave with No Tails

Now consider a modified version of KdV equation in the form(3.1)ut+(u2)x+(u2)xxx=0.
Introducing a complex variable ξ defined as ξ=x-ct, where c is the velocity of traveling wave, integrating once, and we have
(3.2)-cu+u2+(u2)ξξ=D,
where D is an integral constant, for solitary solution, and we set D=0.We rewrite (3.2) in the form
(3.3)νξξ+ν-cν1/2=0,
where u2=ν.In casec=0, we have periodic solution: ν(ξ)=Acosξ+Bsinξ. Periodic solution of nonlinear oscillators can be approximated by sinusoidal function. It helps understanding if an equation can be classified as oscillatory by direct inspection of its terms.We consider two common-order differential equations whose exact solutions are important for physical understanding:(3.4)u′′-k2u=0,(3.5)u′′+ω2u=0.
Both equations have linear terms with constant coefficients.The crucial difference between these two very simple equations is the sign of the coefficient ofu in the second term. This determines whether the solutions are exponential or oscillatory. The general solution of (3.4) is
(3.6)u=Aekt+Be-kt.
The second equation, (3.5), has a positive coefficient of u, and in this case, the general solution reads
(3.7)u=Acosωt+Bsinωt.
This solution describes an oscillation at the angular velocity ω.Equation (3.3) behaves sometimes like an oscillator when 1-cν-1/2>0, that is, u=ν1/2 has a periodic solution, and we assume that ν can be expressed in the form
(3.8)ν=u2=A2cos4ωξ.
Substituting (3.8) into (3.3) results in
(3.9)12A2ω2cos2ωξ-16A2ω2cos4ωξ-A2cos4ωξ-cAcos2ωξ=0.
We, therefore, have
(3.10)12A2ω2-cA=0,-16A2ω2-A2=0.
Solving the above system, (3.10), yields
(3.11)ω=14,A=43c.
We obtain the solution in the form
(3.12)u=ν1/2=4c3cos2[14(x-ct)].
By a careful inspection, ν can tend to a very small value or even zero, and as a result, 1-cν-1/2 tends to negative infinite, and (3.3) behaves like (3.4) with k→∞; the exponential tails vanish completely at the edge of the bell shape (see Figure 3):
(3.13)u={4c3cos[14(x-ct)],|x-ct|≤2π,0,otherwise.
This is a compact wave. Unlike solitons, compacton does not have exponential tails (Figure 3).Figure 3
Compacton wave without tails.


### 3.2. A Criterion for Oscillatory Thermopower Waves

Thermal conduction in fuel/Bi2Te3/Al2O3 or fuel/Bi2Te3/terracotta systems always results in strong oscillation of the output signals. A criterion for oscillatory thermopower waves is much needed.Recently, Walia et al. proposed a theory of thermopower wave oscillations to describe coupled thermal waves in fuel/Bi2Te3/Al2O3 or fuel/Bi2Te3/terracotta systems [11]. The dimensionless governing equations are as follows [11]:
(3.14)∂u∂t=∂2u∂x2+we-1/u-ℓ(u-ua),(3.15)∂w∂t=-βwe-1/u,
where u is nondimensional temperature, w is the concentration of the fuel, β and ℓ are, respectively, parameters related to the properties of the fuel and volumetric heat transfer, and ua is the ambient temperature.Ignoring the nonlinear term in (3.14), we would have a wave solution. Changing the parameters in the system will result in strong oscillation [11], and an analytical criterion to predict oscillatory thermopower waves is very useful for design of Bi2Te3 films.The system, (3.14) and (3.15), is difficult to solve analytically because of strong nonlinearity. In order to obtain a criterion for oscillatory thermopower waves, some necessary approximations are needed. Equation (3.15) is approximately written in the form
(3.16)∂w∂t=-βw(1-1u).u in (3.16) is assumed to be a known function; solving w in (3.16) results in
(3.17)w=exp{-β(1-1u)}.
The nonlinear term, we-1/u, in (3.14) is expressed in an approximate form
(3.18)we-1/u=exp{-β(1-1u)}e-1/u=(1-β(1-1u))(1-1u)=1-β+2β-1u-βu2.
Equation (3.14) is rewritten in the following equivalent form:
(3.19)∂u∂t=∂2u∂x2+1-β+2β-1u-ℓ(u-ua)+f(u,w),
where f(u,w) is defined as
(3.20)f(u,w)=we-1/u{1-β+2β-1u}.
In order to solve thermopower waves, we make a transform
(3.21)ξ=x-ct,
where c is wave speed.By the transform, (3.21), we convert (3.19) into an ordinary differential equation, which is
(3.22)u′′+cu′+2β-1u-ℓu+(ℓua+1-β)+f(u,w)=0.
We rewrite (3.22) in the form
(3.23)u′′+cu′+2β-1u-ℓu=F(u,w),
where
(3.24)F(u,w)=-(ℓua+1-β)-f(u,w).Note.
Equation (3.23) is exactly equivalent to (3.14). In order to solve (3.23) approximately, we write an iteration formulation
(3.25)un+1′′+cun+1′+2β-1un+1-ℓun+1=F(un(ξ),wn(ξ)).
In (3.25), F can be considered as a known function of ξ. Equation (3.25) is, therefore, similar to a forced nonlinear oscillator.
We search for a periodic solution of (3.25). To this end, we assume that its solution can be expressed in the form
(3.26)un+1(ξ)=Acos(ωξ+θ0).
By an analytical method [7], we can obtain the following approximate frequency:
(3.27)ω=2(2β-1)A2-ℓ.
The assumption, (3.26), follows ω>0, that is,
(3.28)2(2β-1)A2-ℓ>0.This is a criterion for oscillatory thermopower waves. When2(2β-1)/A2-ℓ≤0, we can predict thermopower waves without oscillation.


### 3.3. A Criterion for Gaseous Emission Waves

Lin and Hildemann [12] developed a general mathematical model to predict emissions of volatile organic compounds (VOCs) from hazardous or sanitary landfills. The model includes important mechanisms occurring in unsaturated subsurface landfill environments: biogas flow, leachate flow, diffusion, adsorption, degradation, volatilization, and mass transfer limitations through the top cover. Lin-Hildemann equation for gaseous emission can be expressed as follows [12]:
(3.29)∂u∂t+V∂u∂z=D∂2u∂z2-μu,
with the following boundary/initial conditions:
(3.30)Vu-D∂u∂z=-Huatz=0,(3.31)u(∞,t)=0,(3.32)u(z,0)=Aexp(βz),
where u is the total chemical concentration per unit volume of soil, V is the effective emission speed, and D is effective diffusion coefficient. The definitions of V and D are given in [12].This paper aims at a wave solution of (3.29). By the wave transform,
(3.33)ξ=z-ct.Equation (3.29) is converted into an ordinary differential equation, which is
(3.34)Du′′+(c+V)u′-μu=0,
where c is the wave speed.Considering the boundary condition, (3.31), the solution of (3.34) is
(3.35)u=Aexp(aξ)=Aexp{a(z-ct)},
where
(3.36)a=V-c-(V-c)2+4μD2D.
By the boundary condition, (3.30), we have
(3.37)V-Da=-H.
Solving a and c from (3.36) and (3.37) results in (3.38)a=V+HD,(3.39)c=H+μDV+H.
Considering the initial condition, we have (3.40)β=V+HD<0.
Equation (3.40) can be written in the form
(3.41)β=-νGB/RG+νLB/RL+KT/RGDGE/RG+DLE/RL<0,
where νGB, νLB are the bulk (apparent) gas and water velocities, respectively. DGE, DLE are the effective gaseous and aqueous diffusion coefficients in soil, respectively; RG, RL are phase-partitioning coefficients of gas and liquid, respectively.Equation (3.41) is the criterion for gaseous emission waves.For a wave solution, the initial condition should be expressed in the form of (3.32). If the initial condition cannot be expressed in an exponential function, the criterion for gaseous emission waves becomes invalid. The present criterion can easily be extended to various nonlinear cases.


## 3.1. Compacton: An Oscillatory Wave with No Tails

Now consider a modified version of KdV equation in the form(3.1)ut+(u2)x+(u2)xxx=0.
Introducing a complex variable ξ defined as ξ=x-ct, where c is the velocity of traveling wave, integrating once, and we have
(3.2)-cu+u2+(u2)ξξ=D,
where D is an integral constant, for solitary solution, and we set D=0.We rewrite (3.2) in the form
(3.3)νξξ+ν-cν1/2=0,
where u2=ν.In casec=0, we have periodic solution: ν(ξ)=Acosξ+Bsinξ. Periodic solution of nonlinear oscillators can be approximated by sinusoidal function. It helps understanding if an equation can be classified as oscillatory by direct inspection of its terms.We consider two common-order differential equations whose exact solutions are important for physical understanding:(3.4)u′′-k2u=0,(3.5)u′′+ω2u=0.
Both equations have linear terms with constant coefficients.The crucial difference between these two very simple equations is the sign of the coefficient ofu in the second term. This determines whether the solutions are exponential or oscillatory. The general solution of (3.4) is
(3.6)u=Aekt+Be-kt.
The second equation, (3.5), has a positive coefficient of u, and in this case, the general solution reads
(3.7)u=Acosωt+Bsinωt.
This solution describes an oscillation at the angular velocity ω.Equation (3.3) behaves sometimes like an oscillator when 1-cν-1/2>0, that is, u=ν1/2 has a periodic solution, and we assume that ν can be expressed in the form
(3.8)ν=u2=A2cos4ωξ.
Substituting (3.8) into (3.3) results in
(3.9)12A2ω2cos2ωξ-16A2ω2cos4ωξ-A2cos4ωξ-cAcos2ωξ=0.
We, therefore, have
(3.10)12A2ω2-cA=0,-16A2ω2-A2=0.
Solving the above system, (3.10), yields
(3.11)ω=14,A=43c.
We obtain the solution in the form
(3.12)u=ν1/2=4c3cos2[14(x-ct)].
By a careful inspection, ν can tend to a very small value or even zero, and as a result, 1-cν-1/2 tends to negative infinite, and (3.3) behaves like (3.4) with k→∞; the exponential tails vanish completely at the edge of the bell shape (see Figure 3):
(3.13)u={4c3cos[14(x-ct)],|x-ct|≤2π,0,otherwise.
This is a compact wave. Unlike solitons, compacton does not have exponential tails (Figure 3).Figure 3
Compacton wave without tails.


## 3.2. A Criterion for Oscillatory Thermopower Waves

Thermal conduction in fuel/Bi2Te3/Al2O3 or fuel/Bi2Te3/terracotta systems always results in strong oscillation of the output signals. A criterion for oscillatory thermopower waves is much needed.Recently, Walia et al. proposed a theory of thermopower wave oscillations to describe coupled thermal waves in fuel/Bi2Te3/Al2O3 or fuel/Bi2Te3/terracotta systems [11]. The dimensionless governing equations are as follows [11]:
(3.14)∂u∂t=∂2u∂x2+we-1/u-ℓ(u-ua),(3.15)∂w∂t=-βwe-1/u,
where u is nondimensional temperature, w is the concentration of the fuel, β and ℓ are, respectively, parameters related to the properties of the fuel and volumetric heat transfer, and ua is the ambient temperature.Ignoring the nonlinear term in (3.14), we would have a wave solution. Changing the parameters in the system will result in strong oscillation [11], and an analytical criterion to predict oscillatory thermopower waves is very useful for design of Bi2Te3 films.The system, (3.14) and (3.15), is difficult to solve analytically because of strong nonlinearity. In order to obtain a criterion for oscillatory thermopower waves, some necessary approximations are needed. Equation (3.15) is approximately written in the form
(3.16)∂w∂t=-βw(1-1u).u in (3.16) is assumed to be a known function; solving w in (3.16) results in
(3.17)w=exp{-β(1-1u)}.
The nonlinear term, we-1/u, in (3.14) is expressed in an approximate form
(3.18)we-1/u=exp{-β(1-1u)}e-1/u=(1-β(1-1u))(1-1u)=1-β+2β-1u-βu2.
Equation (3.14) is rewritten in the following equivalent form:
(3.19)∂u∂t=∂2u∂x2+1-β+2β-1u-ℓ(u-ua)+f(u,w),
where f(u,w) is defined as
(3.20)f(u,w)=we-1/u{1-β+2β-1u}.
In order to solve thermopower waves, we make a transform
(3.21)ξ=x-ct,
where c is wave speed.By the transform, (3.21), we convert (3.19) into an ordinary differential equation, which is
(3.22)u′′+cu′+2β-1u-ℓu+(ℓua+1-β)+f(u,w)=0.
We rewrite (3.22) in the form
(3.23)u′′+cu′+2β-1u-ℓu=F(u,w),
where
(3.24)F(u,w)=-(ℓua+1-β)-f(u,w).Note.
Equation (3.23) is exactly equivalent to (3.14). In order to solve (3.23) approximately, we write an iteration formulation
(3.25)un+1′′+cun+1′+2β-1un+1-ℓun+1=F(un(ξ),wn(ξ)).
In (3.25), F can be considered as a known function of ξ. Equation (3.25) is, therefore, similar to a forced nonlinear oscillator.
We search for a periodic solution of (3.25). To this end, we assume that its solution can be expressed in the form
(3.26)un+1(ξ)=Acos(ωξ+θ0).
By an analytical method [7], we can obtain the following approximate frequency:
(3.27)ω=2(2β-1)A2-ℓ.
The assumption, (3.26), follows ω>0, that is,
(3.28)2(2β-1)A2-ℓ>0.This is a criterion for oscillatory thermopower waves. When2(2β-1)/A2-ℓ≤0, we can predict thermopower waves without oscillation.


## 3.3. A Criterion for Gaseous Emission Waves

Lin and Hildemann [12] developed a general mathematical model to predict emissions of volatile organic compounds (VOCs) from hazardous or sanitary landfills. The model includes important mechanisms occurring in unsaturated subsurface landfill environments: biogas flow, leachate flow, diffusion, adsorption, degradation, volatilization, and mass transfer limitations through the top cover. Lin-Hildemann equation for gaseous emission can be expressed as follows [12]:
(3.29)∂u∂t+V∂u∂z=D∂2u∂z2-μu,
with the following boundary/initial conditions:
(3.30)Vu-D∂u∂z=-Huatz=0,(3.31)u(∞,t)=0,(3.32)u(z,0)=Aexp(βz),
where u is the total chemical concentration per unit volume of soil, V is the effective emission speed, and D is effective diffusion coefficient. The definitions of V and D are given in [12].This paper aims at a wave solution of (3.29). By the wave transform,
(3.33)ξ=z-ct.Equation (3.29) is converted into an ordinary differential equation, which is
(3.34)Du′′+(c+V)u′-μu=0,
where c is the wave speed.Considering the boundary condition, (3.31), the solution of (3.34) is
(3.35)u=Aexp(aξ)=Aexp{a(z-ct)},
where
(3.36)a=V-c-(V-c)2+4μD2D.
By the boundary condition, (3.30), we have
(3.37)V-Da=-H.
Solving a and c from (3.36) and (3.37) results in (3.38)a=V+HD,(3.39)c=H+μDV+H.
Considering the initial condition, we have (3.40)β=V+HD<0.
Equation (3.40) can be written in the form
(3.41)β=-νGB/RG+νLB/RL+KT/RGDGE/RG+DLE/RL<0,
where νGB, νLB are the bulk (apparent) gas and water velocities, respectively. DGE, DLE are the effective gaseous and aqueous diffusion coefficients in soil, respectively; RG, RL are phase-partitioning coefficients of gas and liquid, respectively.Equation (3.41) is the criterion for gaseous emission waves.For a wave solution, the initial condition should be expressed in the form of (3.32). If the initial condition cannot be expressed in an exponential function, the criterion for gaseous emission waves becomes invalid. The present criterion can easily be extended to various nonlinear cases.


## 4. Exact Solutions versus Asymptotic Solutions

There is plainly a tendency in the modern nonlinear science community to obtain exact solutions for nonlinear equations. There are many results on the exact solutions of nonlinear equations where the initial or boundary conditions are not considered. These solutions are calledmathematical solutions because the physical constraints on the real-world problem that is being modeled are not accounted for. Our main aim, however, is to find solutions of the underlying problem that satisfy all the initial/boundary conditions that exist. These solutions, naturally, are called the physical solutions of the problem. Consider, for example, the well-known KdV equation
(4.1)∂u∂t-6u∂u∂x+∂3u∂x3=0,
with a solitary solution
(4.2)u=-c2sech2c4(x-ct-ξ0).
Many mathematical solutions for (4.1) could be found that carry no physical meaning (u=1, for instance, is an exact solution of (4.1) that has no physical meaning at all). Other researchers, on the other hand, begin with some very good initial conditions, say
(4.3)u(x,0)=-c2sech2c4(x-ξ0),
and find that the condition is in fact too good to solve the equation. For a travelling solution, for example, we might guess a solution of the form
(4.4)u(x,t)=-c2sech2c4(x+at-ξ0),
where the unknown constant a can be determined by substituting (4.4) into (4.1).An asymptotic approach is, however, to search for an asymptotic solution with physical understanding. If, for example, we feel interest in a solitary solution of (4.1), then we can assume that the solution has the form
(4.5)u(x,t)=1aexp{k(x-ct)}+bexp{-k(x-ct)},
or
(4.6)u(x,t)=1aexp{k(x-ct)}+bexp{-k(x-ct)}+d,
where a, b, c, d, and k are unknown constants which can be determined via various methods.ForN-solitary solutions, we can assume that the solution has the following form:
(4.7)u(x,t)=∑i=1N1aiexp{ki(x-cit)}+biexp{-ki(x-cit)}+di,
or
(4.8)u(x,t)=∑nanexp(qnx+pnt)∑mbmexp(αmx+βmt),
where ai,bi,ci,di,andki and αi,βi,qi,andpi are unknown constants to be further determined.For a two-wave solution, we can assume that(4.9)u=a-1exp(-ξ)+b1exp(ξ)+a0+b-1exp(-η)+b1exp(η)c-1exp(-ξ)+c1exp(ξ)+b0+d-1exp(-η)+d1exp(η),
or
(4.10)u=1c-1exp(-ξ)+c1exp(ξ)+b0+d-1exp(-η)+d1exp(η),
where ξ=k1x+w1t,η=k2x+w2t.Some asymptotic methods are easy and accessible to all nonmathematicians using only pencil and paper. Consider a nonlinear differential equation for corneal shape [13]
(4.11)h′′-ah+b1+h′2=0,
with boundary conditions h(1)=0 and h'(0)=0.Hereby we suggest a Taylor series method to find an asymptotic solution [14].We rewrite (4.11) in the form
(4.12)h′′=ah-b(1+h′2)-1/2.
Incorporating the boundary condition, h'(0)=0, we have
(4.13)h′′(0)=ah(0)-b(1+h′2(0))-1/2=ah0-b.
Differentiating (4.12) with respect to x results in
(4.14)h′′′=ah′+b(1+h′2)-3/2h′h′′.
This yields
(4.15)h′′′(0)=0.
Proceeding a similar way as above, we have
(4.16)h(4)(0)=ah′′(0)+bh′′2(0)=a(ah0-b)+b(ah0-b)2.
Applying the Taylor series, we obtain
(4.17)h(x)=h(0)+h′(0)x+12!h′′(0)x2+13!h′′′(0)x3+14!h(4)(0)x4,
or
(4.18)h(x)=h0+12(ah0-b)x2+124[a(ah0-b)+b(ah0-b)2]x4.
Incorporating the boundary condition, h(1)=0, yields
(4.19)h(1)=h0+12(ah0-b)+124[a(ah0-b)+b(ah0-b)2]=0,
or
(4.20)a2bh02+(24+12a+a2-2ab2)h0-12b-ab+b3=0.
From (4.20), h0 can be solved, which reads [14]
(4.21)h0=-(24+12a+a2-2ab2)+(24+12a+a2-2ab2)2+4a2b(12b+ab-b3)2a2b.
To compare with Okrasiński and Płociniczak’s result, setting a=b=1, we have
(4.22)h0=-35+352+482=0.33956,
which is very close to Okrasiński and Płociniczak’s result [13].The accuracy can be further improved if the solution procedure continues.Comparing the Okrasiński and Płociniczak’s method with our pencil-and-paper method, we conclude that the solution process is accessible to nonmathematicians to solve any nonlinear two-point boundary problems.


## 5. Asymptotic Methods for Solitary Solutions

The investigation of soliton solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. There are many analytical approaches to the search for solitary solutions, see [5, 6]. Among various methods, the homotopy perturbation method, the variational iteration method, the exp-function method, and the variational approach have been worked out over a number of years by numerous authors, and they have matured into relatively fledged analytical methods for nonlinear equations thanks to the efforts of many researchers.


### 5.1. Soliton Perturbation

We consider the following perturbed nonlinear evolution equation [5, 15]:
(5.1)uT+L(u)+N(u)=εR(u),0<ε≪1.
When ε=0, we have an unperturbed equation
(5.2)uT+L(u)+N(u)=0,
which is assumed to have a solitary solution.In caseε≠0, but 0<ε≪1, we can use perturbation theory and look for approximate solutions of (5.1) which are close to solitary solutions of (5.2).Using multiple time scales (a slow timeτ and fast time t scale such that ∂T=∂t+ε∂τ), we assume that the soliton solution can be expressed in the form [5, 15]
(5.3)u(x,T)=u0(ξ,τ)+εu1(ξ,τ,t)+ε2u2(ξ,τ,t)+⋯,
where ξ=x-ct, τ is a slow time, and t is a fast time.Substituting (5.3) into (5.1), then equating like powers of ε, we can obtain a series of linear equations for ui(i=0,1,2,3,…), which can be solved sequentially.In most cases, the nonlinear termR(u) in (5.1) plays an import role in understanding various solitary phenomena, and the coefficient ε is not limited to “small parameter.”


### 5.2. Modified Multitime Expansions [16]

In order to overcome the shortcoming arising in the above solution process, hereby we applied the modified multitime expansions (see Section 2.9 of [16]). To illustrate the method, we consider the following equation:
(5.4)∂u∂T+au∂u∂x+b∂3u∂x3+εR(u)=0.
Introducing the time scales Tn=εnt(n=0,1,2,3,…), and using the parameter-expansion method (see Section 5.8), we assume that the solution and the constants a and b can be expressed as [5, 16]
(5.5)u=u0(T0,T1,T2,…)+εu1(T0,T1,T2,…)+ε2u2(T0,T1,T2,…)+⋯,(5.6)a=a0+εa1+ε2a2+⋯,(5.7)b=b0+εb1+ε2b2+⋯,
where the constants ai and bi can be identified by means of no secular terms. Hereby, we define the secular term in a more general form that the term involves time in the form tnf, even in case tnf tends to zero when t→∞.The equation foru0 is
(5.8)∂u0∂t+a0u0∂u0∂x+b0∂3u0∂x3=0.
We can choose suitably the values of a0 and b0, so that the solution of (5.8) can be easily obtained, and involves the basic properties of the original solution.We use the Duffing equation to illustrate the solution procedure [16]
(5.9)u′′+1·u+εu3=0,u(0)=A,u′(0)=0.
Suppose that the solution can be expressed in (5.5), and the coefficient, 1, can be expanded into
(5.10)1=ω2+εω1+ε2ω2+⋯.
Substituting (5.5) and (5.10) into (5.9) and collecting terms of the same powers of ε, we have
(5.11)D02u0+ω2u0=0,(5.12)D02u1+ω2u1+ω1u0+2D0D1u0+u03=0,
where Di=∂2/∂Ti2.Solving (5.11) with the initial conditions u0(0)=A and u0′(0)=0, we have
(5.13)u0=A(T1,T2)cosωT0.Substitutingu0 into (5.12) results in
(5.14)D02u1+ω2u1+A(ω1+34A2)cosωT0-2ωD1(A)sinωT0+14A3cos3ωT0=0.Eliminating secular terms needs(5.15)D1(A)=0,ω1=-34A2.If only the first-order approximate solution is searched for, from (5.10), we have
(5.16)1=ω2+εω1=ω2-34εA2,
or
(5.17)ω=1+34εA2.The obtained frequency-amplitude relationship, (5.17), is valid for the whole solution domain, and the maximal relative error is less than 7% when εA2→∞.


### 5.3. Variational Approach

This section is an elementary introduction to the concepts of the calculus of variations and its applications to solitary solutions. Generally speaking, there exist two basic ways to describe a nonlinear problem: (1) by differential equations (DE) with initial/boundary conditions; (2) by variational principles (VP). The former is widely used, while the later is rarely used in solitary theory. The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration. Variational methods have been, and continue to be, popular tools for nonlinear problems. When contrasted with other approximate analytical methods, variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions.


#### 5.3.1. Inverse Problem of Calculus of Variations

The inverse problem of calculus of variations is to establish a variational formulation directly from governing equations and boundary/initial conditions. We will use the semi-inverse method [17, 18] to establish various variational principles directly from the governing equations.Consider the well-known Korteweg-de Vries (KdV) equation(5.18)ut+auux+buxxx=0,
where a and b are constants, and the subscripts denote partial differentiations.We rewrite it in a conserved form(5.19)∂u∂t+∂∂x(12au2+buxx)=0.
According to the conservation form of (5.19), we can introduce a potential functional Φ defined by
(5.20)Φx=u,(5.21)Φt=-(12au2+buxx).
So the KdV equation can be written in the form
(5.22)Φxt+aΦxΦxx+bΦxxxx=0,
which can be derived from the following variational principle using the semi-inverse method [17]:
(5.23)J(Φ)=∬{12ΦxΦt+a6Φx3-b2Φxx2}dtdx.
In order to obtain a generalized variational principle with two independent fields (Φ,u), we apply the Lagrange multiplier to (5.23)
(5.24)J(Φ,u,λ)=∬{12ΦxΦt+a6Φx3-b2Φxx2+λ(Φx-u)}dtdx.
The stationary condition with respect to u results in
(5.25)λ=0.
The Lagrange multiplier method is not valid for the case. This phenomenon is called Lagrange crisis. Hereby, we suggest three ways to overcome the crisis [18].(
1
) The Semi-Inverse Method [17, 18]
Generally, the multiplier can be expressed in the form after identification(5.26)λ=λ(u,Φ,ut,ux,Φt,Φx,…).
We replace the last term including the Lagrange multiplier by a new variable F, that is,
(5.27)J(Φ,u)=∬{12ΦxΦt+a6Φx3-b2Φxx2+F}dtdx,
where F is unknown, which can be expressed in the form
(5.28)F=F(u,Φ,ut,ux,Φt,Φx,…).
Equation (5.27) is such constructed according to the semi-inverse method [17]. To identify F, making the functional stationary with respect to u, we have
(5.29)δFδu=0,
where δF/δu is the variational derivative defined as
(5.30)δFδu=∂F∂u-∂∂t(∂F∂ut)-∂∂x(∂F∂ux)+∂2∂x2(∂F∂uxx)+⋯.
Equation (5.29) should be equivalent to (5.20), and this requires
(5.31)δFδu=c(Φx-u),
where c is a nonzero constant. From (5.31), we can determine F as follows:
(5.32)F=-12c(Φx-u)2=c~(Φx-u)2,
where c~ is a constant, c~=-c/2.
We, therefore, obtain the following needed variational principle:(5.33)J(Φ,u)=∬{12ΦxΦt+a6Φx3-b2Φxx2+c~(Φx-u)2}dtdx.
Its Euler-Lagrange equations are
(5.34)-2c~(Φx-u)=0,-Φxt-aΦxΦxx-bΦxxxx-2c~(Φx-u)x=0,
which satisfy the field equations (5.20) and (5.22), respectively.(
2
) The Hidden Lagrange Multiplier [18]
Let us come back to (5.25), λ=0, which should be the constraint equation. This means that (5.25) inexplicitly involves a lost constraint equation, so we can identify the multiplier in the form [18]
(5.35)λ=c~(Φx-u).
This results in the same result above.(
3
) Replacement of Some Variables in the Original Functional Using the Constraint Equation
Sometimes the Lagrange crisis can be eliminated by replacing some variables in the original variational principle using the constraint equation; a detailed discussion was systematically given in [18].
We replaceΦxΦt by uΦt in (5.23) and introduce a Lagrange multiplier in the resultant function
(5.36)J(Φ,u,λ)=∬{12uΦt+a6Φx3-b2Φxx2+λ(Φx-u)}dtdx.
Identification of the multiplier yields
(5.37)λ=12Φt.
Submitting the identified multiplier into (5.36) results in
(5.38)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+12Φt(Φx-u)}dtdx=∬{a6Φx3-b2Φxx2+12ΦtΦx}dtdx.
We find that the constraint is not eliminated yet, and this is another Lagrange crisis, which can also be eliminated by the semi-inverse method
(5.39)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+F}dtdx.
The stationary conditions with respect to u and Φ are, respectively, as follows:
(5.40)12Φt+δFδu=0,-12ut-aΦxΦxx-bΦxxxx+δFδΦ=0.
According to (5.20) and (5.21), we have
(5.41)δFδu=-12Φt=14au2+12buxx,δFδΦ=12ut+aΦxΦxx+bΦxxxx=12ut-Φxt=12Φxt-Φxt=-12Φxt.
From (5.41), F can be determined as
(5.42)F=112au3-14b(ux)2+14ΦxΦt.
We, therefore, obtain the following variational principle:
(5.43)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+112au3-14b(ux)2+14ΦxΦt}dtdx.
By the semi-inverse method, we can obtain various different two-field variational principles, and we write here the following one for reference:
(5.44)J(Φ,u)=∬{12ΦxΦt+a(13u3-12u2Φx)-b2Φxx2}dtdx.
The potential in the above functionals (5.23), (5.33), and (5.44) requires second order of differentiation (Φxx), leading to the complications in the finite element calculation. For the purpose of simplification in finite element computation, we often introduce some additional variables to reduce the order of differentiations. This is of course equivalent to introducing some additional constraints in the variational principle. Generally, we can eliminate the introduced constraints by the Lagrange multiplier method, but as illustrated above, the method might fail.Now we introduce a new variableE defined as
(5.45)E=a2u2+buxx.
By the semi-inverse method [17, 18], we obtain the following three-field variational principle:
(5.46)J(Φ,u,E)=∬{uΦt+EΦx+a3u3-bux2-Eu}dtdx.
It is obvious that all variables in the obtained functional (5.46) are in first-order differentiations, leading to much convenience in numerical simulation.It is easy to establish a variational formulation by introducing a potential function, and we can also establish a variational principle without auxiliary special function. To elucidate this, we consider the KdV equation in the form(5.47)ut+uux+uxxx=0,
where subscripts denote partial differentiations. If we introduce a velocity potential Φ defined as u=Φx, then the KdV equation can be derived from the variational principle
(5.48)δ∬Ldtdx=0,
where the Lagrangian L can be expressed in the form
(5.49)L=12ΦxΦt+16Φx3-12Φxx2.
Our aim is to search for a Lagrangian for (5.47). It is easy to establish a variational formulation for differential equations with even orders. The KdV equation has odd-order differentiations, and therefore, no Lagrangian for (5.47). To circumvent this obstacle, we take partial differentiation with respect to x to both sides of KdV equation, which turns out to be the following form:
(5.50)utx+ux2+uuxx+uxxxx=0.
By the semi-inverse method, we construct a trial Lagrangian in the form
(5.51)L=-12uxut+12uxx2+puux2+qu2uxx,
where p and q are constants to be further determined. Its Euler equation can be readily obtained as follows:
(5.52)utx+uxxxx+p[ux2-2(uux)x]+q[2uuxx-(u2)xx]=0,
or
(5.53)utx+uxxxx+(-p-2q)ux2-2puuxx=0.
Setting  p=-1/2,q=-1/4, then (5.53) turns out to be the modification version of the KdV equation (5.50).Finally, we have the following needed Lagrangian in the form of velocity:(5.54)L=-12uxut+12uxx2-12uux2-14u2uxx.
This approach can be extended to many nonlinear equations. Consider the modified KdV equation
(5.55)ut+mu2ux+nuxxx=0.
Similarly, we change the equation so that it has even-order differentiations
(5.56)utx+2muux2+mu2uxx+nuxxxx=0.
By the same manipulation as illustrated above, we construct a trial Lagrange function in the form
(5.57)L=-12uxut+12nuxx2+pu2ux2+qu3uxx,
where p and q are constants to be further determined. Its Euler equation can be readily obtained as follows:
(5.58)utx+nuxxxx+p[2uux2-2(u2ux)x]+q[3u2uxx-(u3)xx]=0,
or
(5.59)utx+nuxxxx+p[2uux2-2(2uux2+u2uxx)]+q[3u2uxx-(6uux2+3u2uxx)]=0,
or
(5.60)utx+nuxxxx+(-2p-6q)uux2-2pu2uxx=0.
Setting
(5.61)p=-m2,q=-m6,
(5.60) becomes (5.56), and we, therefore, obtain the following Lagrangian:
(5.62)L=-12uxut+12nuxx2-m2u2ux2-m6u3uxx.
We can also use the semi-inverse method to establish a family of variational principles for a nonlinear system. We use one-dimensional traffic flow as an example.The research on traffic flow began at the beginning of the 20th century. Lighthill and Whitham first proposed the fluid-dynamical model for traffic flow [19, 20]. The continuum equation for unsteady one-dimensional traffic flow can be, therefore, written as
(5.63)∂∂t(ρA)+∂∂x(ρuA)=q,
where A is the cross-sectional area of the road, u is the velocity, ρ is the density of cars, and q is the source. The deficiency of the model is that the traffic flow actually cannot be considered as a continuum, and to eliminate this deficiency, a fractional differential model can be introduced:
(5.64)∂∂t(ρA)+DαDxα(ρuA)=q,0<α<1,
where Dα/Dxα is the fractional differential, see Section 7.In 1994, Zheng [21] suggested the following traffic flow model:
(5.65)∂∂t(ρ)+∂∂x(ρu)=0,(5.66)∂∂t(ρu)+∂∂x[ρu2+14n(n-1)2uf2ρ01-nρn]=0,
where uf is the possible maximal velocity, m is a constant, and ρ0 is the minimal traffic density when the cars can travel at a maximal velocity.In order to establish a variational principle for the system, we rewrite (5.66) in the following equivalent form:
(5.67)∂∂t(u)+∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]=0.
According to (5.65) and (5.67), we can introduce two functions Φ and Ψ defined as
(5.68)∂Φ∂x=-ρ,∂Φ∂t=ρu,(5.69)∂Ψ∂x=-u,∂Ψ∂t=12u2+14(n-1)uf2ρ01-nρn-1,
so that (5.65) and (5.66) are automatically satisfied.The essence of the semi-inverse method [17, 18] is to construct an energy-like functional with a certain unknown function, which can be identified step by step. An energy-like trial functional for the discussed problem can be constructed in the following form:
(5.70)J(u,ρ,Φ)=∫0t∫0L{u∂Φ∂t+[12u2+14(n-1)uf2ρ01-nρn-1]∂Φ∂x+F}dtdx,
where u, ρ, and Φ are considered as independent variables, and F is an unknown function of u, ρ and/or their derivatives.There exist various approaches to the establishment of energy-like trial functionals for a physical problem, and illustrative examples can be found in [22, 23].The advantage of the above trial functional lies on the fact that the stationary condition with respective toΦ,
(5.71)δΦJ(u,h,Φ)=∫0t∫0L{-∂∂t(u)-∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]}δΦdtdx=0,
leads to (5.67).Calculating the functional (5.71) stationary with respect to u and ρ, we obtain the following Euler equations:
(5.72)∂Φ∂t+u∂Φ∂x+δFδu=0,[14(n-1)2uf2ρ01-nρn-2]∂Φ∂x+δFδρ=0.
We search for such an F, so that the above (5.72) satisfies the two-field equations. To this end, we set
(5.73)δFδu=-∂Φ∂t-u∂Φ∂x=-ρu+uρ=0,δFδρ=-[14(n-1)2uf2ρ01-nρn-2]∂Φ∂x=14(n-1)2uf2ρ01-nρn-1.
From (5.73), we can immediately identify the unknown F, which reads
(5.74)F=14n(n-1)2uf2ρ01-nρn.
So we obtain the following required variational functional:
(5.75)J(u,ρ,Φ)=∫0t∫0L{14n(n-1)2uf2ρ01-nρn+u∂Φ∂t+[12u2+14(n-1)uf2ρ01-nρn-1]∂Φ∂x}dtdx.Proof.
The Euler equations of the above functional (5.75) are
(5.76)δρ:14(n-1)2uf2ρ01-nρn-1+14(n-1)2uf2ρ01-nρn-2∂Φ∂x=0,(5.77)δu:∂Φ∂t+u∂Φ∂x=0,(5.78)δΦ:-∂∂t(u)-∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]=0.
From (5.68), we have ∂Φ/∂x=-ρ. Substituting the result into (5.77) leads to ∂Φ/∂t=ρu.
From the above three-field variational functional, we can easily obtain two-field or one-field variational function by substituting one- or two-field equations into the functional (5.75). For example, substituting ∂Φ/∂x=-ρ into (5.75), we obtain a two-field variational functional
(5.79)J1(u,Φ)=∫0t∫0L{14n(n-1)2uf2ρ01-nρn+u∂Φ∂t-[12ρu2+14(n-1)uf2ρ01-nρn]}dtdx,
where the variable ρ is not now an independent field. Further constraining the two-field functional (5.79) by the equation ∂Φ/∂t=ρu, we have
(5.80)J2(Φ)=14n(n-1)2uf2ρ01-n∫0t∫0Lρndtdx=∫0t∫0LPdtdx,
where P is the traffic pressure defined as
(5.81)P=Cρn=14n(n-1)2uf2ρ01-nρn.
The functional (5.81) has the same form of the well-known Bateman principle in fluid mechanics [18].
By a paralleling operation, we can also establish a variational functional with free fieldsu,  ρ, and Ψ. A trial functional with an unknown function F can be constructed as follows:
(5.82)J~(u,ρ,Ψ)=∫0t∫0L{ρ∂Ψ∂t+ρu∂Ψ∂x+F}dtdx.
Here, the unknown F is free from Ψ and its derivatives. By the same manipulation as illustrated above, we set
(5.83)δFδu=-ρ∂Ψ∂x=ρu,δFδρ=-∂Ψ∂t-u∂Ψ∂x=12u2-14(n-1)uf2ρ01-nρn-1.
From (5.83), we can determine the unknown F as follows:
(5.84)F=12ρu2-14n(n-1)uf2ρ01-nρn.
So we obtain the following needed variational principle:
(5.85)J~(u,ρ,Ψ)=∫0t∫0L{12ρu2-14n(n-1)uf2ρ01-nρn+ρ∂Ψ∂t+ρu∂Ψ∂x}dtdx.
It is easy to prove that the Euler equations of the above functional (5.85) satisfy the field equations (5.67) and (5.69).
Constraining the functional (5.85) by the equation ∂Ψ/∂x=-u, we obtain
(5.86)J~1(ρ,Ψ)=∫0t∫0L{-12ρ(∂Ψ∂x)2-14n(n-1)uf2ρ01-nρn+ρ∂Ψ∂t}dtdx,
which is under the constraint of the equation ∂Ψ/∂x=-u. Further substituting
(5.87)∂Ψ∂t=(12)u2+(14)(n-1)uf2ρ01-nρn-1
into (5.86) results in
(5.88)J~2=∫0t∫0L{12ρu2-14n(n-1)uf2ρ01-nρn}dtdx=∫0t∫0L{12ρu2-P}dtdx,
which is similar to the well-known Hamilton principle.


#### 5.3.2. A Possible Connection between the Uncertain Principle and the Least Action Principle

Maupertuis-Lagrange’s principle of least kinetic potential action for a particle with massm can be expressed as follows [18]:
(5.89)∫t1t212mv2dt→min.
We rewrite (5.89) in the form
(5.90)∫s1s212mv·ds→min,
or
(5.91)∫s1s2p·ds→min.
Equation (5.91) can be approximately written in the form
(5.92)(p2-p1)·ds12=Emin,
where ds12=s2-s1. Equation (5.92) means that given that the particle begins at position s1 at time t1 and ends at position s2 at time t2, the physical trajectory that connects these two endpoints is an extremum of Δp·Δs, where Δs is the standard deviation of the displacement, and Δp is the deviation of the momentum.For arbitraryΔs or Δp, the following inequality holds:
(5.93)Δp·Δs≥Emin.
This is similar to the uncertainty principle.In optics, Fermat’s principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light(5.94)t=∫s1s2dsν→min.
Equation (5.94) can be approximately written in the form
(5.95)(1ν2-1ν1)ds12→min,
where ds12=s2-s1.The light trajectory that connects these two endpointss1 and s2 satisfies the following equation:
(5.96)(1ν2-1ν1)ds=Tmin,
or
(5.97)-1ν1ν2Δv·Δs=Tmin,
where Δs=s2-s1, Δν=ν2-ν1.For arbitraryΔs or Δv, the following inequality holds:
(5.98)-Δv·Δs=ν1ν2Tmin≥c2Tmin,Δv·Δs≤c2Tmin.


#### 5.3.3. Variational Approach to Nonlinear Oscillators [24]

Consider a general nonlinear oscillator in the form(5.99)u′′+f(u)=0.
Its variational principle can be easily established as follows:
(5.100)J(u)=∫0T/4{-12u′2+F(u)}dt,
where T is the period of the nonlinear oscillator, ∂F/∂u=f.Assume that its solution can be expressed as(5.101)u(t)=pcosqt,
where p and q are the amplitude and frequency of the oscillator, respectively.Substituting (5.101) into (5.100) results in
(5.102)J(p,q)=∫0T/4{-12p2q2sin2qt+F(Acosqt)}dt.
Instead of setting ∂J/∂q=0 and ∂J/∂p=0, we only set [8, 24]
(5.103)∂J∂p=0,
from which the relationship between the amplitude and frequency of the oscillator can be obtained.Explanation of (5.103) was given in [8].Consider a nonlinear oscillator with fractional potential [24]:
(5.104)u′′+εu1/3=0.
Its variational formulation can be readily obtained as follows:
(5.105)J(u)=∫0T/4{-12u′2+34εu4/3}dt.
Substituting (5.101) into (5.105), we obtain
(5.106)J=∫0T/4{-12p2q2sin2qt+34p4/3εcos4/3qt}dt.
Setting
(5.107)∂J∂A=∫0T/4{-Aω2sin2ωt+A1/3εcos4/3ωt}dt=0,
we have
(5.108)ω2=∫0T/4εcos4/3ωtdtA2/3∫0T/4sin2ωtdt=∫0π/2εcos4/3tdtA2/3∫0π/2sin2tdt=1.15959526696393εA3/2.
The exact frequency is ω=1.070451ε1/2A-1/3. The 0.597% accuracy is remarkably good.


#### 5.3.4. Variational Approach to Chemical Reactions

As an illustration, consider the following chemical reaction [25]:
(5.109)nA→C+D,
which obeys the equation
(5.110)dxdt=k(a-x)n,x(0)=0,
where a is the number of molecules A at t=0, x is the number of molecules C (or D) after time t, and k is a reaction constant. At the start of reaction (t=0), there are no molecules C (or D) yet formed, so that the initial condition is x(0)=0.In order to obtain a variational model, we differentiate both sides of (5.110) with respect to time, resulting in
(5.111)d2xdt2=-kn(a-x)n-1dxdt.
Substituting (5.111) into (5.110), we obtain the following second-order differential equation:
(5.112)d2xdt2=-k2n(a-x)2n-1,x(0)=0,x′(0)=kan,
which admits a variational expression in the form
(5.113)J(x)=∫0∞{12(dxdt)2+12k2(a-x)2n}dt.
Its Hamiltonian, therefore, can be written in the form
(5.114)H=12(dxdt)2-12k2(a-x)2n,
where H is a Hamiltonian constant, and it can be determined from initial conditions:
(5.115)H=12x˙(0)2-12k2(a-x(0))2n=0.
Equation (5.114) becomes
(5.116)(dxdt)2=k2(a-x)2n.
In view of the initial conditions x(0)=0 and x′(0)=kan, (5.116) is equivalent to (5.110). This means that the variational principle, (5.113), is exactly equivalent to its differential partner, (5.110).Assume that the solution can be expressed in the form(5.117)x=a(1-e-ηt),
where η is an unknown constant to be further determined.Substituting (5.117) into (5.113), and setting dJ/dη=0, we obtain
(5.118)η=1nkan-1.
So we obtain a first-order approximate solution for the discussed problem
(5.119)x=a⌊1-exp(-n-1/2kan-1t)⌋.
In order to improve accuracy, we can assume that the solution can be expressed in a more general form
(5.120)x=a(1-∑i=1mbie-ηit),
which should satisfy initial conditions x(0)=0, and this requires
(5.121)1-∑i=1mbi=0.
Substituting (5.120) into (5.113), we set
(5.122)∂J∂ηi=0(i=1~m),∂J∂bi=0(i=2~m).
Solving (5.121)-(5.122) simultaneously, we can easily determine 2m parameters. The solution procedure is similar to that illustrated in [25], and we will not discuss in details to solve space.We can also choose the following trial function:(5.123)x=a{1-1(1+pt)q},
where p and q are unknown constants to be further determined. It is obvious that (5.123) satisfies the conditions x(0)=0 and x(+∞)=a. Submitting (5.123) into (5.113), and setting
(5.124)∂J∂p=0,∂J∂q=0,
we obtain with ease
(5.125)p=(n-1)kan-1,q=1n-1.
Thus, we obtain the solution
(5.126)x=a{1-1(1+(n-1)kan-1t)1/(n-1)},
which is the exact solution.


#### 5.3.5. Variational Approach to Solitary Solution

In the review article [7], the variational approach to solitons was outlined by few lines, and now the method has been successfully applied to the search for soliton solutions [26, 27] without requirement of small parameter assumption, leading to an extremely simple and elementary but rigorous derivation of soliton solutions.Considering the KdV equation, we seek its traveling wave solutions in the following frame:(5.127)u(x,t)=U(ξ),ν(x,t)=V(ξ),ξ=x-ct,
where c is angular frequency. Substituting (5.127) into (4.1) yields
(5.128)-cu′-6uu′+u′′′=0,
where prime denotes the differential with respect to ξ.Integrating (5.128) yields the result(5.129a)-cu-3u2+u′′=A,
where A is an integration constant, which can be determined from the initial condition. For solitary solutions or limit cycles, the solutions do not depend upon the initial condition, so we always set A=0, and this results in
(5.129b)-cu-3u2+u′′=0.By the semi-inverse method [17], the following variational formulation is established:
(5.130)J=∫0∞(12cu2+u3+12(dudξ)2)dξ.
Equation (5.130) also hints a Hamiltonian invariant, which reads(5.131a)12cu2+u3-12u′2=B,
where B is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set B=0,
(5.131b)12cu2+u3-12u′2=0.Differentiating (5.131a) with respect to ξ, we have
(5.132)cuu′+3u2u′-u′u′′=0,
which is equivalent to (5.129b).The Hamiltonian invariant (5.131a) and (5.131b) can also be used for construction of an asymptotic solution or used as an auxiliary function in the subequation method, see Section 5.10.The semi-inverse method is a powerful mathematical tool to the search for variational formulae for real-life physical problems.By Ritz method, we search for a solitary wave solution in the form(5.133)u=psech2(qξ),
where p and q are constants to be further determined.Substituting (5.133) into (5.130) results in
(5.134)J=∫0∞[12cp2sech4(qξ)+p3sech6(qξ)+12(4p2q2sech4(qξ)tanh2(qξ)]dξ=cp22q∫0∞sech4(z)dz+p3q∫0∞sech6(z)dz+2p2q∫0∞{sech4(z)tanh4(z)}dz=cp23q+8p315q+4p2q15.
Making J stationary with respect to p and q results in
(5.135)∂J∂p=2cp3q+24p215q+8pq15=0,∂J∂q=-cp23q2-8p315q2+4p215=0,
or simplifying
(5.136)5c+12p+4q2=0,-5c-8p+4q2=0.
From (5.136), we can easily obtain the following relations:
(5.137)p=-12c,q=c4.
So the solitary wave solution can be approximated as
(5.138)u=-c2sech2c4(x-ct-ξ0),
which is the exact solitary wave solution of KdV equation.For the KdV equation expressed in (5.47), we have the following variational principle (see (5.54)):
(5.139)J(u)=∬-∞+∞{-12uxut+12uxx2-12uux2-14u2uxx}dxdt.
We assume that the solitary solution can be expressed in the form
(5.140)u(x,t)=1a1ea5x-a5t+a2e-a7x+a8t+a3,
where a1,a2,…,a8 are unknown constants to be further determined.Substituting (5.140) into (5.139), we have
(5.141)J(u)=J(a1,a2,…,a8).
Making the functional, (5.139), stationary can be approximated achieved by making the function (5.141) stationary, and this requires
(5.142)∂J∂ai=0(i=1~8).
Solving the system, (5.142), we can determine the values for a1,a2,…,a8.


### 5.4. Hamiltonian Approach to Solitary Solution

#### 5.4.1. Generalized Action Principles in Mechanics

We begin with the definition of the action functional as time integral over the LagrangianL of a dynamical system
(5.143)J(xi)=∫t1t2Ldt,
and the Lagrangian is defined as follows:
(5.144)L=12mxi′2-V(xi).
Here, xi′=dxi/dt.Newton’s motion equation can be obtained from the stationary condition of the functional (5.143), which reads
(5.145)mxi′′+∂V∂xi=0.We can introduce some constraints to the action functional (5.143), leading to various principles required. For example, if the total energy is a conserved quantity, that is, T+V=const., which is considered as a constraint of the functional (5.143), then we obtain the Euler-Maupertuis principle (principle of least action) [28]:
(5.146)∫t1t2(2T-const.)dt→min,
or
(5.147)∫t1t2Tdt→min.
In this section, we will obtain Hamiltonian and other actions from the Lagrangian (5.144) by introducing some constraints.Now we introduce a generalized velocity(5.148)pi=∂L∂xi′=mxi′.
We consider (5.148) as a constraint of the action functional (5.143), and accordingly the Lagrangian (5.144) can be written as follows:
(5.149)L(xi,pi)=12mpi2-V(xi).
By the Lagrange multiplier, we have the following generalized Lagrangian:
(5.150)L1(xi,pi,λi)=12mpi2-V(xi)+λi(pi-mxi′).
The multiplier can be readily identified, which reads
(5.151)λi=-1mpi.
Substituting the identified multiplier into (5.150) results in
(5.152)L1(xi,pi)=12mpi2-V(xi)-1mpi(pi-mxi′)=pixi′-H,
where H(xi,pi) is a Hamiltonian:
(5.153)H(xi,pi)=12mpi2+V(xi).
If we introduce a new variable ui defined as
(5.154)ui=xi′,
we consider (5.154) as a constraint of the action functional (5.143), and in such case, the Lagrangian (5.144) can be rewritten as
(5.155)L(xi,ui)=12mui2-V(xi).
By the Lagrange multiplier, we have the following generalized Lagrangian:
(5.156)L2(xi,ui,λi)=12mui2-V(xi)+λi(ui-xi′).
The multiplier can be readily identified, which reads
(5.157)λi=-mui.
Substituting the identified multiplier into (5.156) results in
(5.158)L2(xi,ui)=12mui2-V(xi)-mui(ui-xi′)=muixi′-H~(xi,ui),
where H~(xi,ui) is given by
(5.159)H~(xi,pi)=12mui2+V(xi).From (5.157), we know that the multiplier is actually the generalized velocity
(5.160)λi=-mui=-pi.
Substituting (5.160) into (5.156), and keeping pi an independent variable, we have
(5.161)L3(xi,ui,pi)=12mui2-V(xi)-pi(ui-xi′)=pixi′-H-(xi,ui,pi),
where
(5.162)H-(xi,ui,pi)=-12mui2+V(xi)+piui.
Equation (5.161) is called the Schwinger action [28].By the same manipulation, from (5.151), the multiplier can be also determined as
(5.163)λi=-1mpi=-ui.
So we obtain another action like Schwinger’s, which reads
(5.164)L4(xi,pi,ui)=12mpi2-V(xi)-ui(pi-mxi′)=muixi′-H^(xi,ui,pi),
where
(5.165)H^(xi,ui,pi)=-12mpi2+V(xi)+uipi.A more generalized action can be obtained by linear combination ofL1(xi,ui) and L2(xi,pi):
(5.166)L5(xi,ui,pi)=L1(xi,ui)+L2(xi,pi)=-12mpi2-12mui2+(pi+mui)xi′-2V(xi)=(pi+mui)xi′-He(xi,ui,pi),
where
(5.167)He(xi,ui,pi)=12mpi2+12mui2+2V(xi).
The Euler equations can be readily obtained, which read
(5.168)δxi:ddt(pi+mui)+∂He∂xi=ddt(pi+mui)+2∂V∂xi=0,δui:mxi′-∂He∂ui=mxi′-mui=0,δpi:xi′-∂He∂pi=xi′-1mpi=0.
In a more general form, (5.166) can be written as
(5.169)L5(xi,ui,pi)=αL1(xi,ui)+βL2(xi,pi),
where α and β are constants.Linearly combiningLi(i=1,2,3,4,5), we have
(5.170)L6(xi,ui,pi)=∑i=15αiLi,
where αi are constants, and we often set ∑i=15αi=1.


#### 5.4.2. Modified Hamilton Principles for Initial Value Problems

The Hamilton principle can be written in the form(5.171)J(ui)=∫0T(12mu˙i2-U)dt,
with the special requirements δu(0)=δu(T)=0.Hereui is the velocity component in xi, and U is a potential defined as ∂U/∂xi=-Fi, where Fi is the body force component in xi.The Hamilton’s principle holds only for the conditions prescribed at the beginning and at the end of the motion and is therefore useless to deal with the usual initial condition problems, both as an analytical tool and as a basis for approximate solution methods. It is impossible for most real-life physical problems to prescribe terminal conditions.In order to eliminate the unnecessary final condition att=T, Carini and Genna [29] obtained the following functional for the case ui(0)=u˙i(0)=0:
(5.172)J(ui)=∫0T(12mu¨i2-F¨iui)dt+F˙i(T)ui(T)-Fi(T)u˙i(T).
We found that the natural final conditions at t=T satisfy the physical requirements.Liu [30] considered that, in order to deal with the final condition, one term should be added to the Hamilton principle to result in the following functional:
(5.173)J(ui)=∫0T(12mu˙i2-U)dt-mu˙~iu˙i|t=T,
where u˙~i is a restricted variable, that is, δu˙~i=0. The modification, (5.172), is not a classic variational principle due to that the variable u˙~i has to be prescribed at t=T, so that
(5.174)δu˙i(T)=0,
and this requirement is even more overrestricted than δui(T)=0 in Hamilton principle, so no vital innovation was made due to its inconsistent at the final condition. However, Liu [30] obtained some functionals that can successfully deal with the initial conditions, but the final condition still keeps an issue of polemics.It would be a landmark in the history of calculus of variations after Hamilton if we can extend the principle to all initial-value problems without prescribing both initial and final conditions. In order to deal with the final condition, we consider the conserved Hamiltonian:(5.175)12m∑i=13u˙i2+U=12m∑i=13u˙i02+U0.
Here U0=U|t=0, so we have the following identity:
(5.176)u˙i=∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m,i≠j,
for all t≥0.In order to convert the initial conditionsui(0)=ui0 and u˙i(0)=u˙i0 into natural initial conditions and make the natural final condition satisfy the physical requirement, (5.176), we assume that its modified Hamilton principle can be written in the form
(5.177)J(ui)=∫0T(12mu˙i2-U)dt+f(ui,u˙i)|t=0t=T,
where f is an unknown function of ui and/or its derivatives.Making the above functional stationary, we obtain the following stationary conditions att=0 and t=T:
(5.178)[mu˙i+∂f∂ui]t=0t=T=0,∂f∂u˙i|t=0t=T=0.
We search such an f so that the above 4 equations satisfy all initial conditions at t=0 and meet exactly the physical requirement, for example, (5.176), at t=T.Accordingly, we can assume that(5.179)(∂f∂ui)t=0=mu˙i0|t=0,(∂f∂u˙i)t=0=k(ui-ui0)|t=0,(∂f∂ui)t=T=-mu˙i|t=T=-m∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m|t=T,i≠j.
From the above relations, we can identify f in the form
(5.180)f={mu˙i0ui+k(12u˙i2-u˙i0u˙i)}t=0+He|t=T,
where k is a nonzero constant; He is defined as
(5.181)He=m23Fi[∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m]3/2.
So we obtain the following modified Hamilton principle:
(5.182)J(ui)=∫0T(12mu˙i2-U)dt+{mu˙i0ui+k(12u˙i2-u˙i0u˙i)}t=0+He|t=T,
which holds for all initial value problems.Proof.
Making the obtained functional (5.182) stationary, we obtain (1)
in the solution domain (0<t<T):
(5.183)-mu¨i+Fi=0,
which is Newton’s motion equation;(2)
natural initial conditions(t=0):
(5.184)-mu˙i|t=0+mu˙i0|t=0=0,{k(u˙i-u˙i0)}t=0=0,
which satisfy obviously the initial conditions ui(0)=ui0 and u˙i(0)=u˙i0, respectively;(3)
natural final condition(t=T):
(5.185)mu˙i|t=T-m∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m|t=T=0,
which meet the physical requirement of (5.176) at t=T.
We have alternative approaches to identifyingf by different assumptions in (5.178), leading to various new modifications. We write here few modified Hamilton principles for reference
(5.186)J(ui)=∫0T(12mu˙i2-U)dt+{mu˙i0ui+k1(12ui2-ui0ui)+k2(12u˙i2-u˙i0u˙i)}t=0+He|t=T,J(ui)=∫0T(12mu˙i2-U)dt+{(m-k4)ui0ui-k3ui0u˙i+k3ui(u˙i-u˙i0)+k4u˙i(ui-ui0)}t=0+He|t=T,
where k’s are nonzero constants, and it requires that k1-m≠0 and k3+k4=0.
To summarize, we can conclude from the above derivation and strict proof that the obtained modified Hamilton principles, which are first deduced in the history, and valid for all initial-value problems, are extremely important in both pure and applied sciences due to complete elimination of the long-existing shortcomings in Hamilton principle. The stationary conditions of the obtained variational principle satisfy the Newton’s motion equation, and all initial conditions, furthermore, the natural final condition(t=T), satisfy automatically the physical requirement, making a vital innovation of Hamilton principle.


#### 5.4.3. Hamiltonian Approach to Nonlinear Oscillators [31, 32]

In this paper, we consider the following general oscillator:(5.187)u′′+f(u)=0,
with initial conditions u(0)=A and u′(0)=0.It is easy to establish a variational principle for (5.187), which reads [31]
(5.188)J(u)=∫0T/4{12u'2-F(u)}dt,
where T is the period of the oscillator, ∂F/∂u=f(u).In the functional (5.188), (1/2)u′2 is kinetic energy, and F(u) is potential energy, so the functional (5.188) is the least Lagrangian action, from which we can immediately obtain its Hamiltonian, which reads
(5.189)H=12u′2+F(u)=constant=H0,
or
(5.190)12u′2+F(u)-H0=0.
Equation (5.189) replies that the total energy keeps unchanged during the oscillation.Assume that the solution can be expressed as(5.191)u=Acosωt,
where ω is the frequency.Submitting (5.191) into (5.190) results in a residual
(5.192)R(t)=12A2ω2sin2ωt+F(Acosωt)-H0.
According to the energy balance method proposed in [33], locating at some a special point, that is, ωt=π/4, and setting R(t=π/4ω)=0, we can obtain an approximate frequency-amplitude relationship of the studied nonlinear oscillator. Such treatment is much simple and has been widely used by engineers [34–36]. The accuracy of such location method, however, strongly depends upon the chosen location point. To overcome the shortcoming of the energy balance method, in this paper, we suggest a new approach based on Hamiltonian.From (5.192), we have
(5.193)∂H∂A=0.
Introducing a new function, H-(u), defined as
(5.194)H-(u)=∫0T/4{12u'2+F(u)}dt=14TH,
it is obvious that
(5.195)∂H-∂T=14H.
Equation (5.195) is, then, equivalent to the following one:
(5.196)∂∂A(∂H-∂T)=0,
or
(5.197)∂∂A(∂H-∂(1/ω))=0.
From (5.197), we can obtain approximate frequency-amplitude relationship of a nonlinear oscillator.Consider the Duffing equation(5.198)u′′+u+εu3=0,u(0)=A,u′(0)=0.
Its Hamiltonian can be easily obtained, which reads
(5.199)H=12u′2+12u2+14εu4.
Integrating (5.199) with respect to t from 0 to T/4, we have
(5.200)H-(u)=∫0T/4{12u′2+12u2+14εu4}dt.
Assuming that the solution can be expressed as u=Acosωt and substituting it to (5.200), we obtain
(5.201)H-(u)=∫0T/4{12A2ω2sin2ωt+12A2cos2ωt+14εA4cos4ωt}dt=∫0π/2{12A2ωsin2t+1ω[12A2cos2t+14εA4cos4t]}dt=12A2ω·π4+1ω[12A2·π4+14εA4·34·π4].
Setting
(5.202)∂∂A(∂H-∂(1/ω))=-Aω2·π4+[A·π4+εA3·34·π4]=0,
we obtain the following frequency-amplitude relationship:
(5.203)ω=1+34εA3.
Now we consider another nonlinear oscillator with discontinuity:
(5.204)d2udt2+sgn(u)=0,u(0)=A,u′(0)=0.sgn(u) is +1 and −1 for u>0 and u<0, respectively.Its variational formulation can be written as(5.205)J(u)=∫0T/4(-12u′2+u)dt+∫T/4T/2(-12u′2-u)dt,
and H-(u) can be written in the form
(5.206)H-(u)=∫0T/4(12u′2+u)dt+∫T/4T/2(12u′2-u)dt.
Using u=Acosωt as an approximate solution, we have
(5.207)H-=∫0T/4{12A2ω2sin2ωt+Acosωt}dt+∫T/4T/2{12A2ω2sin2ωt-Acosωt}dt=∫0π/2{12A2ωsin2t+1ωAcost}dt+∫π/2π{12A2ωsin2t-1ωAcost}dt=12A2ω·π4+1ωA+12A2ω·π4+1ωA=A2ω·π4+2ωA.
Setting
(5.208)∂∂A(∂H-∂(1/ω))=-Aω2·π2+2=0,
we obtain
(5.209)ω=2πA=1.128379A,
which is very close to the exact one
(5.210)ωexact(A)=π22A=1.110721A.
The 1.59% accuracy is acceptable considering the simple solution procedure.


#### 5.4.4. Hamilton Approach to Solitary Solutions

The Hamiltonian approach to nonlinear oscillators has been now widely used [37–40]; in this section, we will extend the technology to nonlinear wave equations.The Euler-Lagrange equation of (5.130) reads
(5.211)dLdu-ddξ(dLduξ)=0,
where L=(1/2)cu2+u3+(1/2)(du/dξ)2 and uξ=du/dξ.It is obvious that the above Euler-Lagrange equation is the KdV equation.We write the Lagrange function in the formL=T-V, where T is the kinetic energy defined as
(5.212)T=12(dudξ)2,
and V is potential energy defined as
(5.213)V=-12cu2-u3.
The Hamiltonian invariant reads T+V=B, that is,
(5.214)12u′2-12cu2-u3=B,
where B is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set B=0. Equation (5.214) becomes
(5.215)12cu2+u3-12u′2=0.
Differentiating (5.215) with respect to ξ, we have
(5.216)cuu′+3u2u′-u′u′′=0,
which is equivalent to (5.129b) in case u′≠0.If a solitary solution is solved, considering tail property of a soliton, we assume that(5.217)u(ξ)=∑nanexp(qnξ)∑mbmexp(pmξ),
where ai,bi,pi,andqi are constants to be further determined asymptotically. Equation (5.217) should satisfy (2.2).The assumption, (5.217), is also used in the exp-function method (see Section 5.9), where the unknowns are identified exactly, whereas we will determine their values approximated by the weighted residual method.Substituting (5.217) into (5.216) results in the following residual:
(5.218)R(ξ)=12cu2+u3-12u′2.
We can use the method of the least squares to determine the unknown constants involved in (5.217)
(5.219)∫-∞+∞R2(ξ)dξ→min.
We can also use the location method to simply identify the unknowns. To elucidate the solution procedure, we assume its solitary solution has a simple symmetrical form
(5.220)u(ξ)=1a(ekξ+e-kξ)+b,
where a,b, and k are unknown constants.The residual equation is(5.221)R(ξ)=c2[a(ekξ+e-kξ)+b]2+1[a(ekξ+e-kξ)+b]3-[ak(ekξ-e-kξ)]2[a(ekξ+e-kξ)+b]4.
To determine the values of the unknowns in (5.220), we set
(5.222)R(0)=0,R′′(0)=0,R(4)(0)=0.
Note that R′(0)=R′′′(0)≡0.Simplifying (5.222), we obtain
(5.223)2ac+bc+2=0,2ac+bc+3+2ak2=0,32a2c+14abc-b2c+66a+80a2bk2-3b=0.
Solving (5.223) simultaneously, we have
(5.224)a=-12k2,b=-1k2,c=k2.
We, therefore, obtain the following approximate solitary solution:
(5.225)u(x,t)=1-(1/2k2){exp(k(x-k2t)+exp(-k(x-k2t)))}-1/k2=-2k2{expk(x-k2t)+exp(-k(x-k2t))}+2.
Submitting the obtained solution into (4.1), we find it as an exact solution!Consider the following generalized KdV equation:(5.226)ut+6unux+uxxx=0.
This equation admits no variational formulation. If a traveling wave solution is searched for, we can make a transformation ξ=x-ct, and then (5.226) reduces to an ordinary differential equation
(5.227)-cu′+6unu′+u′′′=0.
Integrating the above equation results in
(5.228)-cu+6n+1un+1+u′′=A,
which admits a variational formulation
(5.229)J(u)=∫-∞+∞{-12cu2+6(n+1)(n+2)un+2-Au-12u′2}dη.
We write T=(1/2)u′2 and E=-(1/2)cu2+(6/(n+1)(n+2))un+2-Au, and then T can be interpreted as “kinetic energy” and E “potential energy.” The variational functional, (5.229), can be rewritten in the form of Lagrangian action
(5.230)J(u)=-∫-∞+∞{T-E}dη,
and the Hamiltonian invariant implies the conversation of energy, which requires
(5.231)-12cu2+6(n+1)(n+2)un+2-Au+12u′2=B,
or
(5.232)u′2=cu2-12(n+1)(n+2)un+2+2Au+2B.
For solitary solutions, A=B=0, and (5.232) reduces to
(5.233)u′2=cu2-12(n+1)(n+2)un+2.
If a solitary solution is solved, we assume that
(5.234)u(ξ)=1aekξ+be-kξ+d.
Substituting (5.234) into (5.233) results in the following residual:
(5.235)R(ξ)=u′2-cu2+12(n+1)(n+2)un+2.
Proceeding a similar way as that for (5.222), we can identify a,b, and d in (5.234), and finally, we obtain
(5.236)u(x,t)=-2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]-2.
For a compacton solution, we assume that
(5.237)u(ξ)=acos2ξb+ccos2ξ.
By a similar solution procedure as above, we can determine a, b, and c with ease.


### 5.5. Variational Principles for Fluids

#### 5.5.1. A Bernoulli-Like Equation for Shallow Wave Propagation

Strong storms and cyclones, underwater earthquakes, high-speed ferries, and aerial and submarine landslides can cause giant surface waves approaching the coast and frequently cause extensive coastal flooding, destruction of coastal constructions, and loss of lives. A fast but reliable prediction of a tsunami pulse is of critical importance, and a simple equation like Bernoulli equation is, therefore, much needed.To this end, we use a one-dimensional nonlinear shallow wave propagation for describing runup of irregular waves on a beach. The basic equations are [41]
(5.238)∂u∂t+u∂u∂x+g∂η∂x=0,(5.239)∂η∂t+∂∂x[(h+η)u]=0,
where η is water displacement, u is the depth-averaged velocity, h is the unperturbed water depth, and g is the gravity acceleration.We will establish a variational principle for the system of (5.238) and (5.239) to search for a principle of conservation of energy for the present problem.In order to establish a variational formulation for the system of (5.238) and (5.239), we introduce a function Φ defined as
(5.240)∂Φ∂x=u,∂Φ∂t=-(12u2+gη).
Equation (5.240) is equivalent to (5.238).Using the semi-inverse method [17], we construct a trial functional in the form
(5.241)J(Φ,u,η)=∬{η∂Φ∂t+(h+η)u∂Φ∂x+F(u,η)}dtdx,
where F is an unknown function of u and η.It is obvious that the stationary condition of (5.241) with respect to Φ is (5.239). Now making the functional, (5.241), stationary with respect to η and u, we obtain the following Euler-Lagrange equations:
(5.242)∂Φ∂t+u∂Φ∂x+∂F∂η=0,(h+η)∂Φ∂x+∂F∂u=0.
The above equations should satisfy (5.240). To this end, using (5.240), we set
(5.243)∂F∂η=-∂Φ∂t-u∂Φ∂x=12u2+gη-u2=-12u2+gη,∂F∂u=-(h+η)∂Φ∂x=-(h+η)u.
From (5.243), the unknown F can be identified, which reads
(5.244)F=-12u2η+12gη2-12hu2.
We, therefore, obtain the needed variational formulation, which is
(5.245)J(Φ,u,η)=∬{η∂Φ∂t+(h+η)u∂Φ∂x-12(η+h)u2+12gη2}dtdx.
Submitting (5.240) into (5.245), we obtain a constrained variational formulation
(5.246)J(Φ)=∬{12hu2-12gη2}dtdx,
which is under constraints of (5.240).Proof.
Making the functional, (5.246), stationary, we have
(5.247)δJ(Φ)=∬{huδu-gηδη}dtdx.
According to the constraints, (5.240), we have
(5.248)δ(∂Φ∂x)=δu,(5.249)δ(∂Φ∂t)=-δ(12u2+gη)=-(uδu+gδη).
Replacing δu in (5.249) by δ(∂Φ/∂x) results in
(5.250)gηδη=-ηδ(∂Φ∂t)-ηuδ(∂Φ∂x).
Submitting (5.248) and (5.250) into (5.247), we obtain
(5.251)δJ(Φ)=∬{huδ(∂Φ∂x)+ηδ(∂Φ∂t)+ηuδ(∂Φ∂x)}dtdx=∬{(h+η)u∂∂x(δΦ)+η∂∂t(δΦ)}dtdx=-∬{∂((h+η)u)∂x+∂η∂t}δΦdtdx+∬{∂∂x[(h+η)uδΦ]+∂∂t(ηδΦ)}dtdx=-∬{∂((h+η)u)∂x+∂η∂t}δΦdtdx+∫t(h+η)uδΦdt+∫xηδΦdx.
Setting δJ(Φ)=0, and ignoring the boundary terms, for arbitrary δΦ, we can obtain (5.239) as the needed stationary condition.
We can rewrite (5.246) in the form
(5.252)J(Φ)=∬{T-V}dtdx,
where T=(1/2)hu2 is a generalized kinetic energy of water wave, is V=(1/2)gη2 and a generalized gravitational potential.
Equation (5.252) is a principle of the Lagrange action for the shallow wave propagation, and it also implies conservation of energy T+V=H:
(5.253)12hu2+12gη2=H,
where H is a constant.
We call (5.253) a Bernoulli-like equation for shallow wave propagation.Tsunami prediction, for example, after an earthquake, is of critical importance to save life and property. The extent of catastrophe depends mainly on the height of the Tsunami pulses, that is, the value ofη. It is easy to observe the velocity of a wave in the sea, and we, therefore, can make easy but reliable tsunami prediction using the obtained Bernoulli-like equation.


#### 5.5.2. Lin’s Variational Principle for Ideal Flows

It is well known that the Hamilton’s principle can be applied to a single fluid particle or a closed system by the involutory transformation. For an isoentropy rotational flow, a variational principle can be established using Lin’s constraints [42]. But the essence of Lin’s constraints is not clear yet, and this paper concludes that the functional with Lin’s constraints is not a genuine variational principle, but an approximate one. In addition, a new generalized variational principle with only 6 independent variables (including only one Lin’s constraint) for three-dimensional unsteady compressible rotational flow is established.In 1950s, great progress had been made on the research of variational principle in solid mechanics, with Hu-Washizu variational principle [43] as milestone, and in fluid mechanics, with Lin’s constraints [42] as milestone.Hamilton’s principle was so successfully and powerfully applied to particle mechanics that many attempts have been made to obtain the momentum equations from a variational principle patterned after Hamilton’s principle. These attempts have not all been successful except in the case of isoentropy irrotational flow. For a more general one, Lin’s constraints [44] must be added.Considering a 3D unsteady inviscid compressible rotational flow, we have the following equations [45].(1)
Momentum equation:(5.254)∂ui∂t+ui∂ui∂xi=-1ρ∂P∂xi,
where ui is the flow velocity in xi indirect, P is the pressure, and ρ is the density.(2)
Equation of state:(5.255)P=ρκexp{S-S0m},
where κ is a specific heat ratio, m=(κ-1)-1, and S is entropy.(3)
Continuity equation:(5.256)∂ρ∂t+∂(ρuj)∂xj=0.(4)
Isoentropic equation:(5.257)∂S∂t+uj∂S∂xj=0.(5)
Steady Bernoulli’s equation:(5.258)12ν2+Π=B.For unsteady flow, we should use the following momentum equation:(5.259)∂ν→∂t+∇(12ν2)-ν→×(∇×ν→)+∇Π=0,
where Π=∫(dP/ρ) and ν→=u1i→+u2j→+u3k→.From the equation of state, we can deduce the following equation:(5.260)Π=κκ-1ρκ-1exp{S-S0m}.
In 1955, Herivel first applied Hamilton’s principle to fluid mechanics and deduced the following functional [46]:
(5.261)J(xi)=∫t∫CV{12ρui2-ρE(ρ,S)}dVdt,
where E is specific internal energy. CV is the control volume in Eulerian open space. Hamiltonian fluid mechanics becomes a branch of fluid mechanics and applied mathematics as well [47–56].Applying the Lagrange multipliers to remove the constraint equations (5.257) and (5.256) yields
(5.262)J=∫t∫CV{12ρui2-ρE(ρ,S)-ρη[∂S∂t+uj∂S∂xj]+Φ[∂ρ∂t+∂(ρuj)∂xj]}dVdt.
Making the above functional stationary with respect to ui, we have
(5.263)ν→=∇Φ+η∇S.
However, according to the Crocco equation [45],
(5.264)(∇×ν→)×ν→=T∇S-∇i∙,
where i∙ is enthalpy, T is temperature, and the homentropic flow may be a rotational one. Equation (5.263) implies, however, that homentropic flow (∇S=0) must be a potential flow or irrotational flow (ν→=∇Φ), confiliting with (5.264), and therefore, the functional, equation (5.261) or (5.262), is not reasonable, which was just pointed out by Lin [44].According to Clebsch [57], an arbitrary velocity vector can be expressed as follows:
(5.265)ν→=∇Φ+λi+∇τi,
whereΦ, λi, and τi are Clebsch variables.Lin introduced three-additional constraint equations (Lin’s constraint equations) to the functional (5.262) [44]:
(5.266)DαiDt=0(i=1,2,3),
where αi is Lagrange variables in Lagrange space.Introducing the Lagrange multipliers to remove Lin’s constraints, we have the following general functional:(5.267)J=∫t∫V{12ρui2-ρE(ρ,S)-ρη[∂S∂t+uj∂S∂xj]+Φ[∂ρ∂t+∂(ρuj)∂xj]-ρβi[∂αi∂t+uj∂αi∂xj]}dVdt.
Making the above functional (5.267) stationary with respect to ui, we have
(5.268)ν→=∇Φ+η∇S+βi∇αi.
This expression says that a homentropic flow (∇S=0) can also be an rotational flow, agreeing with (5.264). So the functional (5.267) with Lin’s constraints is acceptable for practical applications. But the essence of the Lin’s constraints has been bewildered for more than half century.Now considering a fluid particle, and applying Hamilton’s principle to construct the following functional:(5.269)J(xi)=∫t(12x˙i2-Π)dt,
where dot means the partial derivation to time alone pathline, that is, material derivation, its Eulerian equation can be easily deduced
(5.270)x¨i=-∂Π∂xi=-1ρ∂P∂xi
that is, the momentum equation for a perfect flow, and it is easly to prove [46] that Π=E+P/ρ, so Π represents enthalpy.The involutory transformation is to convert Lagrange space into Euler space:(5.271)x˙i=ui(i=1,2,3),
which are the pathline equations in the Eulerian form and are equivalent to (5.266) in Lagrangian form, that is, Lin’s constraint equations. So it is very clear that Lin’s constraints are actually the variational constraints introduced by the space exchange. The Hamilton principle is valid for Lagrange space, while all equations in fluid mechanics are derived through Euler space, and Lin’s constraints arise in the space change.Using the Lagrange multipliers to remove the constraints, (5.271), we have
(5.272)J(xi,ui,λi)=∫t(12ν2-Π+λi(x˙i-ui))dt.
Identifying the Lagrange multipliers
(5.273)λi=ui
results in the following variational principle:
(5.274)J(xi,ui)=∫t(12ν2-uix˙i+Π)dt.
Applying Hamilton’s principle to a closed system, we have
(5.275)J(xi)=∫t∫V(t)ρ0(12x˙i2-E)dV(t)dt,
where V(t) is the volume of the closed system in Lagrange space and keeps unchanged, and ∫V(t)ρ0dV is the total mass in the closed system.Making the involutory transformation, we obtain the following functional via Lagrange multiplier method(5.276)J(xi,ui)=∫t∫V(t)ρ0(uix˙i-12ui2-E)dV(t)dt.
It is easy to see that the functional (5.275) or (5.276) ceases to be a stationary variational principle if we transform the closed system in Lagrange space to an open system is Euler space ∫t∫V(t)ρ0dV(t)dt→∫t∫CVρdVdt where a new variable ρ in introduced.In order to guarantee functional (5.276) to be still a stationary principle during space transformation, according to the semi-inverse method [14], let ρ be an independent variable and include an unknown function F in the functional:
(5.277)J(ui,ρ)=∫t∫CV{ρ(12ui2-E)+F(ui,ρ)}dVdt,
where F is an unknown functional of xi,ρ, and CV is the control volume. In the next section the identification of F will be given in details.Rotational equation with Clebsch variables can be expressed as follows:(5.278)ν→=∇Φ+α∇β.
Assuming that ∂α/∂t=0 and Dα/Dt=0, we can deduce the following energy equation:
(5.279)∂Φ∂t+α∂β∂t+12ν2+κκ-1Pρ=R(t).
According to the semi-inverse method [17], we construct a trial functional in the form
(5.280)J(ν→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12ν2-Π)+F}dVdt,
where F is an unknown function of ν→,ρ,Π,Φ,α,  and  β.Making the functional (5.280) stationary with respect to ν→, we have
(5.281)ρν→+∂F∂ν→=0.
Assuming that (5.281) satisfies (5.278), we set
(5.282)∂F∂ν→=-ρν→=-ρ(∇Φ+α∇β).
The unknown function can be identified as follows:
(5.283)F=-ρν→·(∇Φ+α∇β)+f,
where f is an unknown function of ρ,Π,Φ,α,  and  β. Such and trial functional can be updated as follows:
(5.284)J(ν→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12ν2-ν→·(∇Φ+α∇β)-Π)+f}dVdt.
Now the stationary condition with respect to δΦ and continuity equation yield
(5.285)∂f∂Φ=-∇·(ρν→)=∂ρ∂t.
From (5.285), f can be determined in the form
(5.286)f=Φ∂ρ∂t+h,
where h is an unknown function of ρ,Π,α,  and  β. Substituting (5.286) to the trial functional and then taking by part to yield the following new trial functional:
(5.287)J(v→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12v2-v→·(∇Φ+α∇β)-Π-∂Φ∂t)+h}dVdt.
Doing the same as before,
(5.288)δρ:∂h∂ρ=-12v2+v→·(∇Φ+α∇β)+Π+∂Φ∂t=12v2+Π+∂Φ∂t=R-α∂β∂t.
We identifying the unknown function h to yield
(5.289)h=ρ(R-α∂β∂t)+g,
where g is an unknown function of Π,α,  and  β, doing the same way as before,
(5.290)δΠ:∂g∂Π=ρ=(exp{-S-S0m}κ-1κΠ)m.
Identifying the unknown function g to get
(5.291)g=(κ-1κexp(-S-S0m))mκ-1κΠκ/(κ-1),
we get the following generalized variational principle:
(5.292)J(v→,ρ,Π,Φ,α,β)=∫t∫CVLdVdt,
where
(5.293)L=ρ(12ν2-ν→·(∇Φ+α∇β)-Π-∂Φ∂t-α∂β∂t+R)+(κ-1κexp(-S-S0m))mκ-1κΠκ/(κ-1).
The deduced generalized variational principle (GVP) with only 6 independent variables, (5.292), is unknown to the present time and more general and concise than any other known generalized variationals such as the correspondent GVP with Lin’s constraints which needs 11 independent variables (v→,P,ρ and extra three Lin’s constraints together with three Lagrange multipliers, and one Lagrange multiplier for continuity equation, one Lagrange multiplier for isoentropy equation) and actually is of little utility solving the 3D unsteady rotational flow, contrarily the obtained GVP of (5.292) is so beautiful and concise that the theorem makes it possible to deal with the problem via VP-based FEM; and have wide applicability of the solution (DS/Dt≠0). Making the above functional (5.292) stationary, we can get all the Eulerian equations and following additional two equations:
(5.294)DαDt=0,DβDt=1ρ(κ-1κ)m1κΠκ/(κ-1)exp{-(S-S0)}∂S∂α,=1κ-1ρκ-1exp{-S-S0m}∂S∂α=1κPρ∂S∂α.
The first equation of (5.294) is actually Lin’s constraint equation. So via semi-inverse method, the Lin’s constraints equations can reduce to only one, which is the best way to deal with Lin’s constraints in the history.Hereby we have successfully explained the phenomenon of Lin’s constraints via involutory transformation and deduced a generalized variational principle with only 6 independent variables via semi-inverse method, which makes it possible to use FEM to calculate the 3D unsteady compressible rotational flow. The corresponding generalized variational principle with Lin’s constraints is actually an approximate one (for Herrivel’s principle actually is a wrong principle), which, however, can find some application, especially in 2D problems, due to the fact that we can deduce all the Eulerian equations from the functional when making it stationary.


### 5.6. Variational Iteration Method

The variational iteration method [58–61] is an effective method for searching for various wave solutions including periodic solutions, solitons, and compacton solutions without linearization or weak nonlinearity assumptions, see, for example, [62–64].


#### 5.6.1. The Lagrange Multiplier

The variational iteration method has been shown to solve a large class of nonlinear problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions.To illustrate the basic idea of the technique, we consider the following general nonlinear system:(5.295)L[u(t)]+N[u(t)]=0,
where L is a linear operator, and N is a nonlinear operator.The basic concept of the method is to construct a correction functional for the system (5.295), which reads
(5.296)un+1(t)=un(t)+∫t0tλ{Lun(s)+Nu~n(s)}ds,
where λ is a general Lagrange multiplier that can be identified optimally via variational theory, un is the nth approximate solution, and u~n denotes a restricted variation, that is, δu~n=0. To illustrate how restricted variation works in the variational iteration method, we consider a simple algebraic equation
(5.297)x2-3x+2=0.
We rewrite (5.297) in the form
(5.298)x~·x-3x+2=0,
where x~ is called a restricted variable whose value is assumed to be known (5.6.1. the initial guess). Solving x from (5.298) leads to the result
(5.299)x=23-x~,
or in iteration form,
(5.300)xn+1=23-xn.
After identifying the multiplier in (5.296), we have the following iteration algorithm:
(5.301)un+1(t)=un(t)+∫t0tλ{Lun(s)+Nun(s)}ds.
Equation (5.301) is called the variational iteration algorithm-I.Consider the following nonlinear equation ofkth order:
(5.302)u(k)+f(u,u′,u′′,…,u(k))=0.
The variational iteration formulation is constructed as follows:
(5.303)un+1(t)=un(t)+∫t0tλ(un(k)+f~n)ds,
where δf~n=0, fn=f(un,un′,un′′,…). After identifying the multiplier, we have
(5.304)un+1(t)=un(t)+(-1)n∫t0t1(n-1)!(s-t)n-1[un(k)(s)+fn]ds.
This formulation is generally called the variational iteration algorithm-I [11, 61].The main merit of this iteration formula lies in the fact thatu0(t), the initial solution, can be freely chosen, with even unknown parameters contained. However, some repeated and unnecessary iterations are involved in this iteration algorithm at each step.For initial value problems, we can begin with(5.305)u0(t)=u(0)+tu′(0)+12!t2u′′(0)+⋯1k!tku(k)(0).
This leads to a series solution converging to the exact solution.For boundary value problems, the initial guess can be expressed in the form(5.306)u0(t)=a1g1(t)+a2g2(t)+⋯+akgk(t),
where gk(t) are known functions, and ak are unknowns to be further determined after a few iterations by the boundary conditions.After identifying the Lagrange multiplierλ in (5.296), we can construct the iteration formula
(5.307)un+1(t)=u0(t)+(-1)n∫t0t1(n-1)!(s-t)n-1fnds,
instead of the iteration algorithm (5.301). We call (5.307) the variational iteration algorithm-II [11, 61].Note:u0 must satisfy the initial/boundary conditions. This is the main shortcoming of the algorithm.From (5.307), we obtain the following variational iteration algorithm-III
(5.308)un+2(t)=un+1(t)+∫t0tλ{fn+1(s)-fn(s)}ds.
One common property of both the variational iteration algorithm-I and the variational iteration algorithm-III is the allowed dependence of the initial guess on unknown parameters whose values could be identified after a few iterations by using the initial/boundary conditions. the variational iteration algorithm-III, in particular, is highly suitable for boundary value problems of high orders.


#### 5.6.2. Laplace Transform for Identification of the Lagrange Multiplier [65]

We can also apply Laplace transform to identify the Lagrange multiplier. By Laplace transform, we have(5.309)skU(s)+ℓ{f(u,u′,…,u(k))}=0,
or
(5.310)U(s)=-ℓ{f(u,u′,…,u(k))}sk.
The inverse Laplace transform reads
(5.311)u(t)=(-1)k∫0t(η-t)k-1(k-1)!f(un(η),un′(η),…,un(k)(η))dη.
Hence, the following iteration algorithm is derived:
(5.312)un+1(t)=u0(t)+(-1)k∫0t(η-t)k-1(k-1)!f(un(η),un′(η),…,un(k)(η))dη.


#### 5.6.3.  Variational Iteration Method for Solitary Solutions

Here is an incomplete list for variational iteration formulas for various differential equations [8, 9]:
(5.313)u′+f(u,u′)=0,un+1(t)=u0(t)-∫0tf(un,un′)ds,u′+αu+f(u,u′)=0,un+1(t)=u0(t)-∫0teα(s-t)f(un,un′)ds,u′′+f(u,u′,u′′)=0,un+1(t)=u0(t)+∫0t(s-t)f(un,un′,un′′)ds,u′′+ω2u+f(u,u′,u′′)=0,un+1(t)=u0(t)+1ω∫0tsinω(s-t)f(un,un′,un′′)ds,u′′-α2u+f(u,u′,u′′)=0,un+1(t)=u0(t)+∫0t12α(eα(s-t)-eα(t-s))f(un,un′,un′′)ds,u′′′+f(u,u′,u′′,u′′′)=0,un+1(t)=u0(t)-∫0t12(s-t)2f(un,un′,un′′,un′′′)ds,u(4)+f(u,u′,u′′,u′′′,u(4))=0,un+1(t)=u0(t)+∫0t16(s-t)3f(un,un′,un′′,un′′′,un(4))ds,u(k)+f(u,u′,u′′,…,u(k))=0,un+1(t)=u0(t)+(-1)n∫0t1(n-1)!(s-t)n-1f(un,un′,un′′,…,un(k))ds.
The main feature of the method is that the initial solution can be chosen with some unknown parameters in the form of the searched solution. For example, for solitons, we begin with
(5.314)u0=psech2(qξ),ξ=x+ct,
where p and q are the unknown parameters to be further identified after one or few iterations. For a more general form for solitary solutions, we assume that
(5.315)u0(ξ)=∑i=-nmcieiξ∑i=-pqbieiξ,ξ=x+ct,
where ci and bi are constants to be further determined.For discontinuous solitons, we can assume, for example, the following form:(5.316)u(ξ)=pexp(-q|ξ|),
or
(5.317)u0(ξ)=∑i=-nmciei|ξ|∑i=-pqbiei|ξ|,
where p and q are the unknown parameters to be further identified.For compacton-like solution, we assume that the solution has the form(5.318)u0(x,t)=asin2(kx+ωt)b+csin2(kx+ωt),
where a,b,k, and ω are unknown constants further to be determined
(5.319)u0(x,t)=∑iaisin2(kix+ωit)∑j[bj+cjsin2(kjx+ωjt)].As an illustrating example, we consider the following modified KdV equation(5.320)ut+u2ux+uxxx=0.
Its iteration formulation can be constructed as follows:
(5.321)un+1(x,t)=un(x,t)-∫0t{(un)t+un2(un)x+(un)xxx}dt.
To search for its compacton-like solution, we assume that the solution has the form
(5.322)u0(x,t)=asin2(kx+wt)b+csin2(kx+wt),
where a, b, k,and w are unknown constants further to be determined.Substituting (5.322) into (5.321), we can calculate u1 and u2 with ease. In order to identify the constants in the initial solution, we can set
(5.323)∂k∂tkun(x,t)=∂k∂tkun+1(x,t),
setting
(5.324)∂∂tu0(x,t)=∂∂tu1(x,t).
By a simple calculation (see [60] for details), the compacton-like solution is obtained, which reads
(5.325)u(x,t)=±22kcsin2(kx-4k3t)-(3/2)c+csin2(kx-4k3t),
or
(5.326)u(x,t)=±42ksin2(kx-4k3t)-3+2sin2(kx-4k3t).
We can also begin with a more general initial solution in the form
(5.327)u0(x,t)=a+1c+dcos(kx+wt),
where a, b, c, k, and w are unknown constants.Setting∂u0/∂t=∂u1/∂t, we, therefore, obtain the following new compacton-like solution:
(5.328)u(x,t)=a+1-a/3k2±(1/6k2)4a2-6k2cos(kx+(k3-ka2)t).
If we choose k=iK, where K is a constant, then (5.328) becomes
(5.329)u(x,t)=a+1(a/3K2)±(1/6K2)4a2+6K2cosh(Kx+(-K3-Ka2)t)=a+1(a/3K2)±(1/12K2)4a2+6K2[exp(Kx+(-K3-Ka2)t)+exp(-Kx-(-K3-Ka2)t)],
which is a solitary solution. It is interesting that we can convert the compacton-like solution to solitary solution.The initial solution (trial function) can be also constructed in a solitary form. Now we begin with(5.330)u0=a+1b+cexp(kx+wt)+dexp(-kx-wt).
By the same manipulation as illustrated above, we set ∂u0/∂t=∂u1/∂t.So we obtain the following needed solitary solution:(5.331)u(x,t)=a+1a/3k2+cexp(kx+(-k3-ka2)t)+((3k2+2a2)/72k4)(1/c)exp(-kx-(-k3-ka2)t).
Hereby a and c are free parameters.It is also interesting to note that the solitary solution can be converted into a compacton-like solution if we choosek=iK, where K is a constant. If k=iK, (5.331) becomes
(5.332)u(x,t)=a+1-a/3K2+(c+𝒜)cos(Kx+(K3-Ka2)t)+(c-𝒜)isin(Kx+(K3-Ka2)t),
where  𝒜 donates (-3K2+2a2)/72K4c. In the above derivation, we use the relations
(5.333)exp(ikx+iwt)=cos(kx+wt)+isin(kx+wt),exp(-ikx-iwt)=cos(kx+wt)-isin(kx+wt).
In order to convert (5.332) into a compact form, the last term in denominator must be vanished, which requires
(5.334)c--3K2+2a272K4c=0.Solvingc from (5.334) results in
(5.335)c=±112-6K2+4a2K2.Substituting (5.335) into (5.332) yields a compacton-like solution
(5.336)u(x,t)=a+1-a/3K2±(1/6K2)4a2-6K2cos(Kx+(K3-ka2)t).


### 5.7. Homotopy Perturbation Method

#### 5.7.1. Homotopy Technology

Before proceeding the method, we give an interesting application of the homotopy technology to an asymptotic match of Rayleigh grain size distribution and Hillert grain size distribution. Grain growth is a well-known phenomenon in the evolution of crystalline microstructures. A stochastic continuity or Fokker-Planck continuity equation was proposed to accurately predict properties of grain growth [66]. The equation can be expressed in the form [66]
(5.337)εd2fdx2+(α+εx-α+2x)dfdx+(6-α+εx2)f=0.
Let us consider two limiting cases where the driving force is either due to diffusion (Rayleigh) or to the drift velocity (Hillert). In the first case, the Rayleigh grain size distribution occurs when α=0 and ε≠0 [66]:
(5.338)fr(x)=π2xexp(-π4x2).
The other limiting case, α=8 and ε=0, leads to Hillert grain size distribution [66]:
(5.339)fh(x)=8exp(2x/(x-2))(x-2)2.
The actual solution to (5.337) falls between (5.338) and (5.339); in the work of Pande and Cooper [66], a very good but approximate solution was given (see (18) of Pande and Cooper’s publication [66]). Hereby we will suggest a simple matching technology to bridge the limiting cases.An asymptotic match of (5.338) and (5.339) can be expressed in the form [67]
(5.340)f(x,α,ε)=C(α,ε)fr(x)+D(α,ε)fh(x)=C(α,ε)π2xexp(-π4x2)+D(α,ε)8exp(2x/(x-2))(x-2)2,
where C and D are matching parameters.Equation (5.340) should turn out to be exactly (5.338) and (5.339), respectively, for the above two limiting cases. This requires
(5.341)C(0,ε)=1,D(0,ε)=0,C(8,0)=0,D(8,0)=1.
We can freely choose C and D satisfying (5.341) and the following normalization condition
(5.342)C+D=1.
Hereby we give simple expressions for C and D in the forms
(5.343)C(α,ε)=1-18α(1-ε),D(α,ε)=18α(1-ε).
We, therefore, obtain the following approximate solution to (5.337) [67]:
(5.344)f(x)=π2(1-18α(1-ε))xexp(-π4x2)+18α(1-ε)8exp(2x/(x-2))(x-2)2.
This equation is simpler than that given by Pande and Cooper [66]. There are many alternative ways for determination of C and D satisfying (5.341) and (5.342).In this short paper we suggest an asymptotic match to bridge the two limited cases, and the obtained result is valid for the whole case. The asymptotic match is a simple and useful mathematical tool in engineering for reliable treatment of a nonlinear problem whose analytical solution can be easily obtained for two limited cases (e.g.,ε→0 and ε→∞, resp.) by some analytical methods.Consider another example of the relativistic oscillator(5.345)u′′+u1+u2=0,
with initial conditions u(0)=A,u′(0)=0.It is easy to obtain the following approximate frequency [68]:
(5.346)ω=22π(1+4π2·A2)-1/4,forA≫1,ω=(1+12A2)-1/4,forA≪1.In order to match both the casesA→0 and A→∞, we construct the following homotopy [68]:
(5.347)ω~=e-αAf(A)+(1-e-αA)g(A),
where f(A)=(22/π)(1+(4/π2)·A2)-1/4, g(A)=(1+(1/2)A2)-1/4, and α is a free parameter. Now considering the case when A=1, we have exact frequency, which is ω~=0.8736. From this relationship, we can identify α as follows:
(5.348)α=0.4962.
Finally, we obtain the following result:
(5.349)ω~=e-0.4962A·22π(1+4π2·A2)-1/4+(1-e-0.496A)·(1+12A2)-1/4.


#### 5.7.2. Homotopy Perturbation Method with an Auxiliary Term [69]

The two most important steps in application of the homotopy perturbation method [69–72] are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be constructed such that, when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach for construction of the homotopy equation with an auxiliary term Duffing equation is used as example to illustrate the solution procedure.Consider a general nonlinear equation(5.350)Lu+Nu=0,
where L and N are, respectively, the linear operator and nonlinear operator.The first step for the method is to construct a homotopy equation in the form(5.351)L~u+p(Lu-L~u+Nu)=0,
where L~ is a linear operator with a possible unknown constant, and L~u=0 can best describe the solution property. The embedding parameter p monotonically increases from zero to unit as the trivial problem, L~u=0, is continuously deformed to the original one.For example, consider a nonlinear oscillator [9]
(5.352)u′′+εu3=0,u(0)=A,u′(0)=0.
For an oscillator, we can use sine or cosine function. We assume that the approximate solution of (5.352) is
(5.353)u(t)=Acosωt,
where ω is the frequency to be determined later. We, therefore, can choose
(5.354)L~u=u¨+ω2u.
Accordingly, we can construct a homotopy equation in the form
(5.355)u¨+ω2u+p(u3-ω2u)=0.
When p=0, we have
(5.356)u¨+ω2u=0,u(0)=A,u′(0)=0,
which describes the basic solution property of the original nonlinear equation (5.352).Whenp=1, (5.355) becomes the original one. So the solution procedure is to deform from the initial solution, (5.353), to the real one. Due to one unknown parameter in the initial solution, only one iteration is enough. For detailed solution procedure, refer to [9].If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form(5.357)u¨+0·u+pu3=0.
We expand the solution and the coefficient, zero, of the linear term into a series in p:
(5.358)u=u0+pu1+p2u2+⋯,(5.359)0=ω2+pa1+p2a2+⋯,
where the unknown constant, ai, is determined in the (i+1)th iteration. The solution procedure is given in [9].In this paper, we suggest an alternative approach for construction of homotopy equation, which is(5.360)L~u+p(Lu-L~u+Nu)+αp(1-p)u=0,
where α is an auxiliary parameter. When α=0, (5.360) turns out to be that of the classical one, expressed in (5.351). The auxiliary term, αp(1-p)u, vanishes completely when p=0 or p=1, so the auxiliary term will affect neither the initial solution (p=0), nor the real solution (p=1).To illustrate the solution procedure, we consider a nonlinear oscillator in the form(5.361)d2udt2+bu+cu3=0,u(0)=A,u′(0)=0,
where b and c are positive constants.Equation (5.361) admits a periodic solution, and the linearized equation of (5.361) is
(5.362)u′′+ω2u=0,u(0)=A,u′(0)=0,
where ω is the frequency of (5.361).We construct the following homotopy equation with an auxiliary term [69]:
(5.363)u′′+ω2u+p[(b-ω2)u+cu3]+αp(1-p)u=0.
Assume that the solution can be expressed in a power series in p as shown in (5.358). Substituting (5.358) into (5.363), and processing as the standard perturbation method, we have
(5.364)u0′′+ω2u0=0,u0(0)=A,u0′(0)=0,(5.365)u1′′+ω2u1+(b-ω2)u0+cu03+αu0=0,u1(0)=0,u1′(0)=0,(5.366)u2′′+ω2u2+(b-ω2)u1+3cu02u1+α(u1-u0)=0,
with initial conditions
(5.367)∑i=0ui(0)=A,∑i=0ui′(0)=0.
Solving (5.364), we have
(5.368)u0=Acosωt.Substitutingu0 into (5.365) results in
(5.369)u1′′+ω2u1+A(α+b-ω2+(34)cA2)cosωt+14cA3cos3ωt=0.Eliminating the secular term needs(5.370)α+b-ω2+34cA2=0.
A special solution of (5.369) is
(5.371)u1=-cA332ω2cos3ωt.
If only a first-order approximate solution is enough, we just set α=0, and this results in
(5.372)ω=b+34cA2.
The accuracy reaches 7.6% even for the case cA2→∞.The solution procedure continues by submittingu1 into (5.366), and after some simple calculation, we obtain
(5.373)u2′′+ω2u2-(αA+3c2A5128ω2)cosωt-(cA3(b-ω2)32ω2+3c2A564ω2+αcA332ω2)cos3ωt-3c2A5128ω2cos5ω=0.No secular term inu2 requires
(5.374)αA+3c2A5128ω2=0.Solving (5.370) and (5.374) simultaneously, we obtain
(5.375)ω=b+(3/4)cA2+(b+(3/4)cA2)2+(3/32)c2A42,
and the approximate solution is u(t)=Acosωt, where ω is given in (5.375).In order to compare with the perturbation solution and the exact solution, we setb=1. In case c≪1, (5.375) agrees with that obtained by the classical perturbation method; when c→∞, we have
(5.376)limc→∞ω=3/4+(3/4)2+3/322cA2=0.8832cA2.
The exact period reads
(5.377)Tex=41+cA2∫0π/2dx1-ksin2x,
where k=cA2/2(1+cA2).In casec→∞, we have
(5.378)limc→∞Tex=6.743cA2,(5.379)ωex≈2π6.743cA2.
Comparing between (5.376) and (5.378), we find that the accuracy reaches 5.5%, while accuracy of the first-order approximate frequency is 7.6%.If a higher-order approximate solution is needed, we rewrite the homotopy equation in the form(5.380)u′′+ω2u+p[(b-ω2)u+cu3]+1·p(1-p)u=0.
The coefficient, 1, in the auxiliary term is also expanded in a series in p in the form
(5.381)1=α0+pα1+p2α2+⋯,
where αi is identified in the (i+2)th iteration. The solution procedure is similar to that illustrated above.Generally, the homotopy equation can be constructed in the form(5.382)L~u+p(Lu-L~u+Nu)+1·f(p)g(p)h(u,u′,u′′,…)=0,
where f and g are functions of p, satisfying f(0)=0 and g(1)=0, and h can be generally expressed in the form
(5.383)h=u+β1u′+β2u′′+⋯.


#### 5.7.3. Homotopy Perturbation Method for Solitary Solutions

Homotopy perturbation method [69–72] provides a simple mathematical tool for searching for soliton solutions without any small perturbation. Considering the following nonlinear equation:
(5.384)∂u∂t+au∂u∂x+b∂3u∂x3+N(u)=0,a>0,b>0,
we can construct a homotopy in the form
(5.385)(1-p){∂u∂t+6u∂u∂x+∂3u∂x3}+p{∂u∂t+au∂u∂x+b∂3u∂x3+N(u)}=0.
When p=0, we have
(5.386)∂u∂t+6u∂u∂x+∂3u∂x3=0,
a well-known KdV equation, whose soliton solution is known. When p=1, (5.385) turns out to be the original equation. According to the homotopy perturbation method, we assume that
(5.387)u=u0+pu1+p2u2+⋯.
Substituting (5.387) into (5.385), and proceeding the same process as the traditional perturbation method does, we can easily solve u0, u1, and other components. The solution can be expressed finally in the form
(5.388)u=u0+u1+u2+⋯.
The homotopy perturbation method always stops before the second iteration, so the solution can be expressed as
(5.389)u=u0+u1,
for most cases.For traveling wave solutions, we can use the following transformation:(5.390)u(t,x)=u(ξ),ξ=x-Dt.
Equation (5.384) becomes
(5.391)-Duξ+auuξ+buξξξ+N(u)=0.
Integrating the equation with respect to ξ, we have
(5.392)-Du+12au2+buξξ+F(u)=0,
where ∂F/∂u=N(u)+C, and C is an integral constant.We begin with a soliton in the form(5.393)u0=1e-αξ+eαξ,
which is the solution of the following equation:
(5.394)uξξ-α2u+8α2u3=0.
Accordingly, we construct the following homotopy equation:
(5.395)uξξ-Dbu+0·u3+p[a2bu2+1bF(u)]=0.
The solution and the parameters D/b and 0 are expanded in the forms
(5.396)u=u0+pu1+p2u2+⋯,Db=α2+pα1+p2α2+⋯,0=8α2+pβ1+p2β2+⋯.
Substituting (5.396) into (5.395), and proceeding a similar way as the perturbation method, we have
(5.397)u0ξξ-α2u0+8α2u03=0,u1ξξ-α2u1+α1u0+24α2u02u1+β1u03+a2bu02+1bF(u0)=0.
As an example, we consider the following equation:
(5.398)ut+uux+ηuxxx=0.
By the transformation u(t,x)=u(ξ), ξ=x-Dt, we have
(5.399)-Duξ+uuξ+ηuξξξ=0,
or
(5.400)u′′-Dηu+12ηu2+C=0.
In order to make the solution process simple, we construct a homotopy equation in the form
(5.401)u′′-Dηu+p(12ηu2+C)=0.
Assume that the solution and the parameter D/η can be expanded in the forms
(5.402)u=u0+pu1+p2u2+···,Dη=λ02+pλ1+p2λ2+···.
Substituting (5.402) into (5.401), proceeding as the classical perturbation method does, we have
(5.403)u0′′-λ02u0=0,u1′′-λ02u1-λ1u0+12ηu02+C=0,u2′′-λ02u2-λ2u0-λ1u1+1ηu0u1=0.
The solution of the first equation of (5.403) is
(5.404)u0=Aeλ0ξ+Be-λ0ξ.
Substituting (5.404) into the second equation of (5.403) we have
(5.405)u1′′-λ02u1=λ1(Aeλ0ξ+Be-λ0ξ)-12η(Aeλ0ξ+Be-λ0ξ)2-C=λ1(Aeλ0ξ+Be-λ0ξ)-12η(A2e2λ0ξ+B2e-2λ0ξ)-ABη-C.
We set
(5.406)λ1=0,AB=-ηC
in (5.405) to avoid the secular-like terms ξkeλ0ξ, ξke-λ0ξ or ξk (k≥0), and solving the resultant equation, we have the solution for u1:
(5.407)u1=-16η(A2e2λ0ξ+B2e-2λ0ξ).
Substituting u0 and u1 into the third equation of (5.403) results in
(5.408)u2′′-λ02u2=λ2(Aeλ0ξ+Be-λ0ξ)+16η2(Aeλ0ξ+Be-λ0ξ)(A2e2λ0ξ+B2e-2λ0ξ)=A(λ2+AB6η2)eλ0ξ+B(λ2+AB6η2)e-λ0ξ+16η2(A3e3λ0ξ+B3e-3λ0ξ).
Similarly, we set
(5.409)λ2=-AB6η2.
The solution for u2 is
(5.410)u2=148η2(A3e3λ0ξ+B3e-3λ0ξ).
If the second order approximate solution is enough, then we have
(5.411)u=u0+u1+u2=Aeλ0ξ+Be-λ0ξ-16η(A2e2λ0ξ+B2e-2λ0ξ)+148η2(A3e3λ0ξ+B3e-3λ0ξ).
Solving λ0 from the second equation of (5.402), we have
(5.412)λ0=Dη-(pλ1+p2λ2+···)|p=1,λi=0(i≥3)=Dη+AB6η2.Using in the[1,2] exponential Padé approximant [62], we have
(5.413)u=Aeλ0ξ+Be-λ0ξ1+aeλ0ξ+be-λ0ξ+ce2λ0ξ+de-2λ0ξ,
where the parameters A,B,a,b,c, and d can be identified by the exponential Padé approximant similar to the Padé approximant.The[m,n] exponential Padé approximant is defined as follows.Let(5.414)F=F0+εF1+ε2F2+⋯,ε→0.The [m,n] exponential Padé approximant reads
(5.415)Fmn(ε)=∑i=0mciexpdiε∑i=0naiexpbiε.
The coefficients ai,bi,ci,  and  di, are determined from the following condition: the first (2m+2n) components of the expansion of the rational function Fmn(ε) in a Maclaurin series coincide with the first (2m+2n) components of the series F(ε).


#### 5.7.4. Couple Homotopy Perturbation Method with Laplace Transform [73]

Couple of the homotopy perturbation method with the Laplace transform makes the solution procedure much simpler.Consider the following foam drainage equation:(5.416)ut+2u2ux-ux2-12uxxu=0,
with initial conditions
(5.417)u(x,0)=-12+11+ex.
Apply Laplace transformation
(5.418)u(x,s)=1s(-12+11+ex)-1sL(2u2ux-ux2-12uxxu).
Apply inverse Laplace transformation
(5.419)u(x,t)=(-12+11+ex)-L-1(1sL(2u2ux-ux2-12uxxu)).
Apply the convex homotopy
(5.420)u0+pu1+p2u2+⋯=(-12+11+ex)-∫0tL-1(1sL(2(u0+pu1+⋯)2(∂u0∂x+p∂u1∂x+⋯)))dt+∫0tL-1(1sL(u0+pu1+⋯)x2+12(∂2u0∂x2+p∂2u1∂x2+⋯)(u0+pu1+⋯))dt.
Comparing the coefficient of like powers of p,
(5.421)p(0):u0(x,t)=(-12+11+ex),p(1):u1(x,t)=4ext(1+2ex)4+8e2xt(1+2ex)4-5ext(1+2ex)3-2e2xt(1+2ex)3+3ext2(1+2ex)2-ext216(1+2ex)3+e2xt8(1+2ex)3.⋮
The series solution is given by
(5.422)u(x,t)=(-12+11+ex)+4ext(1+2ex)4+8e2xt(1+2ex)4-5ext(1+2ex)3-2e2xt(1+2ex)3+3ext2(1+2ex)2-ext216(1+2ex)3+e2xt8(1+2ex)3⋯.
Applications of the homotopy perturbation method to solitary solutions can be referred to [74–77].


### 5.8. Parameter-Expansion Method

#### 5.8.1. An Example

Slab detachment or break-off is appreciated as an important geological process. Ananalytical solution was given in [78] for the nonlinear dynamics of high amplitude necking in a free layer of power-law fluid extended in layer-parallel direction due to buoyancy stress.The power-law flow law for the layer can be written as(5.423)ε˙=Bτn,
where ε˙, rate of deformation parallel to the layer of a plane in the layer, τ is the mean layer-parallel deviatoric stress, and B and n are a material parameter and the stress exponent, respectively.Schmalholz obtained the following detachment time [78]:
(5.424)t=tcn=1nB((1/2)F)n,
where F is the layer-parallel force due to buoyancy at the top of the layer.The power-law fluid assumption, (5.423), is useful because of its simplicity but only approximately describes the behaviour of a real non-Newtonian fluid. We modify (5.423) in the form
(5.425)ε˙=B1τ+B2τn,
where B1 and B2 are material parameters.The governing equation for the dynamics of necking becomes(5.426)-1DdDdt=B1F2D+B2(F2D)n,D(0)=D0,
where D is the local thickness of the layer, and F is the layer-parallel force.Rewrite (5.426) in the form
(5.427)dDdt+A1+A2D1-n=0,D(0)=D0,
where A1=B1F/2, A2=B2(F/2)n.We use the parameter-expansion method (see Section5.8.2) to solve (5.427). We expand the solution in the form
(5.428)D=D-+pD~+p2D~~+⋯,
where D- is a reference thickness, D~ is a perturbation term, D~~ is an even smaller infinitesimal perturbation term, and p is a bookkeeping parameter, p=1 (see Section  5.8.2).The parametersA1 and A2 in (5.427) are expended in a similar way:
(5.429)A1=A-1+pA~1+p2A~~1+⋯,A2=p(A-2+pA~2+p2A~~2+⋯).
Substituting (5.428)-(5.429) into (5.427), collecting terms of the same power of p, and equating coefficients of like powers of  p yield a series of linear equations. We write only the equations for D- and D~, which read
(5.430)dD-dt+A-1=0,D-(0)=D0,(5.431)dD~dt+A~1+A-2D-1-n=0,D~(0)=0.
The solution of (5.430) reads
(5.432)D-=D0-A-1t.
Substituting the result into (5.431), we have
(5.433)dD~dt+A~1+A-2(D0-A-1t)1-n=0,D~(0)=0.
Its solution is
(5.434)D~=-A-2(D0)2-n(2-n)A-1-A~1t+A-2(D0-A-1t)2-n(2-n)A-1.
Note that D~ is a perturbation term. When t is small, it can be approximated as
(5.435)D~=-A-2(D0)2-n(2-n)A-1-A~1t+A-2(2-n)A-1(D02-n-(2-n)D01-nA-1t)=-(A~1-A-2D01-n)t.
We set
(5.436)A~1=A-2D01-n.
If the first-order approximate solution is enough, setting p=1 in (5.428)-(5.429) and equating higher-order perturbation terms to be zero, we have
(5.437)A2=A-2,A~1=A2D01-n,A-1=A1-A2D01-n,D=D-+D~=D0-A-1t+A2(D0)2-n(2-n)A-1-A~1t-A2(D0-A-1t)2-n(2-n)A-1.
For qualitative analysis, we use D- as an approximate solution
(5.438)D=D0-(A1-A2D01-n)t.
The detachment time is
(5.439)t=D0A1-A2D01-n=D0(B1F/2)-B2(F/2)nD01-n.
Equation (5.439) shows that the detachment time depends upon the initial layer thickness and fluid properties and the buoyancy.The nonlinear equation describing the dynamics of necking is of strong nonlinearity, and in order to solve it analytically, oversimple assumption has to be made. The power-law fluid is simple, but it is not a real non-Newtonian fluid. Recent development of analytical methods increases tantalizing possibility of theoretically seeking the approximate solutions to the present problem.


#### 5.8.2. Parameter-Expansion Method for Solitary Solutions

Parameter-expansion method includes the modified Lindstedt-Poincare method [79] and bookkeeping parameter method [80]. In the review article [7], it was previously called parameter-expanding method. The method does not require to construct a homotopy. To illustrate its solution procedure, we reconsider (5.384), which is rewritten in the form
(5.440)∂u∂t+au∂u∂x+b∂3u∂x3+1·N(u)=0.
Supposing that the parameters, a, b, and 1, can be expressed in the forms
(5.441)a=a0+pa1+p2a2+⋯,b=b0+pb1+p2b2+⋯,1=pc1+p2c2+⋯,
where p is a bookkeeping parameter, p=1, substituting (5.441) into (5.440), proceeding the same way as the perturbation method, we can easily obtain the needed solution.We consider a simple mathematical model in the form(5.442)u′′+u2=0,u(0)=u(1)=0,u′′+0+1·u2=0.We seek an expansion of the form(5.443)u=u0+pu1+p2u2+⋯,
where the ellipsis dots stand for terms proportional to powers of p greater than 2, p is a bookkeeping parameter, p=1.The constants, 0 and 1, in the left-hand side of (5.442) can be, respectively, expanded in a similar way
(5.444)0=2a+a1p+a2p2+⋯,1=b1p+b2p2+⋯.
Substituting (5.444) to (5.442), we have
(5.445)(u0+pu1+p2u2+⋯)′′+(2a+a1p+a2p2+⋯)+(b1p+b2p2+⋯)·(u0+pu1+p2u2+⋯)2=0,
and equating coefficients of like powers of p we obtain the same equations as illustrated in previous examples:
(5.446)u0′′+2a=0,u0(0)=u0(1)=0,u1′′+b1u02+a1=0,u1(0)=u1(1)=0.
Solving u0 and u1, we have the following first-order approximate solution:
(5.447)u(t)=u0(t)+u1(t)=at(1-t)+at2-a2(130t6-110t5+112t4)-(a-160a2)t.
The discussion on how to determine the constant, a, is given in [69]. Applications of the parameter-expansion method to various nonlinear problems are available in [81–86].


### 5.9. Exp-Function Method

#### 5.9.1. Exp-Function Method for Solitary Solutions

Exp-function method [87–89] provides us with a straightforward and concise approach for obtaining generalized solitary solutions and periodic solutions; the solution procedure, by the help of Matlab or Mathematica, is of utter simplicity, see [90–97]. Considering a general nonlinear partial differential equation in the form
(5.448)F(u,ux,uy,uz,ut,uxx,uyy,uzz,utt,uxy,uxt,uyt,…)=0.
The exp-function method assumes that the solitary solution can be expressed in the form
(5.449)u(x,y,z,t)=∑n=-klpnexp(anx+bny+cnz+dnt)∑m=-ijqmexp(amx+bmy+cmz+dmt).
We can also introduce a transformation
(5.450)ξ=ax+by+cz+dt.
We can rewrite (5.448) in the following nonlinear ordinary differential equation:
(5.451)F(u,u′,u′′,u′′′,…)=0,
where the prime denotes the derivation with respect to ξ.According to the exp-function method, the traveling wave solutions can be expressed in the form(5.452)u(ξ)=∑n=-klanexp(nξ)∑m=-ijbmexp(mξ),
where i, j, k, and l are positive integers which could be freely chosen; an and bm are unknown constants to be determined.Consider a fractional Benjamin-Bona-Mahony (BBM) equation [97]
(5.453)∂u∂t-u∂u∂x-∂∂t∂2u∂x2=0,
using a transformation
(5.454)u(x,t)=U(ξ),ξ=kx+wt,
(5.453) becomes an ordinary differential equation,
(5.455)wU′-kUU′-k2wU′′′=0.
By the exp-function method, we assume that the solution of (5.460) can be expressed as
(5.456)U(ξ)=a1exp(ξ)+a0+a-1exp(-ξ)b1exp(ξ)+b0+b-1exp(-ξ),
where ai,bi(i=1,0,-1) are unknown parameters.Substituting (5.456) into (5.455), collecting terms of the same term of exp(iξ), we have
(5.457)1A[C7exp(7ξ)+C6exp(6ξ)+C5exp(5ξ)+C4exp(4ξ)+C3exp(3ξ)+C2exp(2ξ)+C1exp(ξ)]=0.Equating coefficients,Ci(i=1~7), to zero yields a series of linear equations
(5.458)Ci=0(i=1~7).
Solving the system of algebraic equation, we can identify parameters in (5.456). Finally, we obtain [97]
(5.459)u(x,t)=-w(k2-1)k+6k2b0b1exp[(kx+wt)]+b0+(b02/4b1)exp[-(kx+wt)],
where b0 and b1 are free parameters.Byk=iK, w=iW, the obtained solitary solution can be converted into periodic solution.Other expressions are listed as follows:(5.460)u(x,t)=∑nanexp(qnx+pnt)∑mbmexp(αmx+βmt),
where ai,biαi,βiqi,  and  pi are unknown constants to be further determined.For a two-wave solution, we can use the double exp-function method [8], which requires
(5.461)u=a-1exp(-ξ)+b1exp(ξ)+a0+b-1exp(-η)+b1exp(η)c-1exp(-ξ)+c1exp(ξ)+b0+d-1exp(-η)+d1exp(η),
where ξ=a1x+b1y+c1z-d1t and η=a2x+b2y+c2z-d2t.


#### 5.9.2. The Exp-Function Method for Periodic Solutions

The French mathematician J. Fourier (1768–1830) showed that any periodic motion can be represented by a series of sines and cosines that are harmonically related(5.462)u(t)=12a0+a1cosω1t+a2cosω2t+⋯+b1sinω1t+b2sinω2t+⋯,
where ωn=nω1, and an and bn are, respectively, expressed as
(5.463)an=2T∫-T/2T/2u(t)cosωntdt,bn=2T∫-T/2T/2u(t)sinωntdt.
Using Euler’s formula, we have
(5.464)sint=12(eit+e-it),cost=-12i(eit-e-it).
We, therefore, can rewrite the Fourier series, (5.462), in the form
(5.465)u(t)=∑n=-∞∞cneiωnt,
where c0=(1/2)a0 and cn=(1/2)(an-ibn).For periodic solution, we can assume that the solution can be expressed in the form(5.466)u(t)=∑n=-cdpnexp(iωnt)∑m=-fgqmexp(iωmt),
or
(5.467)u(t)=∑0Ncneiωnt.


### 5.10. Subequation Method

There are many subequation methods in the open literature for solitary solution [98–100]. We consider a general nonlinear equation in the form
(5.468)F(u,ut,ux,uy,uz,uxx,uyy,uzz,…)=0.
The sub-equation method follows the following steps.Step 1.
Letu=u(x,y,z,t)=u(ξ), where ξ=ax+by+cz+dt+ξ0. Equation (5.468) turns out to be an ordinary differential equation
(5.469)F(u,uξ,uξξ,uξξξ,…)=0.Step 2.
Assume that the solution can be expressed in the form(5.470)u=∑nanfn(ξ),
or in a more general form,
(5.471)u=U(f(ξ)),
where f is the solution of an auxiliary equation
(5.472)f'2=b0+b1f+b2f2+b3f3+b4f4+⋯.
The exact solution of (5.472) for fixed bi(i=0,1,2,3,…) is known.Step 3.
Use some mathematical software to determinean in (5.470).The auxiliary equation can be constructed using the Hamiltonian invariant as discussed in Section5.3.5. For (5.128), and the following auxiliary equation can be constructed:
(5.473)u′=cu2+2u3.
For (5.227), the following auxiliary equation should be in the form:
(5.474)u′=cu2-12(n+1)(n+2)un+2.
In such case, the solution of the auxiliary equation is the special solution of the problem.


### 5.11. Ancient Chinese Mathematics

#### 5.11.1. Introduction to the Ancient Chinese Mathematics

Can ancient Chinese mathematics still work for nonlinear science? Yes, it can. It is simple but effective.To illustrate the basic idea of the ancient Chinese method, we consider an algebraic equation:(5.475)f(x)=0.Letx1 and x2 be the approximate solutions of the equation, which lead to the remainders f(x1) and f(x2), respectively; the ancient Chinese algorithm leads to the result
(5.476)x=x2f(x1)-x1f(x2)f(x1)-f(x2).Now we consider a generalized nonlinear oscillator in the form(5.477)u′′+f(u)=0,u(0)=A,u′(0)=0.
We use two trial functions u1(t)=Acosω1t and u2=Acosω2t, which are, respectively, the solutions of the following linear oscillator equations:
(5.478)u′′+ω12u=0,u′′+ω22u=0.
The residuals are
(5.479)R1(ω1t)=-ω12cosω1t+f(Acosω1t),R2(ω2t)=-ω22cosω2t+f(Acosω2t).
The residuals depend upon t, and in our previous applications [7], we just use R1(0) and R2(0) for simplicity when we use the ancient Chinese method, resulting in relatively low accuracy [7]
(5.480)ω2=ω12R2(0)-ω22R1(0)R2(0)-R1(0).
Equation (5.480) is called He’s frequency formulation or He’s frequency-amplitude formulation in the literature [101–106].In order to improve its accuracy, we introduce two new residual variablesR~1 and R~2 defined as [105, 106]
(5.481)R~1=4T1∫0T1/4R1(t)cos(2πT1t)dt,(5.482)R~2=4T2∫0T2/4R2(t)cos(2πT2t)dt.
According to the ancient Chinese method, we can approximately determine ω2 in the form
(5.483)ω2=ω12R~2-ω22R~1R~2-R~1.
In this letter, we will use the well-known Duffing equation.Using trial functionsu1(t)=Acost and u2=Acosω2t, respectively, for (5.481), we obtain the following residuals:
(5.484)R1(t)=εA3cos3t,R2(t)=A(1-ω22)cosω2t+εA3cos3ω2t,
where ω2>0 and ω2≠1.By simple calculation, we obtain(5.485)R~1=4T1∫0T1/4R1(t)cos(2πT1t)dt=2π∫0π/2εA3cos4tdt=34π,R~2=4T2∫0T2/4R2(t)cos(2πT2t)dt=2ωπ∫0T2/4{A(1-ω22)cos2ω2t+εA3cos4ω2t}dt=2π∫0π/2{A(1-ω22)cos2s+εA3cos4s}ds=A(1-ω22)1π+34πεA3.
Applying (5.483), we have
(5.486)ω2=ω12R~2-ω22R~1R~2-R~1=A(1-ω22)(1/π)+(3/4π)εA3-(3/4π)εA3ω22A(1-ω22)(1/π)+(3/4π)εA3-(3/4π)εA3=(1-ω22)+(3/4)εA3-(3/4)εA3ω22(1-ω22)=1+34εA2.


#### 5.11.2. Ancient Chinese Mathematics for Optimization [107]

To elucidate the basic idea of application of the ancient Chinese mathematics to optimal problems, we consider first the following simple example:(5.487)f=xy→max,(5.488)s.t.g(x,y)=x+y-1=0.To make the Lagrange functionf  maximal, it requires
(5.489)df=xdy+ydx=0,
where dx and dy are not independent, and according to the constraint equation, (5.490), we have
(5.490)dx+dy=0.
Considering the relationship between dx and dy, (5.492), (5.491) becomes
(5.491)df=(x-y)dy=0.
Due to the arbitrary dy in (5.491), we have the following Euler equation:
(5.492)x-y=0.
Solving (5.488) and (5.492) simultaneously, we find the optimal values
(5.493)x*=y*=12.A general solution procedure is given in [8], and the stationary condition (Euler equation) can be simply obtained using He’s brackets [8, 108]:
(5.494)〈f,g〉x,y=fxgy-fygx=0,
where fx=∂f/∂x=y, fy=∂f/∂y=x, gx=∂g/∂x=1, and gy=∂g/∂y=1.Equation (5.494) is exactly the same with (5.492).According to the ancient Chinese mathematics, to solve a problem, we should choose double trial solutions,x1 and x2. We can choose x1 arbitrarily, for example, x1=2/7, according to the constraint equation (5.488), and we have y1=5/7.From (5.487), we obtain
(5.495)f1=x1y1=1049.
The condition (5.489) can be approximately expressed as
(5.496)Δf=xΔy+yΔx=0,
where Δx=x2-x1, Δy=y2-y1, and Δf=f2-f1.Equation (5.496) implies that
(5.497)f2=f1.
This means that
(5.498)f2=x2y2=1049.
Considering the constraint equation, (5.488), again, we have
(5.499)x2+y2=1.
From (5.498) and (5.499), we can obtain immediately x2=5/7 and y2=2/7.Substitutingx1, x2, y1, and y2 into (5.498), we have
(5.500)Δf=x(y2-y1)+y(x2-x1)=x(27-57)+y(57-27)=0.
Solving (5.488) and (5.500) simultaneously, we find the optimal values given in (5.493).Example 5.1.
Condiser the following [109]:
(5.501)f=x2+y2→min,(5.502)s.t.g(x,y)=y-2-x2=0.
To make (5.501) minimal, it requires that
(5.503)Δf=2xΔx+2yΔy2x2+y2=xΔx+yΔyx2+y2=0.
Choosing x1=1, we have y1=3 and f1=10. We set f2=10, which means that
(5.504)x22+y22=10,(5.505)y2=2+x22.
Solving (5.504) and (5.505) simultaneously, we have x2=-1 and y2=3.
Substitutingx1, x2, y1, and y2 into (5.503), we obtain
(5.506)Δf=x(x2-x1)+y(y2-y1)x2+y2=-2xx2+y2=0.
We, therefore, obtain the optimal valuesx*=0 and y*=2.Example 5.2.
Consider the following problem [109]:
(5.507)f(x,y)=x2+y2→min,(5.508)s.t.y2-(x-1)3=0.
The Lagrange multiplier method becomes invalid for this example [109]. Using the basic idea of the ancient Chinese mathematics, we choose x1=2, y1=1 and x2=2, y2=-1, and both cases lead to f1=f2=5.
Making (5.507) stationary requires
(5.509)Δf=xΔx+yΔyx2+y2=x(x2-x1)+y(y2-y1)x2+y2=-2yx2+y2=0,
which leads to immediately y*=0 and x*=1 if the constraint equation, (5.508), is considered.


#### 5.11.3. Ancient Chinese Mathematics for Nonlinear Wave Equations [110]

Now we consider the nonlinear Schrödinger equation(5.510)iut+uxx+2|u|2u=0,u(x,0)=eikx.
We use the trial functions
(5.511)u1(x,t)=ei(kx+t),u2(x,t)=ei(kx+ωt).
The residuals are
(5.512)R1(x,t)=(1-k2)ei(kx+t),R2(x,t)=(2-ω-k2)ei(kx+ωt).
According to frequency-amplitude formulation, (5.480), we have
(5.513)ω2=ω12R2(x,0)-ω22R1(x,0)R2(x,0)-R1(x,0)=-(ω+ωk2-2ω+k2-2).We get the equation aboutω, and the solution is
(5.514)ω=2-k2,u(x,t)=ei{kx+(2-k2)ωt}.
Now we consider the Schrödinger equation with power law nonlinearity
(5.515)iut+uxx-2|u|2ru=0,u(x,0)=(2(r+1)sech2(2rx))1/2r,r≥1.
Following the analysis presented above, we choose the trial functions
(5.516)u1(x,t)=(2(r+1)sech2(2rx))1/2reit,u2(x,t)=(2(r+1)sech2(2rx))1/2reiωt.
The residuals are
(5.517)R1(x,t)=-(2(r+1)(sech(2rx))2)1/2reit+4(2(r+1)(sech(2rx))2)1/2r(tanh(2rx))2eit-4(2(r+1)(sech(2rx))2)1/2r(1-(tanh(2rx))2)reit+2(2(r+1)(sech(2rx))2)1/2r-1eit,R2(x,t)=-(2(r+1)(sech(2rx))2)1/2reiωtω+4(2(r+1)(sech(2rx))2)1/2r(tanh(2rx))2eiωt-4(2(r+1)(sech(2rx))2)1/2r(1-(tanh(2rx))2)reiωt+2(2(r+1)(sech(2rx))2)1/2reiωt.
According to the frequency-amplitude formulation, (5.480), setting x=0 and t=0,  we have
(5.518)ω2=-4r+2ω|(21/2r(r+1)1/2r)2r|-ω-4ωr+|(21/2r(r+1)1/2r)2r|.
Solving ω from the above equation yields
(5.519)ω=-4r+2|(21/2r(r+1)1/2r)2r|.
We, therefore, obtain the solution
(5.520)u2(x,t)=(2(r+1)sech2(2rx))1/2reiωt.


#### 5.11.4. Solitary-Solution Formulation for Nonlinear Equations

Now consider a general nonlinear equation(5.521)L(u)+N(u)=0,
where L is a linear operator, and N is a nonlinear operator.We choose two trial equations(5.522)L(u)=0,L~(u)=0,
where L~ is a linearized linear operator or a simple nonlinear operator, and its solution L~(u)=0 is known. For example, we consider a nonlinear oscillator: u′′+u3=0, and we can choose L(u)=u′′+u=0 and L~(u)=u′′+ω2u=0, where ω is the frequency to be searched for.Substituting the solutions of (5.522) with same boundary/initial conditions as (5.521) into (5.521), we obtain, respectively, the residuals  R1 and R2.For traveling wave solutions, we can make a transformation(5.523)ξ=ax+by+cz+ωt.
The solitary-solution formulation can be expressed as
(5.524)ω2=R1ω12-R2ω22R1-R2,
or
(5.525)ωn=R1ω1n-R2ω2nR1-R2.The KdV equation is a nonlinear, dispersive partial differential equation for a function of two real variables, spacex and time t:
(5.526)ut+6uux+uxxx=0.
According to the solitary-solution formulation (5.524), we construct two trial functions in the following forms:
(5.527)u1(x,t)=Aexp(kx+t+D)+exp[-(kx+t+D)]+B,u2(x,t)=Aexp(kx+ωt+D)+exp[-(kx+ωt+D)]+B,
where A, B, D are arbitrary constants.Substitutingu1 and u2 into (5.526), respectively, we obtain the following residuals:
(5.528)R1(x,t)=-A{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}2-6A2k{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}3-Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}2-6Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}3{exp(kx+t+D)+exp[-(kx+t+D)]+B}4+6Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}3,R2(x,t)=-Aω{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}2-6A2k{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}3-Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}2-6Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}3{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}4+6Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}3.
Using the solitary-solution formulation (5.480), and choosing the location point (0,0), we obtain
(5.529)ω2=ω12R2(0,0)-ω22R1(0,0)R2(0,0)-R1(0,0).
Simplifying (5.529) results in
(5.530)ω2+ω+(ω+1)k3+[2Bω2+2ωB+(ω+1)(6kA-4k3B)]exp(-D)+[2ω2+B2ω2+2ω+ωB2+(ω+1)(6kAB-22k3+k3B2)]exp(-2D)+[2Bω2+2ωB+(ω+1)(6kA-4k3B)]exp(-3D)+[ω2+ω+(ω+1)k3]exp(-4D)=0.
Setting the coefficients of exp(-nD) (n=0,1,2,3,4) to be zero, we obtain
(5.531)ω2+ω+(ω+1)k3=0,(5.532)2Bω2+2ωB+(ω+1)(6kA-4k3B)=0,(5.533)2ω2+B2ω2+2ω+ωB2+(ω+1)(6kAB-22k3+k3B2)=0.
Solving (5.531)~(5.533) simultaneously, we obtain the following relationships:(1)
the relationship between solitary wavenumber and solitary frequency:(5.534)ω=-k3,(2)
the relationship between solitary amplitude and solitary wavenumber:(5.535)A=k2B,(3)
the morphofactor:(5.536)B2=4, so the solitary wave solution can be readily obtained, which reads
(5.537)u(x,t)=2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]+2,
or
(5.538)u(x,t)=-2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]-2,
which are, by chance, exact solutions.


#### 5.11.5. Ancient Chinese Inequality and Application

In an ancient history book, it is write (see [111]):He Chengtian uses 26/49 as the strong, and 9/17 as the weak. Among the strong and the weak, Chengtian tries to find a more accurate denominator of the fractional day of the Moon. Chengtian obtains 752 as the denominator by using the 15 and 1 as weighting factors, respectively, for the strong and the weak. No other calendar can reach such a high accuracy after Chengtian, who uses heuristically the strong and weak weighting factors.The statement is rather cryptic, and in modern mathematical term, the statement can be explained as follows.According to the observation data, He Chengtian (369?~447AD) finds that(5.539)292649days>1Moon>29917days.
Using the weighting factors (15 and 1), He Chengtian obtains
(5.540)Thefractionalday=26×15+9×149×15+17×1=399752,
so
(5.541)1Moon=29399752days.
He Chengtian actually uses the following inequality.If(5.542)ab<x<dc,
where a,b,c, and d are real numbers, then
(5.543)ab<ma+ndmb+nc<dc,
and x is approximated by
(5.544)x=ma+ndmb+nc,
where m and n are weighting factors.The above inequity was developed by the max-min approach to nonlinear oscillators [112]. Hereby we use the inequity for a chemical problem.A mass balance on a differential volume element of porous medium for a spherical catalyst pellet gives(5.545)∂c′∂t=∇·De∇c′-RA,
where c′ is the chemical reactant concentration, -RA is the rate of reaction per unit volume, and De  is the effective diffusion coefficient for reactant.Assuming the coupled process of diffusion and reaction is steady, we reduce (5.545) to the form
(5.546)De(∂2c′∂r2+2r∂c′∂r)=RA.
The boundary conditions are ∂c′/∂r=0,r=0 (center of catalyst), and c′=cs,r=r0 (surface of catalyst).Introducing dimensionless variablesC=c′/cs and R=r/r0, we have the following nonlinear equation for nth-order and irreversible reaction at isothermal condition:
(5.547)∂2C∂R2+2R∂C∂R=ϕ2Cn,∂C∂R(0)=0,C(1)=1,
where ϕ is the Thiele modulus, defined as ϕ=r02kνcsn-1/De, where  kν is the reaction constant.Equation (5.547) can also be written in Cartesian geometry, and it is easy to transform Cartesian geometry to cylindrical or spherical geometry.To investigate the nature of the solution of (5.547), consider the analogous linear equation
(5.548)∂2C∂R2+2R∂C∂R=k,∂C∂R(0)=0,C(1)=1.MultiplyingR2 on both sides of (5.548), we have
(5.549)∂∂R(R2∂C∂R)=kR2,∂C∂R(0)=0,C(1)=1.Its solution is(5.550)C=1+16k(R2-1).Now we rewrite (5.544) in the form
(5.551)∂2C∂R2+2R∂C∂R=g(C),
where g(C)=ϕ2Cn.The left side of (5.551) is analogous to k in (5.548).Observe that0<C<1, so C is never less than the solution of the following initial value problem:
(5.552)∂2C∂R2+2R∂C∂R=ϕ2Cmaxn=ϕ2.The solution of (5.552) reads
(5.553)C=1+16ϕ2(R2-1).
So it follows that (see Figure 4)
(5.554)6+ϕ2(R2-1)6<C<1.
According to He Chengtian’s interpolation, we have
(5.555)C=6p+pϕ2(R2-1)+q6p+q,
or
(5.556)C=1+ξϕ2(R2-1),
where p and q are weighting factors, ξ=p/(6p+q). The free parameter, ξ, in (5.556) can be identified via various methods. Hereby we illustrate a simple approach by the residual method. Substituting (5.556) into (5.544), we obtain the following residual equation:
(5.557)Π(R,ξ)=∂2C∂R2+2R∂C∂R-ϕ2Cn=2ξϕ2+4ξϕ2-ϕ2[1+ξϕ2(R2-1)]n.Figure 4
Qualitative sketch for the diffusion reaction in spherical porous catalyst. The dashed line presents an initial guessC=1+ϕ2(R2-1)/6, and the continued line presents the real solution.Chemists and technologists always want to have an accurate halfway through the change. Assume that the above residual vanishes completely atC=1/2, that is, 1+ξϕ2(R2-1)=1/2, then we have
(5.558)Π(R,ξ)|R=R*=6ξϕ2-12nϕ2,
where R=R* is the root of the equation 1+ξϕ2(R2-1)=1/2. From (5.558), the unknown ξ can be identified as
(5.559)ξ=2-n6.Finally, we obtain the following design formulation:(5.560)C(R)=1+2-n6ϕ2(R2-1).Comparison with exact solution is illustrated in Figure5.Figure 5
Comparison between exact solution and the approximate solution (n=1, ϕ=1). The dashed line presents an approximate solution, and the continued line presents the exact solution. The maximal error is about 7.8%.In order to identify the weighting factors,p and q, in (5.555), we proceed the same manipulation as illustrated above. Substituting (5.555) into (5.544), we obtain a residual equation:
(5.561)Π(R,p,q)=∂2C∂R2+2R∂C∂R-ϕ2Cn=2pϕ26p+q+4pϕ26p+q-ϕ2(6p+pϕ2(R2-1)+q6p+q)n=0.
Locating at R=0 and R=1/2, we obtain
(5.562)Π(0,p,q)=2pϕ26p+q+4pϕ26p+q-ϕ2(6p-pϕ2+q6p+q)n=0,(5.563)Π(12,p,q)=2pϕ26p+q+4pϕ26p+q-ϕ2(6p-(3/4)pϕ2+q6p+q)n=0.
Solving (5.562) and (5.563) simultaneously, the values of p and q can be readily determined.The accuracy of vanishing residuals depends upon the trial functions, (5.555) and (5.556), which are derived from an analogous linear equation, so the obtained solutions are valid for the whole solution domain, keeping maximal accuracy at the location points.The preceding analysis has the virtue of utter simplicity, and the illustrating example shows that the suggested method is very effective and convenient in solving nonlinear equations. The suggested method can be readily applied to the search for analytical solutions for various nonlinear equations, which should have the character of monotonic increase or monotonic decrease. Many nonlinear problems arising in chemical engineering can be described by, for example, exponential decay/rise functions, so the ancient Chinese method—the He Chengtian interpolation—might be a powerful mathematical tool in chemical engineering.


## 5.1. Soliton Perturbation

We consider the following perturbed nonlinear evolution equation [5, 15]:
(5.1)uT+L(u)+N(u)=εR(u),0<ε≪1.
When ε=0, we have an unperturbed equation
(5.2)uT+L(u)+N(u)=0,
which is assumed to have a solitary solution.In caseε≠0, but 0<ε≪1, we can use perturbation theory and look for approximate solutions of (5.1) which are close to solitary solutions of (5.2).Using multiple time scales (a slow timeτ and fast time t scale such that ∂T=∂t+ε∂τ), we assume that the soliton solution can be expressed in the form [5, 15]
(5.3)u(x,T)=u0(ξ,τ)+εu1(ξ,τ,t)+ε2u2(ξ,τ,t)+⋯,
where ξ=x-ct, τ is a slow time, and t is a fast time.Substituting (5.3) into (5.1), then equating like powers of ε, we can obtain a series of linear equations for ui(i=0,1,2,3,…), which can be solved sequentially.In most cases, the nonlinear termR(u) in (5.1) plays an import role in understanding various solitary phenomena, and the coefficient ε is not limited to “small parameter.”


## 5.2. Modified Multitime Expansions [16]

In order to overcome the shortcoming arising in the above solution process, hereby we applied the modified multitime expansions (see Section 2.9 of [16]). To illustrate the method, we consider the following equation:
(5.4)∂u∂T+au∂u∂x+b∂3u∂x3+εR(u)=0.
Introducing the time scales Tn=εnt(n=0,1,2,3,…), and using the parameter-expansion method (see Section 5.8), we assume that the solution and the constants a and b can be expressed as [5, 16]
(5.5)u=u0(T0,T1,T2,…)+εu1(T0,T1,T2,…)+ε2u2(T0,T1,T2,…)+⋯,(5.6)a=a0+εa1+ε2a2+⋯,(5.7)b=b0+εb1+ε2b2+⋯,
where the constants ai and bi can be identified by means of no secular terms. Hereby, we define the secular term in a more general form that the term involves time in the form tnf, even in case tnf tends to zero when t→∞.The equation foru0 is
(5.8)∂u0∂t+a0u0∂u0∂x+b0∂3u0∂x3=0.
We can choose suitably the values of a0 and b0, so that the solution of (5.8) can be easily obtained, and involves the basic properties of the original solution.We use the Duffing equation to illustrate the solution procedure [16]
(5.9)u′′+1·u+εu3=0,u(0)=A,u′(0)=0.
Suppose that the solution can be expressed in (5.5), and the coefficient, 1, can be expanded into
(5.10)1=ω2+εω1+ε2ω2+⋯.
Substituting (5.5) and (5.10) into (5.9) and collecting terms of the same powers of ε, we have
(5.11)D02u0+ω2u0=0,(5.12)D02u1+ω2u1+ω1u0+2D0D1u0+u03=0,
where Di=∂2/∂Ti2.Solving (5.11) with the initial conditions u0(0)=A and u0′(0)=0, we have
(5.13)u0=A(T1,T2)cosωT0.Substitutingu0 into (5.12) results in
(5.14)D02u1+ω2u1+A(ω1+34A2)cosωT0-2ωD1(A)sinωT0+14A3cos3ωT0=0.Eliminating secular terms needs(5.15)D1(A)=0,ω1=-34A2.If only the first-order approximate solution is searched for, from (5.10), we have
(5.16)1=ω2+εω1=ω2-34εA2,
or
(5.17)ω=1+34εA2.The obtained frequency-amplitude relationship, (5.17), is valid for the whole solution domain, and the maximal relative error is less than 7% when εA2→∞.


## 5.3. Variational Approach

This section is an elementary introduction to the concepts of the calculus of variations and its applications to solitary solutions. Generally speaking, there exist two basic ways to describe a nonlinear problem: (1) by differential equations (DE) with initial/boundary conditions; (2) by variational principles (VP). The former is widely used, while the later is rarely used in solitary theory. The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration. Variational methods have been, and continue to be, popular tools for nonlinear problems. When contrasted with other approximate analytical methods, variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions.


### 5.3.1. Inverse Problem of Calculus of Variations

The inverse problem of calculus of variations is to establish a variational formulation directly from governing equations and boundary/initial conditions. We will use the semi-inverse method [17, 18] to establish various variational principles directly from the governing equations.Consider the well-known Korteweg-de Vries (KdV) equation(5.18)ut+auux+buxxx=0,
where a and b are constants, and the subscripts denote partial differentiations.We rewrite it in a conserved form(5.19)∂u∂t+∂∂x(12au2+buxx)=0.
According to the conservation form of (5.19), we can introduce a potential functional Φ defined by
(5.20)Φx=u,(5.21)Φt=-(12au2+buxx).
So the KdV equation can be written in the form
(5.22)Φxt+aΦxΦxx+bΦxxxx=0,
which can be derived from the following variational principle using the semi-inverse method [17]:
(5.23)J(Φ)=∬{12ΦxΦt+a6Φx3-b2Φxx2}dtdx.
In order to obtain a generalized variational principle with two independent fields (Φ,u), we apply the Lagrange multiplier to (5.23)
(5.24)J(Φ,u,λ)=∬{12ΦxΦt+a6Φx3-b2Φxx2+λ(Φx-u)}dtdx.
The stationary condition with respect to u results in
(5.25)λ=0.
The Lagrange multiplier method is not valid for the case. This phenomenon is called Lagrange crisis. Hereby, we suggest three ways to overcome the crisis [18].(
1
) The Semi-Inverse Method [17, 18]
Generally, the multiplier can be expressed in the form after identification(5.26)λ=λ(u,Φ,ut,ux,Φt,Φx,…).
We replace the last term including the Lagrange multiplier by a new variable F, that is,
(5.27)J(Φ,u)=∬{12ΦxΦt+a6Φx3-b2Φxx2+F}dtdx,
where F is unknown, which can be expressed in the form
(5.28)F=F(u,Φ,ut,ux,Φt,Φx,…).
Equation (5.27) is such constructed according to the semi-inverse method [17]. To identify F, making the functional stationary with respect to u, we have
(5.29)δFδu=0,
where δF/δu is the variational derivative defined as
(5.30)δFδu=∂F∂u-∂∂t(∂F∂ut)-∂∂x(∂F∂ux)+∂2∂x2(∂F∂uxx)+⋯.
Equation (5.29) should be equivalent to (5.20), and this requires
(5.31)δFδu=c(Φx-u),
where c is a nonzero constant. From (5.31), we can determine F as follows:
(5.32)F=-12c(Φx-u)2=c~(Φx-u)2,
where c~ is a constant, c~=-c/2.
We, therefore, obtain the following needed variational principle:(5.33)J(Φ,u)=∬{12ΦxΦt+a6Φx3-b2Φxx2+c~(Φx-u)2}dtdx.
Its Euler-Lagrange equations are
(5.34)-2c~(Φx-u)=0,-Φxt-aΦxΦxx-bΦxxxx-2c~(Φx-u)x=0,
which satisfy the field equations (5.20) and (5.22), respectively.(
2
) The Hidden Lagrange Multiplier [18]
Let us come back to (5.25), λ=0, which should be the constraint equation. This means that (5.25) inexplicitly involves a lost constraint equation, so we can identify the multiplier in the form [18]
(5.35)λ=c~(Φx-u).
This results in the same result above.(
3
) Replacement of Some Variables in the Original Functional Using the Constraint Equation
Sometimes the Lagrange crisis can be eliminated by replacing some variables in the original variational principle using the constraint equation; a detailed discussion was systematically given in [18].
We replaceΦxΦt by uΦt in (5.23) and introduce a Lagrange multiplier in the resultant function
(5.36)J(Φ,u,λ)=∬{12uΦt+a6Φx3-b2Φxx2+λ(Φx-u)}dtdx.
Identification of the multiplier yields
(5.37)λ=12Φt.
Submitting the identified multiplier into (5.36) results in
(5.38)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+12Φt(Φx-u)}dtdx=∬{a6Φx3-b2Φxx2+12ΦtΦx}dtdx.
We find that the constraint is not eliminated yet, and this is another Lagrange crisis, which can also be eliminated by the semi-inverse method
(5.39)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+F}dtdx.
The stationary conditions with respect to u and Φ are, respectively, as follows:
(5.40)12Φt+δFδu=0,-12ut-aΦxΦxx-bΦxxxx+δFδΦ=0.
According to (5.20) and (5.21), we have
(5.41)δFδu=-12Φt=14au2+12buxx,δFδΦ=12ut+aΦxΦxx+bΦxxxx=12ut-Φxt=12Φxt-Φxt=-12Φxt.
From (5.41), F can be determined as
(5.42)F=112au3-14b(ux)2+14ΦxΦt.
We, therefore, obtain the following variational principle:
(5.43)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+112au3-14b(ux)2+14ΦxΦt}dtdx.
By the semi-inverse method, we can obtain various different two-field variational principles, and we write here the following one for reference:
(5.44)J(Φ,u)=∬{12ΦxΦt+a(13u3-12u2Φx)-b2Φxx2}dtdx.
The potential in the above functionals (5.23), (5.33), and (5.44) requires second order of differentiation (Φxx), leading to the complications in the finite element calculation. For the purpose of simplification in finite element computation, we often introduce some additional variables to reduce the order of differentiations. This is of course equivalent to introducing some additional constraints in the variational principle. Generally, we can eliminate the introduced constraints by the Lagrange multiplier method, but as illustrated above, the method might fail.Now we introduce a new variableE defined as
(5.45)E=a2u2+buxx.
By the semi-inverse method [17, 18], we obtain the following three-field variational principle:
(5.46)J(Φ,u,E)=∬{uΦt+EΦx+a3u3-bux2-Eu}dtdx.
It is obvious that all variables in the obtained functional (5.46) are in first-order differentiations, leading to much convenience in numerical simulation.It is easy to establish a variational formulation by introducing a potential function, and we can also establish a variational principle without auxiliary special function. To elucidate this, we consider the KdV equation in the form(5.47)ut+uux+uxxx=0,
where subscripts denote partial differentiations. If we introduce a velocity potential Φ defined as u=Φx, then the KdV equation can be derived from the variational principle
(5.48)δ∬Ldtdx=0,
where the Lagrangian L can be expressed in the form
(5.49)L=12ΦxΦt+16Φx3-12Φxx2.
Our aim is to search for a Lagrangian for (5.47). It is easy to establish a variational formulation for differential equations with even orders. The KdV equation has odd-order differentiations, and therefore, no Lagrangian for (5.47). To circumvent this obstacle, we take partial differentiation with respect to x to both sides of KdV equation, which turns out to be the following form:
(5.50)utx+ux2+uuxx+uxxxx=0.
By the semi-inverse method, we construct a trial Lagrangian in the form
(5.51)L=-12uxut+12uxx2+puux2+qu2uxx,
where p and q are constants to be further determined. Its Euler equation can be readily obtained as follows:
(5.52)utx+uxxxx+p[ux2-2(uux)x]+q[2uuxx-(u2)xx]=0,
or
(5.53)utx+uxxxx+(-p-2q)ux2-2puuxx=0.
Setting  p=-1/2,q=-1/4, then (5.53) turns out to be the modification version of the KdV equation (5.50).Finally, we have the following needed Lagrangian in the form of velocity:(5.54)L=-12uxut+12uxx2-12uux2-14u2uxx.
This approach can be extended to many nonlinear equations. Consider the modified KdV equation
(5.55)ut+mu2ux+nuxxx=0.
Similarly, we change the equation so that it has even-order differentiations
(5.56)utx+2muux2+mu2uxx+nuxxxx=0.
By the same manipulation as illustrated above, we construct a trial Lagrange function in the form
(5.57)L=-12uxut+12nuxx2+pu2ux2+qu3uxx,
where p and q are constants to be further determined. Its Euler equation can be readily obtained as follows:
(5.58)utx+nuxxxx+p[2uux2-2(u2ux)x]+q[3u2uxx-(u3)xx]=0,
or
(5.59)utx+nuxxxx+p[2uux2-2(2uux2+u2uxx)]+q[3u2uxx-(6uux2+3u2uxx)]=0,
or
(5.60)utx+nuxxxx+(-2p-6q)uux2-2pu2uxx=0.
Setting
(5.61)p=-m2,q=-m6,
(5.60) becomes (5.56), and we, therefore, obtain the following Lagrangian:
(5.62)L=-12uxut+12nuxx2-m2u2ux2-m6u3uxx.
We can also use the semi-inverse method to establish a family of variational principles for a nonlinear system. We use one-dimensional traffic flow as an example.The research on traffic flow began at the beginning of the 20th century. Lighthill and Whitham first proposed the fluid-dynamical model for traffic flow [19, 20]. The continuum equation for unsteady one-dimensional traffic flow can be, therefore, written as
(5.63)∂∂t(ρA)+∂∂x(ρuA)=q,
where A is the cross-sectional area of the road, u is the velocity, ρ is the density of cars, and q is the source. The deficiency of the model is that the traffic flow actually cannot be considered as a continuum, and to eliminate this deficiency, a fractional differential model can be introduced:
(5.64)∂∂t(ρA)+DαDxα(ρuA)=q,0<α<1,
where Dα/Dxα is the fractional differential, see Section 7.In 1994, Zheng [21] suggested the following traffic flow model:
(5.65)∂∂t(ρ)+∂∂x(ρu)=0,(5.66)∂∂t(ρu)+∂∂x[ρu2+14n(n-1)2uf2ρ01-nρn]=0,
where uf is the possible maximal velocity, m is a constant, and ρ0 is the minimal traffic density when the cars can travel at a maximal velocity.In order to establish a variational principle for the system, we rewrite (5.66) in the following equivalent form:
(5.67)∂∂t(u)+∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]=0.
According to (5.65) and (5.67), we can introduce two functions Φ and Ψ defined as
(5.68)∂Φ∂x=-ρ,∂Φ∂t=ρu,(5.69)∂Ψ∂x=-u,∂Ψ∂t=12u2+14(n-1)uf2ρ01-nρn-1,
so that (5.65) and (5.66) are automatically satisfied.The essence of the semi-inverse method [17, 18] is to construct an energy-like functional with a certain unknown function, which can be identified step by step. An energy-like trial functional for the discussed problem can be constructed in the following form:
(5.70)J(u,ρ,Φ)=∫0t∫0L{u∂Φ∂t+[12u2+14(n-1)uf2ρ01-nρn-1]∂Φ∂x+F}dtdx,
where u, ρ, and Φ are considered as independent variables, and F is an unknown function of u, ρ and/or their derivatives.There exist various approaches to the establishment of energy-like trial functionals for a physical problem, and illustrative examples can be found in [22, 23].The advantage of the above trial functional lies on the fact that the stationary condition with respective toΦ,
(5.71)δΦJ(u,h,Φ)=∫0t∫0L{-∂∂t(u)-∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]}δΦdtdx=0,
leads to (5.67).Calculating the functional (5.71) stationary with respect to u and ρ, we obtain the following Euler equations:
(5.72)∂Φ∂t+u∂Φ∂x+δFδu=0,[14(n-1)2uf2ρ01-nρn-2]∂Φ∂x+δFδρ=0.
We search for such an F, so that the above (5.72) satisfies the two-field equations. To this end, we set
(5.73)δFδu=-∂Φ∂t-u∂Φ∂x=-ρu+uρ=0,δFδρ=-[14(n-1)2uf2ρ01-nρn-2]∂Φ∂x=14(n-1)2uf2ρ01-nρn-1.
From (5.73), we can immediately identify the unknown F, which reads
(5.74)F=14n(n-1)2uf2ρ01-nρn.
So we obtain the following required variational functional:
(5.75)J(u,ρ,Φ)=∫0t∫0L{14n(n-1)2uf2ρ01-nρn+u∂Φ∂t+[12u2+14(n-1)uf2ρ01-nρn-1]∂Φ∂x}dtdx.Proof.
The Euler equations of the above functional (5.75) are
(5.76)δρ:14(n-1)2uf2ρ01-nρn-1+14(n-1)2uf2ρ01-nρn-2∂Φ∂x=0,(5.77)δu:∂Φ∂t+u∂Φ∂x=0,(5.78)δΦ:-∂∂t(u)-∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]=0.
From (5.68), we have ∂Φ/∂x=-ρ. Substituting the result into (5.77) leads to ∂Φ/∂t=ρu.
From the above three-field variational functional, we can easily obtain two-field or one-field variational function by substituting one- or two-field equations into the functional (5.75). For example, substituting ∂Φ/∂x=-ρ into (5.75), we obtain a two-field variational functional
(5.79)J1(u,Φ)=∫0t∫0L{14n(n-1)2uf2ρ01-nρn+u∂Φ∂t-[12ρu2+14(n-1)uf2ρ01-nρn]}dtdx,
where the variable ρ is not now an independent field. Further constraining the two-field functional (5.79) by the equation ∂Φ/∂t=ρu, we have
(5.80)J2(Φ)=14n(n-1)2uf2ρ01-n∫0t∫0Lρndtdx=∫0t∫0LPdtdx,
where P is the traffic pressure defined as
(5.81)P=Cρn=14n(n-1)2uf2ρ01-nρn.
The functional (5.81) has the same form of the well-known Bateman principle in fluid mechanics [18].
By a paralleling operation, we can also establish a variational functional with free fieldsu,  ρ, and Ψ. A trial functional with an unknown function F can be constructed as follows:
(5.82)J~(u,ρ,Ψ)=∫0t∫0L{ρ∂Ψ∂t+ρu∂Ψ∂x+F}dtdx.
Here, the unknown F is free from Ψ and its derivatives. By the same manipulation as illustrated above, we set
(5.83)δFδu=-ρ∂Ψ∂x=ρu,δFδρ=-∂Ψ∂t-u∂Ψ∂x=12u2-14(n-1)uf2ρ01-nρn-1.
From (5.83), we can determine the unknown F as follows:
(5.84)F=12ρu2-14n(n-1)uf2ρ01-nρn.
So we obtain the following needed variational principle:
(5.85)J~(u,ρ,Ψ)=∫0t∫0L{12ρu2-14n(n-1)uf2ρ01-nρn+ρ∂Ψ∂t+ρu∂Ψ∂x}dtdx.
It is easy to prove that the Euler equations of the above functional (5.85) satisfy the field equations (5.67) and (5.69).
Constraining the functional (5.85) by the equation ∂Ψ/∂x=-u, we obtain
(5.86)J~1(ρ,Ψ)=∫0t∫0L{-12ρ(∂Ψ∂x)2-14n(n-1)uf2ρ01-nρn+ρ∂Ψ∂t}dtdx,
which is under the constraint of the equation ∂Ψ/∂x=-u. Further substituting
(5.87)∂Ψ∂t=(12)u2+(14)(n-1)uf2ρ01-nρn-1
into (5.86) results in
(5.88)J~2=∫0t∫0L{12ρu2-14n(n-1)uf2ρ01-nρn}dtdx=∫0t∫0L{12ρu2-P}dtdx,
which is similar to the well-known Hamilton principle.


### 5.3.2. A Possible Connection between the Uncertain Principle and the Least Action Principle

Maupertuis-Lagrange’s principle of least kinetic potential action for a particle with massm can be expressed as follows [18]:
(5.89)∫t1t212mv2dt→min.
We rewrite (5.89) in the form
(5.90)∫s1s212mv·ds→min,
or
(5.91)∫s1s2p·ds→min.
Equation (5.91) can be approximately written in the form
(5.92)(p2-p1)·ds12=Emin,
where ds12=s2-s1. Equation (5.92) means that given that the particle begins at position s1 at time t1 and ends at position s2 at time t2, the physical trajectory that connects these two endpoints is an extremum of Δp·Δs, where Δs is the standard deviation of the displacement, and Δp is the deviation of the momentum.For arbitraryΔs or Δp, the following inequality holds:
(5.93)Δp·Δs≥Emin.
This is similar to the uncertainty principle.In optics, Fermat’s principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light(5.94)t=∫s1s2dsν→min.
Equation (5.94) can be approximately written in the form
(5.95)(1ν2-1ν1)ds12→min,
where ds12=s2-s1.The light trajectory that connects these two endpointss1 and s2 satisfies the following equation:
(5.96)(1ν2-1ν1)ds=Tmin,
or
(5.97)-1ν1ν2Δv·Δs=Tmin,
where Δs=s2-s1, Δν=ν2-ν1.For arbitraryΔs or Δv, the following inequality holds:
(5.98)-Δv·Δs=ν1ν2Tmin≥c2Tmin,Δv·Δs≤c2Tmin.


### 5.3.3. Variational Approach to Nonlinear Oscillators [24]

Consider a general nonlinear oscillator in the form(5.99)u′′+f(u)=0.
Its variational principle can be easily established as follows:
(5.100)J(u)=∫0T/4{-12u′2+F(u)}dt,
where T is the period of the nonlinear oscillator, ∂F/∂u=f.Assume that its solution can be expressed as(5.101)u(t)=pcosqt,
where p and q are the amplitude and frequency of the oscillator, respectively.Substituting (5.101) into (5.100) results in
(5.102)J(p,q)=∫0T/4{-12p2q2sin2qt+F(Acosqt)}dt.
Instead of setting ∂J/∂q=0 and ∂J/∂p=0, we only set [8, 24]
(5.103)∂J∂p=0,
from which the relationship between the amplitude and frequency of the oscillator can be obtained.Explanation of (5.103) was given in [8].Consider a nonlinear oscillator with fractional potential [24]:
(5.104)u′′+εu1/3=0.
Its variational formulation can be readily obtained as follows:
(5.105)J(u)=∫0T/4{-12u′2+34εu4/3}dt.
Substituting (5.101) into (5.105), we obtain
(5.106)J=∫0T/4{-12p2q2sin2qt+34p4/3εcos4/3qt}dt.
Setting
(5.107)∂J∂A=∫0T/4{-Aω2sin2ωt+A1/3εcos4/3ωt}dt=0,
we have
(5.108)ω2=∫0T/4εcos4/3ωtdtA2/3∫0T/4sin2ωtdt=∫0π/2εcos4/3tdtA2/3∫0π/2sin2tdt=1.15959526696393εA3/2.
The exact frequency is ω=1.070451ε1/2A-1/3. The 0.597% accuracy is remarkably good.


### 5.3.4. Variational Approach to Chemical Reactions

As an illustration, consider the following chemical reaction [25]:
(5.109)nA→C+D,
which obeys the equation
(5.110)dxdt=k(a-x)n,x(0)=0,
where a is the number of molecules A at t=0, x is the number of molecules C (or D) after time t, and k is a reaction constant. At the start of reaction (t=0), there are no molecules C (or D) yet formed, so that the initial condition is x(0)=0.In order to obtain a variational model, we differentiate both sides of (5.110) with respect to time, resulting in
(5.111)d2xdt2=-kn(a-x)n-1dxdt.
Substituting (5.111) into (5.110), we obtain the following second-order differential equation:
(5.112)d2xdt2=-k2n(a-x)2n-1,x(0)=0,x′(0)=kan,
which admits a variational expression in the form
(5.113)J(x)=∫0∞{12(dxdt)2+12k2(a-x)2n}dt.
Its Hamiltonian, therefore, can be written in the form
(5.114)H=12(dxdt)2-12k2(a-x)2n,
where H is a Hamiltonian constant, and it can be determined from initial conditions:
(5.115)H=12x˙(0)2-12k2(a-x(0))2n=0.
Equation (5.114) becomes
(5.116)(dxdt)2=k2(a-x)2n.
In view of the initial conditions x(0)=0 and x′(0)=kan, (5.116) is equivalent to (5.110). This means that the variational principle, (5.113), is exactly equivalent to its differential partner, (5.110).Assume that the solution can be expressed in the form(5.117)x=a(1-e-ηt),
where η is an unknown constant to be further determined.Substituting (5.117) into (5.113), and setting dJ/dη=0, we obtain
(5.118)η=1nkan-1.
So we obtain a first-order approximate solution for the discussed problem
(5.119)x=a⌊1-exp(-n-1/2kan-1t)⌋.
In order to improve accuracy, we can assume that the solution can be expressed in a more general form
(5.120)x=a(1-∑i=1mbie-ηit),
which should satisfy initial conditions x(0)=0, and this requires
(5.121)1-∑i=1mbi=0.
Substituting (5.120) into (5.113), we set
(5.122)∂J∂ηi=0(i=1~m),∂J∂bi=0(i=2~m).
Solving (5.121)-(5.122) simultaneously, we can easily determine 2m parameters. The solution procedure is similar to that illustrated in [25], and we will not discuss in details to solve space.We can also choose the following trial function:(5.123)x=a{1-1(1+pt)q},
where p and q are unknown constants to be further determined. It is obvious that (5.123) satisfies the conditions x(0)=0 and x(+∞)=a. Submitting (5.123) into (5.113), and setting
(5.124)∂J∂p=0,∂J∂q=0,
we obtain with ease
(5.125)p=(n-1)kan-1,q=1n-1.
Thus, we obtain the solution
(5.126)x=a{1-1(1+(n-1)kan-1t)1/(n-1)},
which is the exact solution.


### 5.3.5. Variational Approach to Solitary Solution

In the review article [7], the variational approach to solitons was outlined by few lines, and now the method has been successfully applied to the search for soliton solutions [26, 27] without requirement of small parameter assumption, leading to an extremely simple and elementary but rigorous derivation of soliton solutions.Considering the KdV equation, we seek its traveling wave solutions in the following frame:(5.127)u(x,t)=U(ξ),ν(x,t)=V(ξ),ξ=x-ct,
where c is angular frequency. Substituting (5.127) into (4.1) yields
(5.128)-cu′-6uu′+u′′′=0,
where prime denotes the differential with respect to ξ.Integrating (5.128) yields the result(5.129a)-cu-3u2+u′′=A,
where A is an integration constant, which can be determined from the initial condition. For solitary solutions or limit cycles, the solutions do not depend upon the initial condition, so we always set A=0, and this results in
(5.129b)-cu-3u2+u′′=0.By the semi-inverse method [17], the following variational formulation is established:
(5.130)J=∫0∞(12cu2+u3+12(dudξ)2)dξ.
Equation (5.130) also hints a Hamiltonian invariant, which reads(5.131a)12cu2+u3-12u′2=B,
where B is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set B=0,
(5.131b)12cu2+u3-12u′2=0.Differentiating (5.131a) with respect to ξ, we have
(5.132)cuu′+3u2u′-u′u′′=0,
which is equivalent to (5.129b).The Hamiltonian invariant (5.131a) and (5.131b) can also be used for construction of an asymptotic solution or used as an auxiliary function in the subequation method, see Section 5.10.The semi-inverse method is a powerful mathematical tool to the search for variational formulae for real-life physical problems.By Ritz method, we search for a solitary wave solution in the form(5.133)u=psech2(qξ),
where p and q are constants to be further determined.Substituting (5.133) into (5.130) results in
(5.134)J=∫0∞[12cp2sech4(qξ)+p3sech6(qξ)+12(4p2q2sech4(qξ)tanh2(qξ)]dξ=cp22q∫0∞sech4(z)dz+p3q∫0∞sech6(z)dz+2p2q∫0∞{sech4(z)tanh4(z)}dz=cp23q+8p315q+4p2q15.
Making J stationary with respect to p and q results in
(5.135)∂J∂p=2cp3q+24p215q+8pq15=0,∂J∂q=-cp23q2-8p315q2+4p215=0,
or simplifying
(5.136)5c+12p+4q2=0,-5c-8p+4q2=0.
From (5.136), we can easily obtain the following relations:
(5.137)p=-12c,q=c4.
So the solitary wave solution can be approximated as
(5.138)u=-c2sech2c4(x-ct-ξ0),
which is the exact solitary wave solution of KdV equation.For the KdV equation expressed in (5.47), we have the following variational principle (see (5.54)):
(5.139)J(u)=∬-∞+∞{-12uxut+12uxx2-12uux2-14u2uxx}dxdt.
We assume that the solitary solution can be expressed in the form
(5.140)u(x,t)=1a1ea5x-a5t+a2e-a7x+a8t+a3,
where a1,a2,…,a8 are unknown constants to be further determined.Substituting (5.140) into (5.139), we have
(5.141)J(u)=J(a1,a2,…,a8).
Making the functional, (5.139), stationary can be approximated achieved by making the function (5.141) stationary, and this requires
(5.142)∂J∂ai=0(i=1~8).
Solving the system, (5.142), we can determine the values for a1,a2,…,a8.


## 5.3.1. Inverse Problem of Calculus of Variations

The inverse problem of calculus of variations is to establish a variational formulation directly from governing equations and boundary/initial conditions. We will use the semi-inverse method [17, 18] to establish various variational principles directly from the governing equations.Consider the well-known Korteweg-de Vries (KdV) equation(5.18)ut+auux+buxxx=0,
where a and b are constants, and the subscripts denote partial differentiations.We rewrite it in a conserved form(5.19)∂u∂t+∂∂x(12au2+buxx)=0.
According to the conservation form of (5.19), we can introduce a potential functional Φ defined by
(5.20)Φx=u,(5.21)Φt=-(12au2+buxx).
So the KdV equation can be written in the form
(5.22)Φxt+aΦxΦxx+bΦxxxx=0,
which can be derived from the following variational principle using the semi-inverse method [17]:
(5.23)J(Φ)=∬{12ΦxΦt+a6Φx3-b2Φxx2}dtdx.
In order to obtain a generalized variational principle with two independent fields (Φ,u), we apply the Lagrange multiplier to (5.23)
(5.24)J(Φ,u,λ)=∬{12ΦxΦt+a6Φx3-b2Φxx2+λ(Φx-u)}dtdx.
The stationary condition with respect to u results in
(5.25)λ=0.
The Lagrange multiplier method is not valid for the case. This phenomenon is called Lagrange crisis. Hereby, we suggest three ways to overcome the crisis [18].(
1
) The Semi-Inverse Method [17, 18]
Generally, the multiplier can be expressed in the form after identification(5.26)λ=λ(u,Φ,ut,ux,Φt,Φx,…).
We replace the last term including the Lagrange multiplier by a new variable F, that is,
(5.27)J(Φ,u)=∬{12ΦxΦt+a6Φx3-b2Φxx2+F}dtdx,
where F is unknown, which can be expressed in the form
(5.28)F=F(u,Φ,ut,ux,Φt,Φx,…).
Equation (5.27) is such constructed according to the semi-inverse method [17]. To identify F, making the functional stationary with respect to u, we have
(5.29)δFδu=0,
where δF/δu is the variational derivative defined as
(5.30)δFδu=∂F∂u-∂∂t(∂F∂ut)-∂∂x(∂F∂ux)+∂2∂x2(∂F∂uxx)+⋯.
Equation (5.29) should be equivalent to (5.20), and this requires
(5.31)δFδu=c(Φx-u),
where c is a nonzero constant. From (5.31), we can determine F as follows:
(5.32)F=-12c(Φx-u)2=c~(Φx-u)2,
where c~ is a constant, c~=-c/2.
We, therefore, obtain the following needed variational principle:(5.33)J(Φ,u)=∬{12ΦxΦt+a6Φx3-b2Φxx2+c~(Φx-u)2}dtdx.
Its Euler-Lagrange equations are
(5.34)-2c~(Φx-u)=0,-Φxt-aΦxΦxx-bΦxxxx-2c~(Φx-u)x=0,
which satisfy the field equations (5.20) and (5.22), respectively.(
2
) The Hidden Lagrange Multiplier [18]
Let us come back to (5.25), λ=0, which should be the constraint equation. This means that (5.25) inexplicitly involves a lost constraint equation, so we can identify the multiplier in the form [18]
(5.35)λ=c~(Φx-u).
This results in the same result above.(
3
) Replacement of Some Variables in the Original Functional Using the Constraint Equation
Sometimes the Lagrange crisis can be eliminated by replacing some variables in the original variational principle using the constraint equation; a detailed discussion was systematically given in [18].
We replaceΦxΦt by uΦt in (5.23) and introduce a Lagrange multiplier in the resultant function
(5.36)J(Φ,u,λ)=∬{12uΦt+a6Φx3-b2Φxx2+λ(Φx-u)}dtdx.
Identification of the multiplier yields
(5.37)λ=12Φt.
Submitting the identified multiplier into (5.36) results in
(5.38)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+12Φt(Φx-u)}dtdx=∬{a6Φx3-b2Φxx2+12ΦtΦx}dtdx.
We find that the constraint is not eliminated yet, and this is another Lagrange crisis, which can also be eliminated by the semi-inverse method
(5.39)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+F}dtdx.
The stationary conditions with respect to u and Φ are, respectively, as follows:
(5.40)12Φt+δFδu=0,-12ut-aΦxΦxx-bΦxxxx+δFδΦ=0.
According to (5.20) and (5.21), we have
(5.41)δFδu=-12Φt=14au2+12buxx,δFδΦ=12ut+aΦxΦxx+bΦxxxx=12ut-Φxt=12Φxt-Φxt=-12Φxt.
From (5.41), F can be determined as
(5.42)F=112au3-14b(ux)2+14ΦxΦt.
We, therefore, obtain the following variational principle:
(5.43)J(Φ,u)=∬{12uΦt+a6Φx3-b2Φxx2+112au3-14b(ux)2+14ΦxΦt}dtdx.
By the semi-inverse method, we can obtain various different two-field variational principles, and we write here the following one for reference:
(5.44)J(Φ,u)=∬{12ΦxΦt+a(13u3-12u2Φx)-b2Φxx2}dtdx.
The potential in the above functionals (5.23), (5.33), and (5.44) requires second order of differentiation (Φxx), leading to the complications in the finite element calculation. For the purpose of simplification in finite element computation, we often introduce some additional variables to reduce the order of differentiations. This is of course equivalent to introducing some additional constraints in the variational principle. Generally, we can eliminate the introduced constraints by the Lagrange multiplier method, but as illustrated above, the method might fail.Now we introduce a new variableE defined as
(5.45)E=a2u2+buxx.
By the semi-inverse method [17, 18], we obtain the following three-field variational principle:
(5.46)J(Φ,u,E)=∬{uΦt+EΦx+a3u3-bux2-Eu}dtdx.
It is obvious that all variables in the obtained functional (5.46) are in first-order differentiations, leading to much convenience in numerical simulation.It is easy to establish a variational formulation by introducing a potential function, and we can also establish a variational principle without auxiliary special function. To elucidate this, we consider the KdV equation in the form(5.47)ut+uux+uxxx=0,
where subscripts denote partial differentiations. If we introduce a velocity potential Φ defined as u=Φx, then the KdV equation can be derived from the variational principle
(5.48)δ∬Ldtdx=0,
where the Lagrangian L can be expressed in the form
(5.49)L=12ΦxΦt+16Φx3-12Φxx2.
Our aim is to search for a Lagrangian for (5.47). It is easy to establish a variational formulation for differential equations with even orders. The KdV equation has odd-order differentiations, and therefore, no Lagrangian for (5.47). To circumvent this obstacle, we take partial differentiation with respect to x to both sides of KdV equation, which turns out to be the following form:
(5.50)utx+ux2+uuxx+uxxxx=0.
By the semi-inverse method, we construct a trial Lagrangian in the form
(5.51)L=-12uxut+12uxx2+puux2+qu2uxx,
where p and q are constants to be further determined. Its Euler equation can be readily obtained as follows:
(5.52)utx+uxxxx+p[ux2-2(uux)x]+q[2uuxx-(u2)xx]=0,
or
(5.53)utx+uxxxx+(-p-2q)ux2-2puuxx=0.
Setting  p=-1/2,q=-1/4, then (5.53) turns out to be the modification version of the KdV equation (5.50).Finally, we have the following needed Lagrangian in the form of velocity:(5.54)L=-12uxut+12uxx2-12uux2-14u2uxx.
This approach can be extended to many nonlinear equations. Consider the modified KdV equation
(5.55)ut+mu2ux+nuxxx=0.
Similarly, we change the equation so that it has even-order differentiations
(5.56)utx+2muux2+mu2uxx+nuxxxx=0.
By the same manipulation as illustrated above, we construct a trial Lagrange function in the form
(5.57)L=-12uxut+12nuxx2+pu2ux2+qu3uxx,
where p and q are constants to be further determined. Its Euler equation can be readily obtained as follows:
(5.58)utx+nuxxxx+p[2uux2-2(u2ux)x]+q[3u2uxx-(u3)xx]=0,
or
(5.59)utx+nuxxxx+p[2uux2-2(2uux2+u2uxx)]+q[3u2uxx-(6uux2+3u2uxx)]=0,
or
(5.60)utx+nuxxxx+(-2p-6q)uux2-2pu2uxx=0.
Setting
(5.61)p=-m2,q=-m6,
(5.60) becomes (5.56), and we, therefore, obtain the following Lagrangian:
(5.62)L=-12uxut+12nuxx2-m2u2ux2-m6u3uxx.
We can also use the semi-inverse method to establish a family of variational principles for a nonlinear system. We use one-dimensional traffic flow as an example.The research on traffic flow began at the beginning of the 20th century. Lighthill and Whitham first proposed the fluid-dynamical model for traffic flow [19, 20]. The continuum equation for unsteady one-dimensional traffic flow can be, therefore, written as
(5.63)∂∂t(ρA)+∂∂x(ρuA)=q,
where A is the cross-sectional area of the road, u is the velocity, ρ is the density of cars, and q is the source. The deficiency of the model is that the traffic flow actually cannot be considered as a continuum, and to eliminate this deficiency, a fractional differential model can be introduced:
(5.64)∂∂t(ρA)+DαDxα(ρuA)=q,0<α<1,
where Dα/Dxα is the fractional differential, see Section 7.In 1994, Zheng [21] suggested the following traffic flow model:
(5.65)∂∂t(ρ)+∂∂x(ρu)=0,(5.66)∂∂t(ρu)+∂∂x[ρu2+14n(n-1)2uf2ρ01-nρn]=0,
where uf is the possible maximal velocity, m is a constant, and ρ0 is the minimal traffic density when the cars can travel at a maximal velocity.In order to establish a variational principle for the system, we rewrite (5.66) in the following equivalent form:
(5.67)∂∂t(u)+∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]=0.
According to (5.65) and (5.67), we can introduce two functions Φ and Ψ defined as
(5.68)∂Φ∂x=-ρ,∂Φ∂t=ρu,(5.69)∂Ψ∂x=-u,∂Ψ∂t=12u2+14(n-1)uf2ρ01-nρn-1,
so that (5.65) and (5.66) are automatically satisfied.The essence of the semi-inverse method [17, 18] is to construct an energy-like functional with a certain unknown function, which can be identified step by step. An energy-like trial functional for the discussed problem can be constructed in the following form:
(5.70)J(u,ρ,Φ)=∫0t∫0L{u∂Φ∂t+[12u2+14(n-1)uf2ρ01-nρn-1]∂Φ∂x+F}dtdx,
where u, ρ, and Φ are considered as independent variables, and F is an unknown function of u, ρ and/or their derivatives.There exist various approaches to the establishment of energy-like trial functionals for a physical problem, and illustrative examples can be found in [22, 23].The advantage of the above trial functional lies on the fact that the stationary condition with respective toΦ,
(5.71)δΦJ(u,h,Φ)=∫0t∫0L{-∂∂t(u)-∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]}δΦdtdx=0,
leads to (5.67).Calculating the functional (5.71) stationary with respect to u and ρ, we obtain the following Euler equations:
(5.72)∂Φ∂t+u∂Φ∂x+δFδu=0,[14(n-1)2uf2ρ01-nρn-2]∂Φ∂x+δFδρ=0.
We search for such an F, so that the above (5.72) satisfies the two-field equations. To this end, we set
(5.73)δFδu=-∂Φ∂t-u∂Φ∂x=-ρu+uρ=0,δFδρ=-[14(n-1)2uf2ρ01-nρn-2]∂Φ∂x=14(n-1)2uf2ρ01-nρn-1.
From (5.73), we can immediately identify the unknown F, which reads
(5.74)F=14n(n-1)2uf2ρ01-nρn.
So we obtain the following required variational functional:
(5.75)J(u,ρ,Φ)=∫0t∫0L{14n(n-1)2uf2ρ01-nρn+u∂Φ∂t+[12u2+14(n-1)uf2ρ01-nρn-1]∂Φ∂x}dtdx.Proof.
The Euler equations of the above functional (5.75) are
(5.76)δρ:14(n-1)2uf2ρ01-nρn-1+14(n-1)2uf2ρ01-nρn-2∂Φ∂x=0,(5.77)δu:∂Φ∂t+u∂Φ∂x=0,(5.78)δΦ:-∂∂t(u)-∂∂x[12u2+14(n-1)uf2ρ01-nρn-1]=0.
From (5.68), we have ∂Φ/∂x=-ρ. Substituting the result into (5.77) leads to ∂Φ/∂t=ρu.
From the above three-field variational functional, we can easily obtain two-field or one-field variational function by substituting one- or two-field equations into the functional (5.75). For example, substituting ∂Φ/∂x=-ρ into (5.75), we obtain a two-field variational functional
(5.79)J1(u,Φ)=∫0t∫0L{14n(n-1)2uf2ρ01-nρn+u∂Φ∂t-[12ρu2+14(n-1)uf2ρ01-nρn]}dtdx,
where the variable ρ is not now an independent field. Further constraining the two-field functional (5.79) by the equation ∂Φ/∂t=ρu, we have
(5.80)J2(Φ)=14n(n-1)2uf2ρ01-n∫0t∫0Lρndtdx=∫0t∫0LPdtdx,
where P is the traffic pressure defined as
(5.81)P=Cρn=14n(n-1)2uf2ρ01-nρn.
The functional (5.81) has the same form of the well-known Bateman principle in fluid mechanics [18].
By a paralleling operation, we can also establish a variational functional with free fieldsu,  ρ, and Ψ. A trial functional with an unknown function F can be constructed as follows:
(5.82)J~(u,ρ,Ψ)=∫0t∫0L{ρ∂Ψ∂t+ρu∂Ψ∂x+F}dtdx.
Here, the unknown F is free from Ψ and its derivatives. By the same manipulation as illustrated above, we set
(5.83)δFδu=-ρ∂Ψ∂x=ρu,δFδρ=-∂Ψ∂t-u∂Ψ∂x=12u2-14(n-1)uf2ρ01-nρn-1.
From (5.83), we can determine the unknown F as follows:
(5.84)F=12ρu2-14n(n-1)uf2ρ01-nρn.
So we obtain the following needed variational principle:
(5.85)J~(u,ρ,Ψ)=∫0t∫0L{12ρu2-14n(n-1)uf2ρ01-nρn+ρ∂Ψ∂t+ρu∂Ψ∂x}dtdx.
It is easy to prove that the Euler equations of the above functional (5.85) satisfy the field equations (5.67) and (5.69).
Constraining the functional (5.85) by the equation ∂Ψ/∂x=-u, we obtain
(5.86)J~1(ρ,Ψ)=∫0t∫0L{-12ρ(∂Ψ∂x)2-14n(n-1)uf2ρ01-nρn+ρ∂Ψ∂t}dtdx,
which is under the constraint of the equation ∂Ψ/∂x=-u. Further substituting
(5.87)∂Ψ∂t=(12)u2+(14)(n-1)uf2ρ01-nρn-1
into (5.86) results in
(5.88)J~2=∫0t∫0L{12ρu2-14n(n-1)uf2ρ01-nρn}dtdx=∫0t∫0L{12ρu2-P}dtdx,
which is similar to the well-known Hamilton principle.


## 5.3.2. A Possible Connection between the Uncertain Principle and the Least Action Principle

Maupertuis-Lagrange’s principle of least kinetic potential action for a particle with massm can be expressed as follows [18]:
(5.89)∫t1t212mv2dt→min.
We rewrite (5.89) in the form
(5.90)∫s1s212mv·ds→min,
or
(5.91)∫s1s2p·ds→min.
Equation (5.91) can be approximately written in the form
(5.92)(p2-p1)·ds12=Emin,
where ds12=s2-s1. Equation (5.92) means that given that the particle begins at position s1 at time t1 and ends at position s2 at time t2, the physical trajectory that connects these two endpoints is an extremum of Δp·Δs, where Δs is the standard deviation of the displacement, and Δp is the deviation of the momentum.For arbitraryΔs or Δp, the following inequality holds:
(5.93)Δp·Δs≥Emin.
This is similar to the uncertainty principle.In optics, Fermat’s principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light(5.94)t=∫s1s2dsν→min.
Equation (5.94) can be approximately written in the form
(5.95)(1ν2-1ν1)ds12→min,
where ds12=s2-s1.The light trajectory that connects these two endpointss1 and s2 satisfies the following equation:
(5.96)(1ν2-1ν1)ds=Tmin,
or
(5.97)-1ν1ν2Δv·Δs=Tmin,
where Δs=s2-s1, Δν=ν2-ν1.For arbitraryΔs or Δv, the following inequality holds:
(5.98)-Δv·Δs=ν1ν2Tmin≥c2Tmin,Δv·Δs≤c2Tmin.


## 5.3.3. Variational Approach to Nonlinear Oscillators [24]

Consider a general nonlinear oscillator in the form(5.99)u′′+f(u)=0.
Its variational principle can be easily established as follows:
(5.100)J(u)=∫0T/4{-12u′2+F(u)}dt,
where T is the period of the nonlinear oscillator, ∂F/∂u=f.Assume that its solution can be expressed as(5.101)u(t)=pcosqt,
where p and q are the amplitude and frequency of the oscillator, respectively.Substituting (5.101) into (5.100) results in
(5.102)J(p,q)=∫0T/4{-12p2q2sin2qt+F(Acosqt)}dt.
Instead of setting ∂J/∂q=0 and ∂J/∂p=0, we only set [8, 24]
(5.103)∂J∂p=0,
from which the relationship between the amplitude and frequency of the oscillator can be obtained.Explanation of (5.103) was given in [8].Consider a nonlinear oscillator with fractional potential [24]:
(5.104)u′′+εu1/3=0.
Its variational formulation can be readily obtained as follows:
(5.105)J(u)=∫0T/4{-12u′2+34εu4/3}dt.
Substituting (5.101) into (5.105), we obtain
(5.106)J=∫0T/4{-12p2q2sin2qt+34p4/3εcos4/3qt}dt.
Setting
(5.107)∂J∂A=∫0T/4{-Aω2sin2ωt+A1/3εcos4/3ωt}dt=0,
we have
(5.108)ω2=∫0T/4εcos4/3ωtdtA2/3∫0T/4sin2ωtdt=∫0π/2εcos4/3tdtA2/3∫0π/2sin2tdt=1.15959526696393εA3/2.
The exact frequency is ω=1.070451ε1/2A-1/3. The 0.597% accuracy is remarkably good.


## 5.3.4. Variational Approach to Chemical Reactions

As an illustration, consider the following chemical reaction [25]:
(5.109)nA→C+D,
which obeys the equation
(5.110)dxdt=k(a-x)n,x(0)=0,
where a is the number of molecules A at t=0, x is the number of molecules C (or D) after time t, and k is a reaction constant. At the start of reaction (t=0), there are no molecules C (or D) yet formed, so that the initial condition is x(0)=0.In order to obtain a variational model, we differentiate both sides of (5.110) with respect to time, resulting in
(5.111)d2xdt2=-kn(a-x)n-1dxdt.
Substituting (5.111) into (5.110), we obtain the following second-order differential equation:
(5.112)d2xdt2=-k2n(a-x)2n-1,x(0)=0,x′(0)=kan,
which admits a variational expression in the form
(5.113)J(x)=∫0∞{12(dxdt)2+12k2(a-x)2n}dt.
Its Hamiltonian, therefore, can be written in the form
(5.114)H=12(dxdt)2-12k2(a-x)2n,
where H is a Hamiltonian constant, and it can be determined from initial conditions:
(5.115)H=12x˙(0)2-12k2(a-x(0))2n=0.
Equation (5.114) becomes
(5.116)(dxdt)2=k2(a-x)2n.
In view of the initial conditions x(0)=0 and x′(0)=kan, (5.116) is equivalent to (5.110). This means that the variational principle, (5.113), is exactly equivalent to its differential partner, (5.110).Assume that the solution can be expressed in the form(5.117)x=a(1-e-ηt),
where η is an unknown constant to be further determined.Substituting (5.117) into (5.113), and setting dJ/dη=0, we obtain
(5.118)η=1nkan-1.
So we obtain a first-order approximate solution for the discussed problem
(5.119)x=a⌊1-exp(-n-1/2kan-1t)⌋.
In order to improve accuracy, we can assume that the solution can be expressed in a more general form
(5.120)x=a(1-∑i=1mbie-ηit),
which should satisfy initial conditions x(0)=0, and this requires
(5.121)1-∑i=1mbi=0.
Substituting (5.120) into (5.113), we set
(5.122)∂J∂ηi=0(i=1~m),∂J∂bi=0(i=2~m).
Solving (5.121)-(5.122) simultaneously, we can easily determine 2m parameters. The solution procedure is similar to that illustrated in [25], and we will not discuss in details to solve space.We can also choose the following trial function:(5.123)x=a{1-1(1+pt)q},
where p and q are unknown constants to be further determined. It is obvious that (5.123) satisfies the conditions x(0)=0 and x(+∞)=a. Submitting (5.123) into (5.113), and setting
(5.124)∂J∂p=0,∂J∂q=0,
we obtain with ease
(5.125)p=(n-1)kan-1,q=1n-1.
Thus, we obtain the solution
(5.126)x=a{1-1(1+(n-1)kan-1t)1/(n-1)},
which is the exact solution.


## 5.3.5. Variational Approach to Solitary Solution

In the review article [7], the variational approach to solitons was outlined by few lines, and now the method has been successfully applied to the search for soliton solutions [26, 27] without requirement of small parameter assumption, leading to an extremely simple and elementary but rigorous derivation of soliton solutions.Considering the KdV equation, we seek its traveling wave solutions in the following frame:(5.127)u(x,t)=U(ξ),ν(x,t)=V(ξ),ξ=x-ct,
where c is angular frequency. Substituting (5.127) into (4.1) yields
(5.128)-cu′-6uu′+u′′′=0,
where prime denotes the differential with respect to ξ.Integrating (5.128) yields the result(5.129a)-cu-3u2+u′′=A,
where A is an integration constant, which can be determined from the initial condition. For solitary solutions or limit cycles, the solutions do not depend upon the initial condition, so we always set A=0, and this results in
(5.129b)-cu-3u2+u′′=0.By the semi-inverse method [17], the following variational formulation is established:
(5.130)J=∫0∞(12cu2+u3+12(dudξ)2)dξ.
Equation (5.130) also hints a Hamiltonian invariant, which reads(5.131a)12cu2+u3-12u′2=B,
where B is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set B=0,
(5.131b)12cu2+u3-12u′2=0.Differentiating (5.131a) with respect to ξ, we have
(5.132)cuu′+3u2u′-u′u′′=0,
which is equivalent to (5.129b).The Hamiltonian invariant (5.131a) and (5.131b) can also be used for construction of an asymptotic solution or used as an auxiliary function in the subequation method, see Section 5.10.The semi-inverse method is a powerful mathematical tool to the search for variational formulae for real-life physical problems.By Ritz method, we search for a solitary wave solution in the form(5.133)u=psech2(qξ),
where p and q are constants to be further determined.Substituting (5.133) into (5.130) results in
(5.134)J=∫0∞[12cp2sech4(qξ)+p3sech6(qξ)+12(4p2q2sech4(qξ)tanh2(qξ)]dξ=cp22q∫0∞sech4(z)dz+p3q∫0∞sech6(z)dz+2p2q∫0∞{sech4(z)tanh4(z)}dz=cp23q+8p315q+4p2q15.
Making J stationary with respect to p and q results in
(5.135)∂J∂p=2cp3q+24p215q+8pq15=0,∂J∂q=-cp23q2-8p315q2+4p215=0,
or simplifying
(5.136)5c+12p+4q2=0,-5c-8p+4q2=0.
From (5.136), we can easily obtain the following relations:
(5.137)p=-12c,q=c4.
So the solitary wave solution can be approximated as
(5.138)u=-c2sech2c4(x-ct-ξ0),
which is the exact solitary wave solution of KdV equation.For the KdV equation expressed in (5.47), we have the following variational principle (see (5.54)):
(5.139)J(u)=∬-∞+∞{-12uxut+12uxx2-12uux2-14u2uxx}dxdt.
We assume that the solitary solution can be expressed in the form
(5.140)u(x,t)=1a1ea5x-a5t+a2e-a7x+a8t+a3,
where a1,a2,…,a8 are unknown constants to be further determined.Substituting (5.140) into (5.139), we have
(5.141)J(u)=J(a1,a2,…,a8).
Making the functional, (5.139), stationary can be approximated achieved by making the function (5.141) stationary, and this requires
(5.142)∂J∂ai=0(i=1~8).
Solving the system, (5.142), we can determine the values for a1,a2,…,a8.


## 5.4. Hamiltonian Approach to Solitary Solution

### 5.4.1. Generalized Action Principles in Mechanics

We begin with the definition of the action functional as time integral over the LagrangianL of a dynamical system
(5.143)J(xi)=∫t1t2Ldt,
and the Lagrangian is defined as follows:
(5.144)L=12mxi′2-V(xi).
Here, xi′=dxi/dt.Newton’s motion equation can be obtained from the stationary condition of the functional (5.143), which reads
(5.145)mxi′′+∂V∂xi=0.We can introduce some constraints to the action functional (5.143), leading to various principles required. For example, if the total energy is a conserved quantity, that is, T+V=const., which is considered as a constraint of the functional (5.143), then we obtain the Euler-Maupertuis principle (principle of least action) [28]:
(5.146)∫t1t2(2T-const.)dt→min,
or
(5.147)∫t1t2Tdt→min.
In this section, we will obtain Hamiltonian and other actions from the Lagrangian (5.144) by introducing some constraints.Now we introduce a generalized velocity(5.148)pi=∂L∂xi′=mxi′.
We consider (5.148) as a constraint of the action functional (5.143), and accordingly the Lagrangian (5.144) can be written as follows:
(5.149)L(xi,pi)=12mpi2-V(xi).
By the Lagrange multiplier, we have the following generalized Lagrangian:
(5.150)L1(xi,pi,λi)=12mpi2-V(xi)+λi(pi-mxi′).
The multiplier can be readily identified, which reads
(5.151)λi=-1mpi.
Substituting the identified multiplier into (5.150) results in
(5.152)L1(xi,pi)=12mpi2-V(xi)-1mpi(pi-mxi′)=pixi′-H,
where H(xi,pi) is a Hamiltonian:
(5.153)H(xi,pi)=12mpi2+V(xi).
If we introduce a new variable ui defined as
(5.154)ui=xi′,
we consider (5.154) as a constraint of the action functional (5.143), and in such case, the Lagrangian (5.144) can be rewritten as
(5.155)L(xi,ui)=12mui2-V(xi).
By the Lagrange multiplier, we have the following generalized Lagrangian:
(5.156)L2(xi,ui,λi)=12mui2-V(xi)+λi(ui-xi′).
The multiplier can be readily identified, which reads
(5.157)λi=-mui.
Substituting the identified multiplier into (5.156) results in
(5.158)L2(xi,ui)=12mui2-V(xi)-mui(ui-xi′)=muixi′-H~(xi,ui),
where H~(xi,ui) is given by
(5.159)H~(xi,pi)=12mui2+V(xi).From (5.157), we know that the multiplier is actually the generalized velocity
(5.160)λi=-mui=-pi.
Substituting (5.160) into (5.156), and keeping pi an independent variable, we have
(5.161)L3(xi,ui,pi)=12mui2-V(xi)-pi(ui-xi′)=pixi′-H-(xi,ui,pi),
where
(5.162)H-(xi,ui,pi)=-12mui2+V(xi)+piui.
Equation (5.161) is called the Schwinger action [28].By the same manipulation, from (5.151), the multiplier can be also determined as
(5.163)λi=-1mpi=-ui.
So we obtain another action like Schwinger’s, which reads
(5.164)L4(xi,pi,ui)=12mpi2-V(xi)-ui(pi-mxi′)=muixi′-H^(xi,ui,pi),
where
(5.165)H^(xi,ui,pi)=-12mpi2+V(xi)+uipi.A more generalized action can be obtained by linear combination ofL1(xi,ui) and L2(xi,pi):
(5.166)L5(xi,ui,pi)=L1(xi,ui)+L2(xi,pi)=-12mpi2-12mui2+(pi+mui)xi′-2V(xi)=(pi+mui)xi′-He(xi,ui,pi),
where
(5.167)He(xi,ui,pi)=12mpi2+12mui2+2V(xi).
The Euler equations can be readily obtained, which read
(5.168)δxi:ddt(pi+mui)+∂He∂xi=ddt(pi+mui)+2∂V∂xi=0,δui:mxi′-∂He∂ui=mxi′-mui=0,δpi:xi′-∂He∂pi=xi′-1mpi=0.
In a more general form, (5.166) can be written as
(5.169)L5(xi,ui,pi)=αL1(xi,ui)+βL2(xi,pi),
where α and β are constants.Linearly combiningLi(i=1,2,3,4,5), we have
(5.170)L6(xi,ui,pi)=∑i=15αiLi,
where αi are constants, and we often set ∑i=15αi=1.


### 5.4.2. Modified Hamilton Principles for Initial Value Problems

The Hamilton principle can be written in the form(5.171)J(ui)=∫0T(12mu˙i2-U)dt,
with the special requirements δu(0)=δu(T)=0.Hereui is the velocity component in xi, and U is a potential defined as ∂U/∂xi=-Fi, where Fi is the body force component in xi.The Hamilton’s principle holds only for the conditions prescribed at the beginning and at the end of the motion and is therefore useless to deal with the usual initial condition problems, both as an analytical tool and as a basis for approximate solution methods. It is impossible for most real-life physical problems to prescribe terminal conditions.In order to eliminate the unnecessary final condition att=T, Carini and Genna [29] obtained the following functional for the case ui(0)=u˙i(0)=0:
(5.172)J(ui)=∫0T(12mu¨i2-F¨iui)dt+F˙i(T)ui(T)-Fi(T)u˙i(T).
We found that the natural final conditions at t=T satisfy the physical requirements.Liu [30] considered that, in order to deal with the final condition, one term should be added to the Hamilton principle to result in the following functional:
(5.173)J(ui)=∫0T(12mu˙i2-U)dt-mu˙~iu˙i|t=T,
where u˙~i is a restricted variable, that is, δu˙~i=0. The modification, (5.172), is not a classic variational principle due to that the variable u˙~i has to be prescribed at t=T, so that
(5.174)δu˙i(T)=0,
and this requirement is even more overrestricted than δui(T)=0 in Hamilton principle, so no vital innovation was made due to its inconsistent at the final condition. However, Liu [30] obtained some functionals that can successfully deal with the initial conditions, but the final condition still keeps an issue of polemics.It would be a landmark in the history of calculus of variations after Hamilton if we can extend the principle to all initial-value problems without prescribing both initial and final conditions. In order to deal with the final condition, we consider the conserved Hamiltonian:(5.175)12m∑i=13u˙i2+U=12m∑i=13u˙i02+U0.
Here U0=U|t=0, so we have the following identity:
(5.176)u˙i=∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m,i≠j,
for all t≥0.In order to convert the initial conditionsui(0)=ui0 and u˙i(0)=u˙i0 into natural initial conditions and make the natural final condition satisfy the physical requirement, (5.176), we assume that its modified Hamilton principle can be written in the form
(5.177)J(ui)=∫0T(12mu˙i2-U)dt+f(ui,u˙i)|t=0t=T,
where f is an unknown function of ui and/or its derivatives.Making the above functional stationary, we obtain the following stationary conditions att=0 and t=T:
(5.178)[mu˙i+∂f∂ui]t=0t=T=0,∂f∂u˙i|t=0t=T=0.
We search such an f so that the above 4 equations satisfy all initial conditions at t=0 and meet exactly the physical requirement, for example, (5.176), at t=T.Accordingly, we can assume that(5.179)(∂f∂ui)t=0=mu˙i0|t=0,(∂f∂u˙i)t=0=k(ui-ui0)|t=0,(∂f∂ui)t=T=-mu˙i|t=T=-m∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m|t=T,i≠j.
From the above relations, we can identify f in the form
(5.180)f={mu˙i0ui+k(12u˙i2-u˙i0u˙i)}t=0+He|t=T,
where k is a nonzero constant; He is defined as
(5.181)He=m23Fi[∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m]3/2.
So we obtain the following modified Hamilton principle:
(5.182)J(ui)=∫0T(12mu˙i2-U)dt+{mu˙i0ui+k(12u˙i2-u˙i0u˙i)}t=0+He|t=T,
which holds for all initial value problems.Proof.
Making the obtained functional (5.182) stationary, we obtain (1)
in the solution domain (0<t<T):
(5.183)-mu¨i+Fi=0,
which is Newton’s motion equation;(2)
natural initial conditions(t=0):
(5.184)-mu˙i|t=0+mu˙i0|t=0=0,{k(u˙i-u˙i0)}t=0=0,
which satisfy obviously the initial conditions ui(0)=ui0 and u˙i(0)=u˙i0, respectively;(3)
natural final condition(t=T):
(5.185)mu˙i|t=T-m∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m|t=T=0,
which meet the physical requirement of (5.176) at t=T.
We have alternative approaches to identifyingf by different assumptions in (5.178), leading to various new modifications. We write here few modified Hamilton principles for reference
(5.186)J(ui)=∫0T(12mu˙i2-U)dt+{mu˙i0ui+k1(12ui2-ui0ui)+k2(12u˙i2-u˙i0u˙i)}t=0+He|t=T,J(ui)=∫0T(12mu˙i2-U)dt+{(m-k4)ui0ui-k3ui0u˙i+k3ui(u˙i-u˙i0)+k4u˙i(ui-ui0)}t=0+He|t=T,
where k’s are nonzero constants, and it requires that k1-m≠0 and k3+k4=0.
To summarize, we can conclude from the above derivation and strict proof that the obtained modified Hamilton principles, which are first deduced in the history, and valid for all initial-value problems, are extremely important in both pure and applied sciences due to complete elimination of the long-existing shortcomings in Hamilton principle. The stationary conditions of the obtained variational principle satisfy the Newton’s motion equation, and all initial conditions, furthermore, the natural final condition(t=T), satisfy automatically the physical requirement, making a vital innovation of Hamilton principle.


### 5.4.3. Hamiltonian Approach to Nonlinear Oscillators [31, 32]

In this paper, we consider the following general oscillator:(5.187)u′′+f(u)=0,
with initial conditions u(0)=A and u′(0)=0.It is easy to establish a variational principle for (5.187), which reads [31]
(5.188)J(u)=∫0T/4{12u'2-F(u)}dt,
where T is the period of the oscillator, ∂F/∂u=f(u).In the functional (5.188), (1/2)u′2 is kinetic energy, and F(u) is potential energy, so the functional (5.188) is the least Lagrangian action, from which we can immediately obtain its Hamiltonian, which reads
(5.189)H=12u′2+F(u)=constant=H0,
or
(5.190)12u′2+F(u)-H0=0.
Equation (5.189) replies that the total energy keeps unchanged during the oscillation.Assume that the solution can be expressed as(5.191)u=Acosωt,
where ω is the frequency.Submitting (5.191) into (5.190) results in a residual
(5.192)R(t)=12A2ω2sin2ωt+F(Acosωt)-H0.
According to the energy balance method proposed in [33], locating at some a special point, that is, ωt=π/4, and setting R(t=π/4ω)=0, we can obtain an approximate frequency-amplitude relationship of the studied nonlinear oscillator. Such treatment is much simple and has been widely used by engineers [34–36]. The accuracy of such location method, however, strongly depends upon the chosen location point. To overcome the shortcoming of the energy balance method, in this paper, we suggest a new approach based on Hamiltonian.From (5.192), we have
(5.193)∂H∂A=0.
Introducing a new function, H-(u), defined as
(5.194)H-(u)=∫0T/4{12u'2+F(u)}dt=14TH,
it is obvious that
(5.195)∂H-∂T=14H.
Equation (5.195) is, then, equivalent to the following one:
(5.196)∂∂A(∂H-∂T)=0,
or
(5.197)∂∂A(∂H-∂(1/ω))=0.
From (5.197), we can obtain approximate frequency-amplitude relationship of a nonlinear oscillator.Consider the Duffing equation(5.198)u′′+u+εu3=0,u(0)=A,u′(0)=0.
Its Hamiltonian can be easily obtained, which reads
(5.199)H=12u′2+12u2+14εu4.
Integrating (5.199) with respect to t from 0 to T/4, we have
(5.200)H-(u)=∫0T/4{12u′2+12u2+14εu4}dt.
Assuming that the solution can be expressed as u=Acosωt and substituting it to (5.200), we obtain
(5.201)H-(u)=∫0T/4{12A2ω2sin2ωt+12A2cos2ωt+14εA4cos4ωt}dt=∫0π/2{12A2ωsin2t+1ω[12A2cos2t+14εA4cos4t]}dt=12A2ω·π4+1ω[12A2·π4+14εA4·34·π4].
Setting
(5.202)∂∂A(∂H-∂(1/ω))=-Aω2·π4+[A·π4+εA3·34·π4]=0,
we obtain the following frequency-amplitude relationship:
(5.203)ω=1+34εA3.
Now we consider another nonlinear oscillator with discontinuity:
(5.204)d2udt2+sgn(u)=0,u(0)=A,u′(0)=0.sgn(u) is +1 and −1 for u>0 and u<0, respectively.Its variational formulation can be written as(5.205)J(u)=∫0T/4(-12u′2+u)dt+∫T/4T/2(-12u′2-u)dt,
and H-(u) can be written in the form
(5.206)H-(u)=∫0T/4(12u′2+u)dt+∫T/4T/2(12u′2-u)dt.
Using u=Acosωt as an approximate solution, we have
(5.207)H-=∫0T/4{12A2ω2sin2ωt+Acosωt}dt+∫T/4T/2{12A2ω2sin2ωt-Acosωt}dt=∫0π/2{12A2ωsin2t+1ωAcost}dt+∫π/2π{12A2ωsin2t-1ωAcost}dt=12A2ω·π4+1ωA+12A2ω·π4+1ωA=A2ω·π4+2ωA.
Setting
(5.208)∂∂A(∂H-∂(1/ω))=-Aω2·π2+2=0,
we obtain
(5.209)ω=2πA=1.128379A,
which is very close to the exact one
(5.210)ωexact(A)=π22A=1.110721A.
The 1.59% accuracy is acceptable considering the simple solution procedure.


### 5.4.4. Hamilton Approach to Solitary Solutions

The Hamiltonian approach to nonlinear oscillators has been now widely used [37–40]; in this section, we will extend the technology to nonlinear wave equations.The Euler-Lagrange equation of (5.130) reads
(5.211)dLdu-ddξ(dLduξ)=0,
where L=(1/2)cu2+u3+(1/2)(du/dξ)2 and uξ=du/dξ.It is obvious that the above Euler-Lagrange equation is the KdV equation.We write the Lagrange function in the formL=T-V, where T is the kinetic energy defined as
(5.212)T=12(dudξ)2,
and V is potential energy defined as
(5.213)V=-12cu2-u3.
The Hamiltonian invariant reads T+V=B, that is,
(5.214)12u′2-12cu2-u3=B,
where B is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set B=0. Equation (5.214) becomes
(5.215)12cu2+u3-12u′2=0.
Differentiating (5.215) with respect to ξ, we have
(5.216)cuu′+3u2u′-u′u′′=0,
which is equivalent to (5.129b) in case u′≠0.If a solitary solution is solved, considering tail property of a soliton, we assume that(5.217)u(ξ)=∑nanexp(qnξ)∑mbmexp(pmξ),
where ai,bi,pi,andqi are constants to be further determined asymptotically. Equation (5.217) should satisfy (2.2).The assumption, (5.217), is also used in the exp-function method (see Section 5.9), where the unknowns are identified exactly, whereas we will determine their values approximated by the weighted residual method.Substituting (5.217) into (5.216) results in the following residual:
(5.218)R(ξ)=12cu2+u3-12u′2.
We can use the method of the least squares to determine the unknown constants involved in (5.217)
(5.219)∫-∞+∞R2(ξ)dξ→min.
We can also use the location method to simply identify the unknowns. To elucidate the solution procedure, we assume its solitary solution has a simple symmetrical form
(5.220)u(ξ)=1a(ekξ+e-kξ)+b,
where a,b, and k are unknown constants.The residual equation is(5.221)R(ξ)=c2[a(ekξ+e-kξ)+b]2+1[a(ekξ+e-kξ)+b]3-[ak(ekξ-e-kξ)]2[a(ekξ+e-kξ)+b]4.
To determine the values of the unknowns in (5.220), we set
(5.222)R(0)=0,R′′(0)=0,R(4)(0)=0.
Note that R′(0)=R′′′(0)≡0.Simplifying (5.222), we obtain
(5.223)2ac+bc+2=0,2ac+bc+3+2ak2=0,32a2c+14abc-b2c+66a+80a2bk2-3b=0.
Solving (5.223) simultaneously, we have
(5.224)a=-12k2,b=-1k2,c=k2.
We, therefore, obtain the following approximate solitary solution:
(5.225)u(x,t)=1-(1/2k2){exp(k(x-k2t)+exp(-k(x-k2t)))}-1/k2=-2k2{expk(x-k2t)+exp(-k(x-k2t))}+2.
Submitting the obtained solution into (4.1), we find it as an exact solution!Consider the following generalized KdV equation:(5.226)ut+6unux+uxxx=0.
This equation admits no variational formulation. If a traveling wave solution is searched for, we can make a transformation ξ=x-ct, and then (5.226) reduces to an ordinary differential equation
(5.227)-cu′+6unu′+u′′′=0.
Integrating the above equation results in
(5.228)-cu+6n+1un+1+u′′=A,
which admits a variational formulation
(5.229)J(u)=∫-∞+∞{-12cu2+6(n+1)(n+2)un+2-Au-12u′2}dη.
We write T=(1/2)u′2 and E=-(1/2)cu2+(6/(n+1)(n+2))un+2-Au, and then T can be interpreted as “kinetic energy” and E “potential energy.” The variational functional, (5.229), can be rewritten in the form of Lagrangian action
(5.230)J(u)=-∫-∞+∞{T-E}dη,
and the Hamiltonian invariant implies the conversation of energy, which requires
(5.231)-12cu2+6(n+1)(n+2)un+2-Au+12u′2=B,
or
(5.232)u′2=cu2-12(n+1)(n+2)un+2+2Au+2B.
For solitary solutions, A=B=0, and (5.232) reduces to
(5.233)u′2=cu2-12(n+1)(n+2)un+2.
If a solitary solution is solved, we assume that
(5.234)u(ξ)=1aekξ+be-kξ+d.
Substituting (5.234) into (5.233) results in the following residual:
(5.235)R(ξ)=u′2-cu2+12(n+1)(n+2)un+2.
Proceeding a similar way as that for (5.222), we can identify a,b, and d in (5.234), and finally, we obtain
(5.236)u(x,t)=-2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]-2.
For a compacton solution, we assume that
(5.237)u(ξ)=acos2ξb+ccos2ξ.
By a similar solution procedure as above, we can determine a, b, and c with ease.


## 5.4.1. Generalized Action Principles in Mechanics

We begin with the definition of the action functional as time integral over the LagrangianL of a dynamical system
(5.143)J(xi)=∫t1t2Ldt,
and the Lagrangian is defined as follows:
(5.144)L=12mxi′2-V(xi).
Here, xi′=dxi/dt.Newton’s motion equation can be obtained from the stationary condition of the functional (5.143), which reads
(5.145)mxi′′+∂V∂xi=0.We can introduce some constraints to the action functional (5.143), leading to various principles required. For example, if the total energy is a conserved quantity, that is, T+V=const., which is considered as a constraint of the functional (5.143), then we obtain the Euler-Maupertuis principle (principle of least action) [28]:
(5.146)∫t1t2(2T-const.)dt→min,
or
(5.147)∫t1t2Tdt→min.
In this section, we will obtain Hamiltonian and other actions from the Lagrangian (5.144) by introducing some constraints.Now we introduce a generalized velocity(5.148)pi=∂L∂xi′=mxi′.
We consider (5.148) as a constraint of the action functional (5.143), and accordingly the Lagrangian (5.144) can be written as follows:
(5.149)L(xi,pi)=12mpi2-V(xi).
By the Lagrange multiplier, we have the following generalized Lagrangian:
(5.150)L1(xi,pi,λi)=12mpi2-V(xi)+λi(pi-mxi′).
The multiplier can be readily identified, which reads
(5.151)λi=-1mpi.
Substituting the identified multiplier into (5.150) results in
(5.152)L1(xi,pi)=12mpi2-V(xi)-1mpi(pi-mxi′)=pixi′-H,
where H(xi,pi) is a Hamiltonian:
(5.153)H(xi,pi)=12mpi2+V(xi).
If we introduce a new variable ui defined as
(5.154)ui=xi′,
we consider (5.154) as a constraint of the action functional (5.143), and in such case, the Lagrangian (5.144) can be rewritten as
(5.155)L(xi,ui)=12mui2-V(xi).
By the Lagrange multiplier, we have the following generalized Lagrangian:
(5.156)L2(xi,ui,λi)=12mui2-V(xi)+λi(ui-xi′).
The multiplier can be readily identified, which reads
(5.157)λi=-mui.
Substituting the identified multiplier into (5.156) results in
(5.158)L2(xi,ui)=12mui2-V(xi)-mui(ui-xi′)=muixi′-H~(xi,ui),
where H~(xi,ui) is given by
(5.159)H~(xi,pi)=12mui2+V(xi).From (5.157), we know that the multiplier is actually the generalized velocity
(5.160)λi=-mui=-pi.
Substituting (5.160) into (5.156), and keeping pi an independent variable, we have
(5.161)L3(xi,ui,pi)=12mui2-V(xi)-pi(ui-xi′)=pixi′-H-(xi,ui,pi),
where
(5.162)H-(xi,ui,pi)=-12mui2+V(xi)+piui.
Equation (5.161) is called the Schwinger action [28].By the same manipulation, from (5.151), the multiplier can be also determined as
(5.163)λi=-1mpi=-ui.
So we obtain another action like Schwinger’s, which reads
(5.164)L4(xi,pi,ui)=12mpi2-V(xi)-ui(pi-mxi′)=muixi′-H^(xi,ui,pi),
where
(5.165)H^(xi,ui,pi)=-12mpi2+V(xi)+uipi.A more generalized action can be obtained by linear combination ofL1(xi,ui) and L2(xi,pi):
(5.166)L5(xi,ui,pi)=L1(xi,ui)+L2(xi,pi)=-12mpi2-12mui2+(pi+mui)xi′-2V(xi)=(pi+mui)xi′-He(xi,ui,pi),
where
(5.167)He(xi,ui,pi)=12mpi2+12mui2+2V(xi).
The Euler equations can be readily obtained, which read
(5.168)δxi:ddt(pi+mui)+∂He∂xi=ddt(pi+mui)+2∂V∂xi=0,δui:mxi′-∂He∂ui=mxi′-mui=0,δpi:xi′-∂He∂pi=xi′-1mpi=0.
In a more general form, (5.166) can be written as
(5.169)L5(xi,ui,pi)=αL1(xi,ui)+βL2(xi,pi),
where α and β are constants.Linearly combiningLi(i=1,2,3,4,5), we have
(5.170)L6(xi,ui,pi)=∑i=15αiLi,
where αi are constants, and we often set ∑i=15αi=1.


## 5.4.2. Modified Hamilton Principles for Initial Value Problems

The Hamilton principle can be written in the form(5.171)J(ui)=∫0T(12mu˙i2-U)dt,
with the special requirements δu(0)=δu(T)=0.Hereui is the velocity component in xi, and U is a potential defined as ∂U/∂xi=-Fi, where Fi is the body force component in xi.The Hamilton’s principle holds only for the conditions prescribed at the beginning and at the end of the motion and is therefore useless to deal with the usual initial condition problems, both as an analytical tool and as a basis for approximate solution methods. It is impossible for most real-life physical problems to prescribe terminal conditions.In order to eliminate the unnecessary final condition att=T, Carini and Genna [29] obtained the following functional for the case ui(0)=u˙i(0)=0:
(5.172)J(ui)=∫0T(12mu¨i2-F¨iui)dt+F˙i(T)ui(T)-Fi(T)u˙i(T).
We found that the natural final conditions at t=T satisfy the physical requirements.Liu [30] considered that, in order to deal with the final condition, one term should be added to the Hamilton principle to result in the following functional:
(5.173)J(ui)=∫0T(12mu˙i2-U)dt-mu˙~iu˙i|t=T,
where u˙~i is a restricted variable, that is, δu˙~i=0. The modification, (5.172), is not a classic variational principle due to that the variable u˙~i has to be prescribed at t=T, so that
(5.174)δu˙i(T)=0,
and this requirement is even more overrestricted than δui(T)=0 in Hamilton principle, so no vital innovation was made due to its inconsistent at the final condition. However, Liu [30] obtained some functionals that can successfully deal with the initial conditions, but the final condition still keeps an issue of polemics.It would be a landmark in the history of calculus of variations after Hamilton if we can extend the principle to all initial-value problems without prescribing both initial and final conditions. In order to deal with the final condition, we consider the conserved Hamiltonian:(5.175)12m∑i=13u˙i2+U=12m∑i=13u˙i02+U0.
Here U0=U|t=0, so we have the following identity:
(5.176)u˙i=∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m,i≠j,
for all t≥0.In order to convert the initial conditionsui(0)=ui0 and u˙i(0)=u˙i0 into natural initial conditions and make the natural final condition satisfy the physical requirement, (5.176), we assume that its modified Hamilton principle can be written in the form
(5.177)J(ui)=∫0T(12mu˙i2-U)dt+f(ui,u˙i)|t=0t=T,
where f is an unknown function of ui and/or its derivatives.Making the above functional stationary, we obtain the following stationary conditions att=0 and t=T:
(5.178)[mu˙i+∂f∂ui]t=0t=T=0,∂f∂u˙i|t=0t=T=0.
We search such an f so that the above 4 equations satisfy all initial conditions at t=0 and meet exactly the physical requirement, for example, (5.176), at t=T.Accordingly, we can assume that(5.179)(∂f∂ui)t=0=mu˙i0|t=0,(∂f∂u˙i)t=0=k(ui-ui0)|t=0,(∂f∂ui)t=T=-mu˙i|t=T=-m∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m|t=T,i≠j.
From the above relations, we can identify f in the form
(5.180)f={mu˙i0ui+k(12u˙i2-u˙i0u˙i)}t=0+He|t=T,
where k is a nonzero constant; He is defined as
(5.181)He=m23Fi[∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m]3/2.
So we obtain the following modified Hamilton principle:
(5.182)J(ui)=∫0T(12mu˙i2-U)dt+{mu˙i0ui+k(12u˙i2-u˙i0u˙i)}t=0+He|t=T,
which holds for all initial value problems.Proof.
Making the obtained functional (5.182) stationary, we obtain (1)
in the solution domain (0<t<T):
(5.183)-mu¨i+Fi=0,
which is Newton’s motion equation;(2)
natural initial conditions(t=0):
(5.184)-mu˙i|t=0+mu˙i0|t=0=0,{k(u˙i-u˙i0)}t=0=0,
which satisfy obviously the initial conditions ui(0)=ui0 and u˙i(0)=u˙i0, respectively;(3)
natural final condition(t=T):
(5.185)mu˙i|t=T-m∑i=13u˙i02-∑j=13u˙j2+2(U0-U)m|t=T=0,
which meet the physical requirement of (5.176) at t=T.
We have alternative approaches to identifyingf by different assumptions in (5.178), leading to various new modifications. We write here few modified Hamilton principles for reference
(5.186)J(ui)=∫0T(12mu˙i2-U)dt+{mu˙i0ui+k1(12ui2-ui0ui)+k2(12u˙i2-u˙i0u˙i)}t=0+He|t=T,J(ui)=∫0T(12mu˙i2-U)dt+{(m-k4)ui0ui-k3ui0u˙i+k3ui(u˙i-u˙i0)+k4u˙i(ui-ui0)}t=0+He|t=T,
where k’s are nonzero constants, and it requires that k1-m≠0 and k3+k4=0.
To summarize, we can conclude from the above derivation and strict proof that the obtained modified Hamilton principles, which are first deduced in the history, and valid for all initial-value problems, are extremely important in both pure and applied sciences due to complete elimination of the long-existing shortcomings in Hamilton principle. The stationary conditions of the obtained variational principle satisfy the Newton’s motion equation, and all initial conditions, furthermore, the natural final condition(t=T), satisfy automatically the physical requirement, making a vital innovation of Hamilton principle.


## 5.4.3. Hamiltonian Approach to Nonlinear Oscillators [31, 32]

In this paper, we consider the following general oscillator:(5.187)u′′+f(u)=0,
with initial conditions u(0)=A and u′(0)=0.It is easy to establish a variational principle for (5.187), which reads [31]
(5.188)J(u)=∫0T/4{12u'2-F(u)}dt,
where T is the period of the oscillator, ∂F/∂u=f(u).In the functional (5.188), (1/2)u′2 is kinetic energy, and F(u) is potential energy, so the functional (5.188) is the least Lagrangian action, from which we can immediately obtain its Hamiltonian, which reads
(5.189)H=12u′2+F(u)=constant=H0,
or
(5.190)12u′2+F(u)-H0=0.
Equation (5.189) replies that the total energy keeps unchanged during the oscillation.Assume that the solution can be expressed as(5.191)u=Acosωt,
where ω is the frequency.Submitting (5.191) into (5.190) results in a residual
(5.192)R(t)=12A2ω2sin2ωt+F(Acosωt)-H0.
According to the energy balance method proposed in [33], locating at some a special point, that is, ωt=π/4, and setting R(t=π/4ω)=0, we can obtain an approximate frequency-amplitude relationship of the studied nonlinear oscillator. Such treatment is much simple and has been widely used by engineers [34–36]. The accuracy of such location method, however, strongly depends upon the chosen location point. To overcome the shortcoming of the energy balance method, in this paper, we suggest a new approach based on Hamiltonian.From (5.192), we have
(5.193)∂H∂A=0.
Introducing a new function, H-(u), defined as
(5.194)H-(u)=∫0T/4{12u'2+F(u)}dt=14TH,
it is obvious that
(5.195)∂H-∂T=14H.
Equation (5.195) is, then, equivalent to the following one:
(5.196)∂∂A(∂H-∂T)=0,
or
(5.197)∂∂A(∂H-∂(1/ω))=0.
From (5.197), we can obtain approximate frequency-amplitude relationship of a nonlinear oscillator.Consider the Duffing equation(5.198)u′′+u+εu3=0,u(0)=A,u′(0)=0.
Its Hamiltonian can be easily obtained, which reads
(5.199)H=12u′2+12u2+14εu4.
Integrating (5.199) with respect to t from 0 to T/4, we have
(5.200)H-(u)=∫0T/4{12u′2+12u2+14εu4}dt.
Assuming that the solution can be expressed as u=Acosωt and substituting it to (5.200), we obtain
(5.201)H-(u)=∫0T/4{12A2ω2sin2ωt+12A2cos2ωt+14εA4cos4ωt}dt=∫0π/2{12A2ωsin2t+1ω[12A2cos2t+14εA4cos4t]}dt=12A2ω·π4+1ω[12A2·π4+14εA4·34·π4].
Setting
(5.202)∂∂A(∂H-∂(1/ω))=-Aω2·π4+[A·π4+εA3·34·π4]=0,
we obtain the following frequency-amplitude relationship:
(5.203)ω=1+34εA3.
Now we consider another nonlinear oscillator with discontinuity:
(5.204)d2udt2+sgn(u)=0,u(0)=A,u′(0)=0.sgn(u) is +1 and −1 for u>0 and u<0, respectively.Its variational formulation can be written as(5.205)J(u)=∫0T/4(-12u′2+u)dt+∫T/4T/2(-12u′2-u)dt,
and H-(u) can be written in the form
(5.206)H-(u)=∫0T/4(12u′2+u)dt+∫T/4T/2(12u′2-u)dt.
Using u=Acosωt as an approximate solution, we have
(5.207)H-=∫0T/4{12A2ω2sin2ωt+Acosωt}dt+∫T/4T/2{12A2ω2sin2ωt-Acosωt}dt=∫0π/2{12A2ωsin2t+1ωAcost}dt+∫π/2π{12A2ωsin2t-1ωAcost}dt=12A2ω·π4+1ωA+12A2ω·π4+1ωA=A2ω·π4+2ωA.
Setting
(5.208)∂∂A(∂H-∂(1/ω))=-Aω2·π2+2=0,
we obtain
(5.209)ω=2πA=1.128379A,
which is very close to the exact one
(5.210)ωexact(A)=π22A=1.110721A.
The 1.59% accuracy is acceptable considering the simple solution procedure.


## 5.4.4. Hamilton Approach to Solitary Solutions

The Hamiltonian approach to nonlinear oscillators has been now widely used [37–40]; in this section, we will extend the technology to nonlinear wave equations.The Euler-Lagrange equation of (5.130) reads
(5.211)dLdu-ddξ(dLduξ)=0,
where L=(1/2)cu2+u3+(1/2)(du/dξ)2 and uξ=du/dξ.It is obvious that the above Euler-Lagrange equation is the KdV equation.We write the Lagrange function in the formL=T-V, where T is the kinetic energy defined as
(5.212)T=12(dudξ)2,
and V is potential energy defined as
(5.213)V=-12cu2-u3.
The Hamiltonian invariant reads T+V=B, that is,
(5.214)12u′2-12cu2-u3=B,
where B is a constant, which can be determined by incorporating the initial condition. For solitary solutions or limit cycles, we set B=0. Equation (5.214) becomes
(5.215)12cu2+u3-12u′2=0.
Differentiating (5.215) with respect to ξ, we have
(5.216)cuu′+3u2u′-u′u′′=0,
which is equivalent to (5.129b) in case u′≠0.If a solitary solution is solved, considering tail property of a soliton, we assume that(5.217)u(ξ)=∑nanexp(qnξ)∑mbmexp(pmξ),
where ai,bi,pi,andqi are constants to be further determined asymptotically. Equation (5.217) should satisfy (2.2).The assumption, (5.217), is also used in the exp-function method (see Section 5.9), where the unknowns are identified exactly, whereas we will determine their values approximated by the weighted residual method.Substituting (5.217) into (5.216) results in the following residual:
(5.218)R(ξ)=12cu2+u3-12u′2.
We can use the method of the least squares to determine the unknown constants involved in (5.217)
(5.219)∫-∞+∞R2(ξ)dξ→min.
We can also use the location method to simply identify the unknowns. To elucidate the solution procedure, we assume its solitary solution has a simple symmetrical form
(5.220)u(ξ)=1a(ekξ+e-kξ)+b,
where a,b, and k are unknown constants.The residual equation is(5.221)R(ξ)=c2[a(ekξ+e-kξ)+b]2+1[a(ekξ+e-kξ)+b]3-[ak(ekξ-e-kξ)]2[a(ekξ+e-kξ)+b]4.
To determine the values of the unknowns in (5.220), we set
(5.222)R(0)=0,R′′(0)=0,R(4)(0)=0.
Note that R′(0)=R′′′(0)≡0.Simplifying (5.222), we obtain
(5.223)2ac+bc+2=0,2ac+bc+3+2ak2=0,32a2c+14abc-b2c+66a+80a2bk2-3b=0.
Solving (5.223) simultaneously, we have
(5.224)a=-12k2,b=-1k2,c=k2.
We, therefore, obtain the following approximate solitary solution:
(5.225)u(x,t)=1-(1/2k2){exp(k(x-k2t)+exp(-k(x-k2t)))}-1/k2=-2k2{expk(x-k2t)+exp(-k(x-k2t))}+2.
Submitting the obtained solution into (4.1), we find it as an exact solution!Consider the following generalized KdV equation:(5.226)ut+6unux+uxxx=0.
This equation admits no variational formulation. If a traveling wave solution is searched for, we can make a transformation ξ=x-ct, and then (5.226) reduces to an ordinary differential equation
(5.227)-cu′+6unu′+u′′′=0.
Integrating the above equation results in
(5.228)-cu+6n+1un+1+u′′=A,
which admits a variational formulation
(5.229)J(u)=∫-∞+∞{-12cu2+6(n+1)(n+2)un+2-Au-12u′2}dη.
We write T=(1/2)u′2 and E=-(1/2)cu2+(6/(n+1)(n+2))un+2-Au, and then T can be interpreted as “kinetic energy” and E “potential energy.” The variational functional, (5.229), can be rewritten in the form of Lagrangian action
(5.230)J(u)=-∫-∞+∞{T-E}dη,
and the Hamiltonian invariant implies the conversation of energy, which requires
(5.231)-12cu2+6(n+1)(n+2)un+2-Au+12u′2=B,
or
(5.232)u′2=cu2-12(n+1)(n+2)un+2+2Au+2B.
For solitary solutions, A=B=0, and (5.232) reduces to
(5.233)u′2=cu2-12(n+1)(n+2)un+2.
If a solitary solution is solved, we assume that
(5.234)u(ξ)=1aekξ+be-kξ+d.
Substituting (5.234) into (5.233) results in the following residual:
(5.235)R(ξ)=u′2-cu2+12(n+1)(n+2)un+2.
Proceeding a similar way as that for (5.222), we can identify a,b, and d in (5.234), and finally, we obtain
(5.236)u(x,t)=-2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]-2.
For a compacton solution, we assume that
(5.237)u(ξ)=acos2ξb+ccos2ξ.
By a similar solution procedure as above, we can determine a, b, and c with ease.


## 5.5. Variational Principles for Fluids

### 5.5.1. A Bernoulli-Like Equation for Shallow Wave Propagation

Strong storms and cyclones, underwater earthquakes, high-speed ferries, and aerial and submarine landslides can cause giant surface waves approaching the coast and frequently cause extensive coastal flooding, destruction of coastal constructions, and loss of lives. A fast but reliable prediction of a tsunami pulse is of critical importance, and a simple equation like Bernoulli equation is, therefore, much needed.To this end, we use a one-dimensional nonlinear shallow wave propagation for describing runup of irregular waves on a beach. The basic equations are [41]
(5.238)∂u∂t+u∂u∂x+g∂η∂x=0,(5.239)∂η∂t+∂∂x[(h+η)u]=0,
where η is water displacement, u is the depth-averaged velocity, h is the unperturbed water depth, and g is the gravity acceleration.We will establish a variational principle for the system of (5.238) and (5.239) to search for a principle of conservation of energy for the present problem.In order to establish a variational formulation for the system of (5.238) and (5.239), we introduce a function Φ defined as
(5.240)∂Φ∂x=u,∂Φ∂t=-(12u2+gη).
Equation (5.240) is equivalent to (5.238).Using the semi-inverse method [17], we construct a trial functional in the form
(5.241)J(Φ,u,η)=∬{η∂Φ∂t+(h+η)u∂Φ∂x+F(u,η)}dtdx,
where F is an unknown function of u and η.It is obvious that the stationary condition of (5.241) with respect to Φ is (5.239). Now making the functional, (5.241), stationary with respect to η and u, we obtain the following Euler-Lagrange equations:
(5.242)∂Φ∂t+u∂Φ∂x+∂F∂η=0,(h+η)∂Φ∂x+∂F∂u=0.
The above equations should satisfy (5.240). To this end, using (5.240), we set
(5.243)∂F∂η=-∂Φ∂t-u∂Φ∂x=12u2+gη-u2=-12u2+gη,∂F∂u=-(h+η)∂Φ∂x=-(h+η)u.
From (5.243), the unknown F can be identified, which reads
(5.244)F=-12u2η+12gη2-12hu2.
We, therefore, obtain the needed variational formulation, which is
(5.245)J(Φ,u,η)=∬{η∂Φ∂t+(h+η)u∂Φ∂x-12(η+h)u2+12gη2}dtdx.
Submitting (5.240) into (5.245), we obtain a constrained variational formulation
(5.246)J(Φ)=∬{12hu2-12gη2}dtdx,
which is under constraints of (5.240).Proof.
Making the functional, (5.246), stationary, we have
(5.247)δJ(Φ)=∬{huδu-gηδη}dtdx.
According to the constraints, (5.240), we have
(5.248)δ(∂Φ∂x)=δu,(5.249)δ(∂Φ∂t)=-δ(12u2+gη)=-(uδu+gδη).
Replacing δu in (5.249) by δ(∂Φ/∂x) results in
(5.250)gηδη=-ηδ(∂Φ∂t)-ηuδ(∂Φ∂x).
Submitting (5.248) and (5.250) into (5.247), we obtain
(5.251)δJ(Φ)=∬{huδ(∂Φ∂x)+ηδ(∂Φ∂t)+ηuδ(∂Φ∂x)}dtdx=∬{(h+η)u∂∂x(δΦ)+η∂∂t(δΦ)}dtdx=-∬{∂((h+η)u)∂x+∂η∂t}δΦdtdx+∬{∂∂x[(h+η)uδΦ]+∂∂t(ηδΦ)}dtdx=-∬{∂((h+η)u)∂x+∂η∂t}δΦdtdx+∫t(h+η)uδΦdt+∫xηδΦdx.
Setting δJ(Φ)=0, and ignoring the boundary terms, for arbitrary δΦ, we can obtain (5.239) as the needed stationary condition.
We can rewrite (5.246) in the form
(5.252)J(Φ)=∬{T-V}dtdx,
where T=(1/2)hu2 is a generalized kinetic energy of water wave, is V=(1/2)gη2 and a generalized gravitational potential.
Equation (5.252) is a principle of the Lagrange action for the shallow wave propagation, and it also implies conservation of energy T+V=H:
(5.253)12hu2+12gη2=H,
where H is a constant.
We call (5.253) a Bernoulli-like equation for shallow wave propagation.Tsunami prediction, for example, after an earthquake, is of critical importance to save life and property. The extent of catastrophe depends mainly on the height of the Tsunami pulses, that is, the value ofη. It is easy to observe the velocity of a wave in the sea, and we, therefore, can make easy but reliable tsunami prediction using the obtained Bernoulli-like equation.


### 5.5.2. Lin’s Variational Principle for Ideal Flows

It is well known that the Hamilton’s principle can be applied to a single fluid particle or a closed system by the involutory transformation. For an isoentropy rotational flow, a variational principle can be established using Lin’s constraints [42]. But the essence of Lin’s constraints is not clear yet, and this paper concludes that the functional with Lin’s constraints is not a genuine variational principle, but an approximate one. In addition, a new generalized variational principle with only 6 independent variables (including only one Lin’s constraint) for three-dimensional unsteady compressible rotational flow is established.In 1950s, great progress had been made on the research of variational principle in solid mechanics, with Hu-Washizu variational principle [43] as milestone, and in fluid mechanics, with Lin’s constraints [42] as milestone.Hamilton’s principle was so successfully and powerfully applied to particle mechanics that many attempts have been made to obtain the momentum equations from a variational principle patterned after Hamilton’s principle. These attempts have not all been successful except in the case of isoentropy irrotational flow. For a more general one, Lin’s constraints [44] must be added.Considering a 3D unsteady inviscid compressible rotational flow, we have the following equations [45].(1)
Momentum equation:(5.254)∂ui∂t+ui∂ui∂xi=-1ρ∂P∂xi,
where ui is the flow velocity in xi indirect, P is the pressure, and ρ is the density.(2)
Equation of state:(5.255)P=ρκexp{S-S0m},
where κ is a specific heat ratio, m=(κ-1)-1, and S is entropy.(3)
Continuity equation:(5.256)∂ρ∂t+∂(ρuj)∂xj=0.(4)
Isoentropic equation:(5.257)∂S∂t+uj∂S∂xj=0.(5)
Steady Bernoulli’s equation:(5.258)12ν2+Π=B.For unsteady flow, we should use the following momentum equation:(5.259)∂ν→∂t+∇(12ν2)-ν→×(∇×ν→)+∇Π=0,
where Π=∫(dP/ρ) and ν→=u1i→+u2j→+u3k→.From the equation of state, we can deduce the following equation:(5.260)Π=κκ-1ρκ-1exp{S-S0m}.
In 1955, Herivel first applied Hamilton’s principle to fluid mechanics and deduced the following functional [46]:
(5.261)J(xi)=∫t∫CV{12ρui2-ρE(ρ,S)}dVdt,
where E is specific internal energy. CV is the control volume in Eulerian open space. Hamiltonian fluid mechanics becomes a branch of fluid mechanics and applied mathematics as well [47–56].Applying the Lagrange multipliers to remove the constraint equations (5.257) and (5.256) yields
(5.262)J=∫t∫CV{12ρui2-ρE(ρ,S)-ρη[∂S∂t+uj∂S∂xj]+Φ[∂ρ∂t+∂(ρuj)∂xj]}dVdt.
Making the above functional stationary with respect to ui, we have
(5.263)ν→=∇Φ+η∇S.
However, according to the Crocco equation [45],
(5.264)(∇×ν→)×ν→=T∇S-∇i∙,
where i∙ is enthalpy, T is temperature, and the homentropic flow may be a rotational one. Equation (5.263) implies, however, that homentropic flow (∇S=0) must be a potential flow or irrotational flow (ν→=∇Φ), confiliting with (5.264), and therefore, the functional, equation (5.261) or (5.262), is not reasonable, which was just pointed out by Lin [44].According to Clebsch [57], an arbitrary velocity vector can be expressed as follows:
(5.265)ν→=∇Φ+λi+∇τi,
whereΦ, λi, and τi are Clebsch variables.Lin introduced three-additional constraint equations (Lin’s constraint equations) to the functional (5.262) [44]:
(5.266)DαiDt=0(i=1,2,3),
where αi is Lagrange variables in Lagrange space.Introducing the Lagrange multipliers to remove Lin’s constraints, we have the following general functional:(5.267)J=∫t∫V{12ρui2-ρE(ρ,S)-ρη[∂S∂t+uj∂S∂xj]+Φ[∂ρ∂t+∂(ρuj)∂xj]-ρβi[∂αi∂t+uj∂αi∂xj]}dVdt.
Making the above functional (5.267) stationary with respect to ui, we have
(5.268)ν→=∇Φ+η∇S+βi∇αi.
This expression says that a homentropic flow (∇S=0) can also be an rotational flow, agreeing with (5.264). So the functional (5.267) with Lin’s constraints is acceptable for practical applications. But the essence of the Lin’s constraints has been bewildered for more than half century.Now considering a fluid particle, and applying Hamilton’s principle to construct the following functional:(5.269)J(xi)=∫t(12x˙i2-Π)dt,
where dot means the partial derivation to time alone pathline, that is, material derivation, its Eulerian equation can be easily deduced
(5.270)x¨i=-∂Π∂xi=-1ρ∂P∂xi
that is, the momentum equation for a perfect flow, and it is easly to prove [46] that Π=E+P/ρ, so Π represents enthalpy.The involutory transformation is to convert Lagrange space into Euler space:(5.271)x˙i=ui(i=1,2,3),
which are the pathline equations in the Eulerian form and are equivalent to (5.266) in Lagrangian form, that is, Lin’s constraint equations. So it is very clear that Lin’s constraints are actually the variational constraints introduced by the space exchange. The Hamilton principle is valid for Lagrange space, while all equations in fluid mechanics are derived through Euler space, and Lin’s constraints arise in the space change.Using the Lagrange multipliers to remove the constraints, (5.271), we have
(5.272)J(xi,ui,λi)=∫t(12ν2-Π+λi(x˙i-ui))dt.
Identifying the Lagrange multipliers
(5.273)λi=ui
results in the following variational principle:
(5.274)J(xi,ui)=∫t(12ν2-uix˙i+Π)dt.
Applying Hamilton’s principle to a closed system, we have
(5.275)J(xi)=∫t∫V(t)ρ0(12x˙i2-E)dV(t)dt,
where V(t) is the volume of the closed system in Lagrange space and keeps unchanged, and ∫V(t)ρ0dV is the total mass in the closed system.Making the involutory transformation, we obtain the following functional via Lagrange multiplier method(5.276)J(xi,ui)=∫t∫V(t)ρ0(uix˙i-12ui2-E)dV(t)dt.
It is easy to see that the functional (5.275) or (5.276) ceases to be a stationary variational principle if we transform the closed system in Lagrange space to an open system is Euler space ∫t∫V(t)ρ0dV(t)dt→∫t∫CVρdVdt where a new variable ρ in introduced.In order to guarantee functional (5.276) to be still a stationary principle during space transformation, according to the semi-inverse method [14], let ρ be an independent variable and include an unknown function F in the functional:
(5.277)J(ui,ρ)=∫t∫CV{ρ(12ui2-E)+F(ui,ρ)}dVdt,
where F is an unknown functional of xi,ρ, and CV is the control volume. In the next section the identification of F will be given in details.Rotational equation with Clebsch variables can be expressed as follows:(5.278)ν→=∇Φ+α∇β.
Assuming that ∂α/∂t=0 and Dα/Dt=0, we can deduce the following energy equation:
(5.279)∂Φ∂t+α∂β∂t+12ν2+κκ-1Pρ=R(t).
According to the semi-inverse method [17], we construct a trial functional in the form
(5.280)J(ν→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12ν2-Π)+F}dVdt,
where F is an unknown function of ν→,ρ,Π,Φ,α,  and  β.Making the functional (5.280) stationary with respect to ν→, we have
(5.281)ρν→+∂F∂ν→=0.
Assuming that (5.281) satisfies (5.278), we set
(5.282)∂F∂ν→=-ρν→=-ρ(∇Φ+α∇β).
The unknown function can be identified as follows:
(5.283)F=-ρν→·(∇Φ+α∇β)+f,
where f is an unknown function of ρ,Π,Φ,α,  and  β. Such and trial functional can be updated as follows:
(5.284)J(ν→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12ν2-ν→·(∇Φ+α∇β)-Π)+f}dVdt.
Now the stationary condition with respect to δΦ and continuity equation yield
(5.285)∂f∂Φ=-∇·(ρν→)=∂ρ∂t.
From (5.285), f can be determined in the form
(5.286)f=Φ∂ρ∂t+h,
where h is an unknown function of ρ,Π,α,  and  β. Substituting (5.286) to the trial functional and then taking by part to yield the following new trial functional:
(5.287)J(v→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12v2-v→·(∇Φ+α∇β)-Π-∂Φ∂t)+h}dVdt.
Doing the same as before,
(5.288)δρ:∂h∂ρ=-12v2+v→·(∇Φ+α∇β)+Π+∂Φ∂t=12v2+Π+∂Φ∂t=R-α∂β∂t.
We identifying the unknown function h to yield
(5.289)h=ρ(R-α∂β∂t)+g,
where g is an unknown function of Π,α,  and  β, doing the same way as before,
(5.290)δΠ:∂g∂Π=ρ=(exp{-S-S0m}κ-1κΠ)m.
Identifying the unknown function g to get
(5.291)g=(κ-1κexp(-S-S0m))mκ-1κΠκ/(κ-1),
we get the following generalized variational principle:
(5.292)J(v→,ρ,Π,Φ,α,β)=∫t∫CVLdVdt,
where
(5.293)L=ρ(12ν2-ν→·(∇Φ+α∇β)-Π-∂Φ∂t-α∂β∂t+R)+(κ-1κexp(-S-S0m))mκ-1κΠκ/(κ-1).
The deduced generalized variational principle (GVP) with only 6 independent variables, (5.292), is unknown to the present time and more general and concise than any other known generalized variationals such as the correspondent GVP with Lin’s constraints which needs 11 independent variables (v→,P,ρ and extra three Lin’s constraints together with three Lagrange multipliers, and one Lagrange multiplier for continuity equation, one Lagrange multiplier for isoentropy equation) and actually is of little utility solving the 3D unsteady rotational flow, contrarily the obtained GVP of (5.292) is so beautiful and concise that the theorem makes it possible to deal with the problem via VP-based FEM; and have wide applicability of the solution (DS/Dt≠0). Making the above functional (5.292) stationary, we can get all the Eulerian equations and following additional two equations:
(5.294)DαDt=0,DβDt=1ρ(κ-1κ)m1κΠκ/(κ-1)exp{-(S-S0)}∂S∂α,=1κ-1ρκ-1exp{-S-S0m}∂S∂α=1κPρ∂S∂α.
The first equation of (5.294) is actually Lin’s constraint equation. So via semi-inverse method, the Lin’s constraints equations can reduce to only one, which is the best way to deal with Lin’s constraints in the history.Hereby we have successfully explained the phenomenon of Lin’s constraints via involutory transformation and deduced a generalized variational principle with only 6 independent variables via semi-inverse method, which makes it possible to use FEM to calculate the 3D unsteady compressible rotational flow. The corresponding generalized variational principle with Lin’s constraints is actually an approximate one (for Herrivel’s principle actually is a wrong principle), which, however, can find some application, especially in 2D problems, due to the fact that we can deduce all the Eulerian equations from the functional when making it stationary.


## 5.5.1. A Bernoulli-Like Equation for Shallow Wave Propagation

Strong storms and cyclones, underwater earthquakes, high-speed ferries, and aerial and submarine landslides can cause giant surface waves approaching the coast and frequently cause extensive coastal flooding, destruction of coastal constructions, and loss of lives. A fast but reliable prediction of a tsunami pulse is of critical importance, and a simple equation like Bernoulli equation is, therefore, much needed.To this end, we use a one-dimensional nonlinear shallow wave propagation for describing runup of irregular waves on a beach. The basic equations are [41]
(5.238)∂u∂t+u∂u∂x+g∂η∂x=0,(5.239)∂η∂t+∂∂x[(h+η)u]=0,
where η is water displacement, u is the depth-averaged velocity, h is the unperturbed water depth, and g is the gravity acceleration.We will establish a variational principle for the system of (5.238) and (5.239) to search for a principle of conservation of energy for the present problem.In order to establish a variational formulation for the system of (5.238) and (5.239), we introduce a function Φ defined as
(5.240)∂Φ∂x=u,∂Φ∂t=-(12u2+gη).
Equation (5.240) is equivalent to (5.238).Using the semi-inverse method [17], we construct a trial functional in the form
(5.241)J(Φ,u,η)=∬{η∂Φ∂t+(h+η)u∂Φ∂x+F(u,η)}dtdx,
where F is an unknown function of u and η.It is obvious that the stationary condition of (5.241) with respect to Φ is (5.239). Now making the functional, (5.241), stationary with respect to η and u, we obtain the following Euler-Lagrange equations:
(5.242)∂Φ∂t+u∂Φ∂x+∂F∂η=0,(h+η)∂Φ∂x+∂F∂u=0.
The above equations should satisfy (5.240). To this end, using (5.240), we set
(5.243)∂F∂η=-∂Φ∂t-u∂Φ∂x=12u2+gη-u2=-12u2+gη,∂F∂u=-(h+η)∂Φ∂x=-(h+η)u.
From (5.243), the unknown F can be identified, which reads
(5.244)F=-12u2η+12gη2-12hu2.
We, therefore, obtain the needed variational formulation, which is
(5.245)J(Φ,u,η)=∬{η∂Φ∂t+(h+η)u∂Φ∂x-12(η+h)u2+12gη2}dtdx.
Submitting (5.240) into (5.245), we obtain a constrained variational formulation
(5.246)J(Φ)=∬{12hu2-12gη2}dtdx,
which is under constraints of (5.240).Proof.
Making the functional, (5.246), stationary, we have
(5.247)δJ(Φ)=∬{huδu-gηδη}dtdx.
According to the constraints, (5.240), we have
(5.248)δ(∂Φ∂x)=δu,(5.249)δ(∂Φ∂t)=-δ(12u2+gη)=-(uδu+gδη).
Replacing δu in (5.249) by δ(∂Φ/∂x) results in
(5.250)gηδη=-ηδ(∂Φ∂t)-ηuδ(∂Φ∂x).
Submitting (5.248) and (5.250) into (5.247), we obtain
(5.251)δJ(Φ)=∬{huδ(∂Φ∂x)+ηδ(∂Φ∂t)+ηuδ(∂Φ∂x)}dtdx=∬{(h+η)u∂∂x(δΦ)+η∂∂t(δΦ)}dtdx=-∬{∂((h+η)u)∂x+∂η∂t}δΦdtdx+∬{∂∂x[(h+η)uδΦ]+∂∂t(ηδΦ)}dtdx=-∬{∂((h+η)u)∂x+∂η∂t}δΦdtdx+∫t(h+η)uδΦdt+∫xηδΦdx.
Setting δJ(Φ)=0, and ignoring the boundary terms, for arbitrary δΦ, we can obtain (5.239) as the needed stationary condition.
We can rewrite (5.246) in the form
(5.252)J(Φ)=∬{T-V}dtdx,
where T=(1/2)hu2 is a generalized kinetic energy of water wave, is V=(1/2)gη2 and a generalized gravitational potential.
Equation (5.252) is a principle of the Lagrange action for the shallow wave propagation, and it also implies conservation of energy T+V=H:
(5.253)12hu2+12gη2=H,
where H is a constant.
We call (5.253) a Bernoulli-like equation for shallow wave propagation.Tsunami prediction, for example, after an earthquake, is of critical importance to save life and property. The extent of catastrophe depends mainly on the height of the Tsunami pulses, that is, the value ofη. It is easy to observe the velocity of a wave in the sea, and we, therefore, can make easy but reliable tsunami prediction using the obtained Bernoulli-like equation.


## 5.5.2. Lin’s Variational Principle for Ideal Flows

It is well known that the Hamilton’s principle can be applied to a single fluid particle or a closed system by the involutory transformation. For an isoentropy rotational flow, a variational principle can be established using Lin’s constraints [42]. But the essence of Lin’s constraints is not clear yet, and this paper concludes that the functional with Lin’s constraints is not a genuine variational principle, but an approximate one. In addition, a new generalized variational principle with only 6 independent variables (including only one Lin’s constraint) for three-dimensional unsteady compressible rotational flow is established.In 1950s, great progress had been made on the research of variational principle in solid mechanics, with Hu-Washizu variational principle [43] as milestone, and in fluid mechanics, with Lin’s constraints [42] as milestone.Hamilton’s principle was so successfully and powerfully applied to particle mechanics that many attempts have been made to obtain the momentum equations from a variational principle patterned after Hamilton’s principle. These attempts have not all been successful except in the case of isoentropy irrotational flow. For a more general one, Lin’s constraints [44] must be added.Considering a 3D unsteady inviscid compressible rotational flow, we have the following equations [45].(1)
Momentum equation:(5.254)∂ui∂t+ui∂ui∂xi=-1ρ∂P∂xi,
where ui is the flow velocity in xi indirect, P is the pressure, and ρ is the density.(2)
Equation of state:(5.255)P=ρκexp{S-S0m},
where κ is a specific heat ratio, m=(κ-1)-1, and S is entropy.(3)
Continuity equation:(5.256)∂ρ∂t+∂(ρuj)∂xj=0.(4)
Isoentropic equation:(5.257)∂S∂t+uj∂S∂xj=0.(5)
Steady Bernoulli’s equation:(5.258)12ν2+Π=B.For unsteady flow, we should use the following momentum equation:(5.259)∂ν→∂t+∇(12ν2)-ν→×(∇×ν→)+∇Π=0,
where Π=∫(dP/ρ) and ν→=u1i→+u2j→+u3k→.From the equation of state, we can deduce the following equation:(5.260)Π=κκ-1ρκ-1exp{S-S0m}.
In 1955, Herivel first applied Hamilton’s principle to fluid mechanics and deduced the following functional [46]:
(5.261)J(xi)=∫t∫CV{12ρui2-ρE(ρ,S)}dVdt,
where E is specific internal energy. CV is the control volume in Eulerian open space. Hamiltonian fluid mechanics becomes a branch of fluid mechanics and applied mathematics as well [47–56].Applying the Lagrange multipliers to remove the constraint equations (5.257) and (5.256) yields
(5.262)J=∫t∫CV{12ρui2-ρE(ρ,S)-ρη[∂S∂t+uj∂S∂xj]+Φ[∂ρ∂t+∂(ρuj)∂xj]}dVdt.
Making the above functional stationary with respect to ui, we have
(5.263)ν→=∇Φ+η∇S.
However, according to the Crocco equation [45],
(5.264)(∇×ν→)×ν→=T∇S-∇i∙,
where i∙ is enthalpy, T is temperature, and the homentropic flow may be a rotational one. Equation (5.263) implies, however, that homentropic flow (∇S=0) must be a potential flow or irrotational flow (ν→=∇Φ), confiliting with (5.264), and therefore, the functional, equation (5.261) or (5.262), is not reasonable, which was just pointed out by Lin [44].According to Clebsch [57], an arbitrary velocity vector can be expressed as follows:
(5.265)ν→=∇Φ+λi+∇τi,
whereΦ, λi, and τi are Clebsch variables.Lin introduced three-additional constraint equations (Lin’s constraint equations) to the functional (5.262) [44]:
(5.266)DαiDt=0(i=1,2,3),
where αi is Lagrange variables in Lagrange space.Introducing the Lagrange multipliers to remove Lin’s constraints, we have the following general functional:(5.267)J=∫t∫V{12ρui2-ρE(ρ,S)-ρη[∂S∂t+uj∂S∂xj]+Φ[∂ρ∂t+∂(ρuj)∂xj]-ρβi[∂αi∂t+uj∂αi∂xj]}dVdt.
Making the above functional (5.267) stationary with respect to ui, we have
(5.268)ν→=∇Φ+η∇S+βi∇αi.
This expression says that a homentropic flow (∇S=0) can also be an rotational flow, agreeing with (5.264). So the functional (5.267) with Lin’s constraints is acceptable for practical applications. But the essence of the Lin’s constraints has been bewildered for more than half century.Now considering a fluid particle, and applying Hamilton’s principle to construct the following functional:(5.269)J(xi)=∫t(12x˙i2-Π)dt,
where dot means the partial derivation to time alone pathline, that is, material derivation, its Eulerian equation can be easily deduced
(5.270)x¨i=-∂Π∂xi=-1ρ∂P∂xi
that is, the momentum equation for a perfect flow, and it is easly to prove [46] that Π=E+P/ρ, so Π represents enthalpy.The involutory transformation is to convert Lagrange space into Euler space:(5.271)x˙i=ui(i=1,2,3),
which are the pathline equations in the Eulerian form and are equivalent to (5.266) in Lagrangian form, that is, Lin’s constraint equations. So it is very clear that Lin’s constraints are actually the variational constraints introduced by the space exchange. The Hamilton principle is valid for Lagrange space, while all equations in fluid mechanics are derived through Euler space, and Lin’s constraints arise in the space change.Using the Lagrange multipliers to remove the constraints, (5.271), we have
(5.272)J(xi,ui,λi)=∫t(12ν2-Π+λi(x˙i-ui))dt.
Identifying the Lagrange multipliers
(5.273)λi=ui
results in the following variational principle:
(5.274)J(xi,ui)=∫t(12ν2-uix˙i+Π)dt.
Applying Hamilton’s principle to a closed system, we have
(5.275)J(xi)=∫t∫V(t)ρ0(12x˙i2-E)dV(t)dt,
where V(t) is the volume of the closed system in Lagrange space and keeps unchanged, and ∫V(t)ρ0dV is the total mass in the closed system.Making the involutory transformation, we obtain the following functional via Lagrange multiplier method(5.276)J(xi,ui)=∫t∫V(t)ρ0(uix˙i-12ui2-E)dV(t)dt.
It is easy to see that the functional (5.275) or (5.276) ceases to be a stationary variational principle if we transform the closed system in Lagrange space to an open system is Euler space ∫t∫V(t)ρ0dV(t)dt→∫t∫CVρdVdt where a new variable ρ in introduced.In order to guarantee functional (5.276) to be still a stationary principle during space transformation, according to the semi-inverse method [14], let ρ be an independent variable and include an unknown function F in the functional:
(5.277)J(ui,ρ)=∫t∫CV{ρ(12ui2-E)+F(ui,ρ)}dVdt,
where F is an unknown functional of xi,ρ, and CV is the control volume. In the next section the identification of F will be given in details.Rotational equation with Clebsch variables can be expressed as follows:(5.278)ν→=∇Φ+α∇β.
Assuming that ∂α/∂t=0 and Dα/Dt=0, we can deduce the following energy equation:
(5.279)∂Φ∂t+α∂β∂t+12ν2+κκ-1Pρ=R(t).
According to the semi-inverse method [17], we construct a trial functional in the form
(5.280)J(ν→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12ν2-Π)+F}dVdt,
where F is an unknown function of ν→,ρ,Π,Φ,α,  and  β.Making the functional (5.280) stationary with respect to ν→, we have
(5.281)ρν→+∂F∂ν→=0.
Assuming that (5.281) satisfies (5.278), we set
(5.282)∂F∂ν→=-ρν→=-ρ(∇Φ+α∇β).
The unknown function can be identified as follows:
(5.283)F=-ρν→·(∇Φ+α∇β)+f,
where f is an unknown function of ρ,Π,Φ,α,  and  β. Such and trial functional can be updated as follows:
(5.284)J(ν→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12ν2-ν→·(∇Φ+α∇β)-Π)+f}dVdt.
Now the stationary condition with respect to δΦ and continuity equation yield
(5.285)∂f∂Φ=-∇·(ρν→)=∂ρ∂t.
From (5.285), f can be determined in the form
(5.286)f=Φ∂ρ∂t+h,
where h is an unknown function of ρ,Π,α,  and  β. Substituting (5.286) to the trial functional and then taking by part to yield the following new trial functional:
(5.287)J(v→,ρ,Π,Φ,α,β)=∫t∫CV{ρ(12v2-v→·(∇Φ+α∇β)-Π-∂Φ∂t)+h}dVdt.
Doing the same as before,
(5.288)δρ:∂h∂ρ=-12v2+v→·(∇Φ+α∇β)+Π+∂Φ∂t=12v2+Π+∂Φ∂t=R-α∂β∂t.
We identifying the unknown function h to yield
(5.289)h=ρ(R-α∂β∂t)+g,
where g is an unknown function of Π,α,  and  β, doing the same way as before,
(5.290)δΠ:∂g∂Π=ρ=(exp{-S-S0m}κ-1κΠ)m.
Identifying the unknown function g to get
(5.291)g=(κ-1κexp(-S-S0m))mκ-1κΠκ/(κ-1),
we get the following generalized variational principle:
(5.292)J(v→,ρ,Π,Φ,α,β)=∫t∫CVLdVdt,
where
(5.293)L=ρ(12ν2-ν→·(∇Φ+α∇β)-Π-∂Φ∂t-α∂β∂t+R)+(κ-1κexp(-S-S0m))mκ-1κΠκ/(κ-1).
The deduced generalized variational principle (GVP) with only 6 independent variables, (5.292), is unknown to the present time and more general and concise than any other known generalized variationals such as the correspondent GVP with Lin’s constraints which needs 11 independent variables (v→,P,ρ and extra three Lin’s constraints together with three Lagrange multipliers, and one Lagrange multiplier for continuity equation, one Lagrange multiplier for isoentropy equation) and actually is of little utility solving the 3D unsteady rotational flow, contrarily the obtained GVP of (5.292) is so beautiful and concise that the theorem makes it possible to deal with the problem via VP-based FEM; and have wide applicability of the solution (DS/Dt≠0). Making the above functional (5.292) stationary, we can get all the Eulerian equations and following additional two equations:
(5.294)DαDt=0,DβDt=1ρ(κ-1κ)m1κΠκ/(κ-1)exp{-(S-S0)}∂S∂α,=1κ-1ρκ-1exp{-S-S0m}∂S∂α=1κPρ∂S∂α.
The first equation of (5.294) is actually Lin’s constraint equation. So via semi-inverse method, the Lin’s constraints equations can reduce to only one, which is the best way to deal with Lin’s constraints in the history.Hereby we have successfully explained the phenomenon of Lin’s constraints via involutory transformation and deduced a generalized variational principle with only 6 independent variables via semi-inverse method, which makes it possible to use FEM to calculate the 3D unsteady compressible rotational flow. The corresponding generalized variational principle with Lin’s constraints is actually an approximate one (for Herrivel’s principle actually is a wrong principle), which, however, can find some application, especially in 2D problems, due to the fact that we can deduce all the Eulerian equations from the functional when making it stationary.


## 5.6. Variational Iteration Method

The variational iteration method [58–61] is an effective method for searching for various wave solutions including periodic solutions, solitons, and compacton solutions without linearization or weak nonlinearity assumptions, see, for example, [62–64].


### 5.6.1. The Lagrange Multiplier

The variational iteration method has been shown to solve a large class of nonlinear problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions.To illustrate the basic idea of the technique, we consider the following general nonlinear system:(5.295)L[u(t)]+N[u(t)]=0,
where L is a linear operator, and N is a nonlinear operator.The basic concept of the method is to construct a correction functional for the system (5.295), which reads
(5.296)un+1(t)=un(t)+∫t0tλ{Lun(s)+Nu~n(s)}ds,
where λ is a general Lagrange multiplier that can be identified optimally via variational theory, un is the nth approximate solution, and u~n denotes a restricted variation, that is, δu~n=0. To illustrate how restricted variation works in the variational iteration method, we consider a simple algebraic equation
(5.297)x2-3x+2=0.
We rewrite (5.297) in the form
(5.298)x~·x-3x+2=0,
where x~ is called a restricted variable whose value is assumed to be known (5.6.1. the initial guess). Solving x from (5.298) leads to the result
(5.299)x=23-x~,
or in iteration form,
(5.300)xn+1=23-xn.
After identifying the multiplier in (5.296), we have the following iteration algorithm:
(5.301)un+1(t)=un(t)+∫t0tλ{Lun(s)+Nun(s)}ds.
Equation (5.301) is called the variational iteration algorithm-I.Consider the following nonlinear equation ofkth order:
(5.302)u(k)+f(u,u′,u′′,…,u(k))=0.
The variational iteration formulation is constructed as follows:
(5.303)un+1(t)=un(t)+∫t0tλ(un(k)+f~n)ds,
where δf~n=0, fn=f(un,un′,un′′,…). After identifying the multiplier, we have
(5.304)un+1(t)=un(t)+(-1)n∫t0t1(n-1)!(s-t)n-1[un(k)(s)+fn]ds.
This formulation is generally called the variational iteration algorithm-I [11, 61].The main merit of this iteration formula lies in the fact thatu0(t), the initial solution, can be freely chosen, with even unknown parameters contained. However, some repeated and unnecessary iterations are involved in this iteration algorithm at each step.For initial value problems, we can begin with(5.305)u0(t)=u(0)+tu′(0)+12!t2u′′(0)+⋯1k!tku(k)(0).
This leads to a series solution converging to the exact solution.For boundary value problems, the initial guess can be expressed in the form(5.306)u0(t)=a1g1(t)+a2g2(t)+⋯+akgk(t),
where gk(t) are known functions, and ak are unknowns to be further determined after a few iterations by the boundary conditions.After identifying the Lagrange multiplierλ in (5.296), we can construct the iteration formula
(5.307)un+1(t)=u0(t)+(-1)n∫t0t1(n-1)!(s-t)n-1fnds,
instead of the iteration algorithm (5.301). We call (5.307) the variational iteration algorithm-II [11, 61].Note:u0 must satisfy the initial/boundary conditions. This is the main shortcoming of the algorithm.From (5.307), we obtain the following variational iteration algorithm-III
(5.308)un+2(t)=un+1(t)+∫t0tλ{fn+1(s)-fn(s)}ds.
One common property of both the variational iteration algorithm-I and the variational iteration algorithm-III is the allowed dependence of the initial guess on unknown parameters whose values could be identified after a few iterations by using the initial/boundary conditions. the variational iteration algorithm-III, in particular, is highly suitable for boundary value problems of high orders.


### 5.6.2. Laplace Transform for Identification of the Lagrange Multiplier [65]

We can also apply Laplace transform to identify the Lagrange multiplier. By Laplace transform, we have(5.309)skU(s)+ℓ{f(u,u′,…,u(k))}=0,
or
(5.310)U(s)=-ℓ{f(u,u′,…,u(k))}sk.
The inverse Laplace transform reads
(5.311)u(t)=(-1)k∫0t(η-t)k-1(k-1)!f(un(η),un′(η),…,un(k)(η))dη.
Hence, the following iteration algorithm is derived:
(5.312)un+1(t)=u0(t)+(-1)k∫0t(η-t)k-1(k-1)!f(un(η),un′(η),…,un(k)(η))dη.


### 5.6.3.  Variational Iteration Method for Solitary Solutions

Here is an incomplete list for variational iteration formulas for various differential equations [8, 9]:
(5.313)u′+f(u,u′)=0,un+1(t)=u0(t)-∫0tf(un,un′)ds,u′+αu+f(u,u′)=0,un+1(t)=u0(t)-∫0teα(s-t)f(un,un′)ds,u′′+f(u,u′,u′′)=0,un+1(t)=u0(t)+∫0t(s-t)f(un,un′,un′′)ds,u′′+ω2u+f(u,u′,u′′)=0,un+1(t)=u0(t)+1ω∫0tsinω(s-t)f(un,un′,un′′)ds,u′′-α2u+f(u,u′,u′′)=0,un+1(t)=u0(t)+∫0t12α(eα(s-t)-eα(t-s))f(un,un′,un′′)ds,u′′′+f(u,u′,u′′,u′′′)=0,un+1(t)=u0(t)-∫0t12(s-t)2f(un,un′,un′′,un′′′)ds,u(4)+f(u,u′,u′′,u′′′,u(4))=0,un+1(t)=u0(t)+∫0t16(s-t)3f(un,un′,un′′,un′′′,un(4))ds,u(k)+f(u,u′,u′′,…,u(k))=0,un+1(t)=u0(t)+(-1)n∫0t1(n-1)!(s-t)n-1f(un,un′,un′′,…,un(k))ds.
The main feature of the method is that the initial solution can be chosen with some unknown parameters in the form of the searched solution. For example, for solitons, we begin with
(5.314)u0=psech2(qξ),ξ=x+ct,
where p and q are the unknown parameters to be further identified after one or few iterations. For a more general form for solitary solutions, we assume that
(5.315)u0(ξ)=∑i=-nmcieiξ∑i=-pqbieiξ,ξ=x+ct,
where ci and bi are constants to be further determined.For discontinuous solitons, we can assume, for example, the following form:(5.316)u(ξ)=pexp(-q|ξ|),
or
(5.317)u0(ξ)=∑i=-nmciei|ξ|∑i=-pqbiei|ξ|,
where p and q are the unknown parameters to be further identified.For compacton-like solution, we assume that the solution has the form(5.318)u0(x,t)=asin2(kx+ωt)b+csin2(kx+ωt),
where a,b,k, and ω are unknown constants further to be determined
(5.319)u0(x,t)=∑iaisin2(kix+ωit)∑j[bj+cjsin2(kjx+ωjt)].As an illustrating example, we consider the following modified KdV equation(5.320)ut+u2ux+uxxx=0.
Its iteration formulation can be constructed as follows:
(5.321)un+1(x,t)=un(x,t)-∫0t{(un)t+un2(un)x+(un)xxx}dt.
To search for its compacton-like solution, we assume that the solution has the form
(5.322)u0(x,t)=asin2(kx+wt)b+csin2(kx+wt),
where a, b, k,and w are unknown constants further to be determined.Substituting (5.322) into (5.321), we can calculate u1 and u2 with ease. In order to identify the constants in the initial solution, we can set
(5.323)∂k∂tkun(x,t)=∂k∂tkun+1(x,t),
setting
(5.324)∂∂tu0(x,t)=∂∂tu1(x,t).
By a simple calculation (see [60] for details), the compacton-like solution is obtained, which reads
(5.325)u(x,t)=±22kcsin2(kx-4k3t)-(3/2)c+csin2(kx-4k3t),
or
(5.326)u(x,t)=±42ksin2(kx-4k3t)-3+2sin2(kx-4k3t).
We can also begin with a more general initial solution in the form
(5.327)u0(x,t)=a+1c+dcos(kx+wt),
where a, b, c, k, and w are unknown constants.Setting∂u0/∂t=∂u1/∂t, we, therefore, obtain the following new compacton-like solution:
(5.328)u(x,t)=a+1-a/3k2±(1/6k2)4a2-6k2cos(kx+(k3-ka2)t).
If we choose k=iK, where K is a constant, then (5.328) becomes
(5.329)u(x,t)=a+1(a/3K2)±(1/6K2)4a2+6K2cosh(Kx+(-K3-Ka2)t)=a+1(a/3K2)±(1/12K2)4a2+6K2[exp(Kx+(-K3-Ka2)t)+exp(-Kx-(-K3-Ka2)t)],
which is a solitary solution. It is interesting that we can convert the compacton-like solution to solitary solution.The initial solution (trial function) can be also constructed in a solitary form. Now we begin with(5.330)u0=a+1b+cexp(kx+wt)+dexp(-kx-wt).
By the same manipulation as illustrated above, we set ∂u0/∂t=∂u1/∂t.So we obtain the following needed solitary solution:(5.331)u(x,t)=a+1a/3k2+cexp(kx+(-k3-ka2)t)+((3k2+2a2)/72k4)(1/c)exp(-kx-(-k3-ka2)t).
Hereby a and c are free parameters.It is also interesting to note that the solitary solution can be converted into a compacton-like solution if we choosek=iK, where K is a constant. If k=iK, (5.331) becomes
(5.332)u(x,t)=a+1-a/3K2+(c+𝒜)cos(Kx+(K3-Ka2)t)+(c-𝒜)isin(Kx+(K3-Ka2)t),
where  𝒜 donates (-3K2+2a2)/72K4c. In the above derivation, we use the relations
(5.333)exp(ikx+iwt)=cos(kx+wt)+isin(kx+wt),exp(-ikx-iwt)=cos(kx+wt)-isin(kx+wt).
In order to convert (5.332) into a compact form, the last term in denominator must be vanished, which requires
(5.334)c--3K2+2a272K4c=0.Solvingc from (5.334) results in
(5.335)c=±112-6K2+4a2K2.Substituting (5.335) into (5.332) yields a compacton-like solution
(5.336)u(x,t)=a+1-a/3K2±(1/6K2)4a2-6K2cos(Kx+(K3-ka2)t).


## 5.6.1. The Lagrange Multiplier

The variational iteration method has been shown to solve a large class of nonlinear problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions.To illustrate the basic idea of the technique, we consider the following general nonlinear system:(5.295)L[u(t)]+N[u(t)]=0,
where L is a linear operator, and N is a nonlinear operator.The basic concept of the method is to construct a correction functional for the system (5.295), which reads
(5.296)un+1(t)=un(t)+∫t0tλ{Lun(s)+Nu~n(s)}ds,
where λ is a general Lagrange multiplier that can be identified optimally via variational theory, un is the nth approximate solution, and u~n denotes a restricted variation, that is, δu~n=0. To illustrate how restricted variation works in the variational iteration method, we consider a simple algebraic equation
(5.297)x2-3x+2=0.
We rewrite (5.297) in the form
(5.298)x~·x-3x+2=0,
where x~ is called a restricted variable whose value is assumed to be known (5.6.1. the initial guess). Solving x from (5.298) leads to the result
(5.299)x=23-x~,
or in iteration form,
(5.300)xn+1=23-xn.
After identifying the multiplier in (5.296), we have the following iteration algorithm:
(5.301)un+1(t)=un(t)+∫t0tλ{Lun(s)+Nun(s)}ds.
Equation (5.301) is called the variational iteration algorithm-I.Consider the following nonlinear equation ofkth order:
(5.302)u(k)+f(u,u′,u′′,…,u(k))=0.
The variational iteration formulation is constructed as follows:
(5.303)un+1(t)=un(t)+∫t0tλ(un(k)+f~n)ds,
where δf~n=0, fn=f(un,un′,un′′,…). After identifying the multiplier, we have
(5.304)un+1(t)=un(t)+(-1)n∫t0t1(n-1)!(s-t)n-1[un(k)(s)+fn]ds.
This formulation is generally called the variational iteration algorithm-I [11, 61].The main merit of this iteration formula lies in the fact thatu0(t), the initial solution, can be freely chosen, with even unknown parameters contained. However, some repeated and unnecessary iterations are involved in this iteration algorithm at each step.For initial value problems, we can begin with(5.305)u0(t)=u(0)+tu′(0)+12!t2u′′(0)+⋯1k!tku(k)(0).
This leads to a series solution converging to the exact solution.For boundary value problems, the initial guess can be expressed in the form(5.306)u0(t)=a1g1(t)+a2g2(t)+⋯+akgk(t),
where gk(t) are known functions, and ak are unknowns to be further determined after a few iterations by the boundary conditions.After identifying the Lagrange multiplierλ in (5.296), we can construct the iteration formula
(5.307)un+1(t)=u0(t)+(-1)n∫t0t1(n-1)!(s-t)n-1fnds,
instead of the iteration algorithm (5.301). We call (5.307) the variational iteration algorithm-II [11, 61].Note:u0 must satisfy the initial/boundary conditions. This is the main shortcoming of the algorithm.From (5.307), we obtain the following variational iteration algorithm-III
(5.308)un+2(t)=un+1(t)+∫t0tλ{fn+1(s)-fn(s)}ds.
One common property of both the variational iteration algorithm-I and the variational iteration algorithm-III is the allowed dependence of the initial guess on unknown parameters whose values could be identified after a few iterations by using the initial/boundary conditions. the variational iteration algorithm-III, in particular, is highly suitable for boundary value problems of high orders.


## 5.6.2. Laplace Transform for Identification of the Lagrange Multiplier [65]

We can also apply Laplace transform to identify the Lagrange multiplier. By Laplace transform, we have(5.309)skU(s)+ℓ{f(u,u′,…,u(k))}=0,
or
(5.310)U(s)=-ℓ{f(u,u′,…,u(k))}sk.
The inverse Laplace transform reads
(5.311)u(t)=(-1)k∫0t(η-t)k-1(k-1)!f(un(η),un′(η),…,un(k)(η))dη.
Hence, the following iteration algorithm is derived:
(5.312)un+1(t)=u0(t)+(-1)k∫0t(η-t)k-1(k-1)!f(un(η),un′(η),…,un(k)(η))dη.


## 5.6.3.  Variational Iteration Method for Solitary Solutions

Here is an incomplete list for variational iteration formulas for various differential equations [8, 9]:
(5.313)u′+f(u,u′)=0,un+1(t)=u0(t)-∫0tf(un,un′)ds,u′+αu+f(u,u′)=0,un+1(t)=u0(t)-∫0teα(s-t)f(un,un′)ds,u′′+f(u,u′,u′′)=0,un+1(t)=u0(t)+∫0t(s-t)f(un,un′,un′′)ds,u′′+ω2u+f(u,u′,u′′)=0,un+1(t)=u0(t)+1ω∫0tsinω(s-t)f(un,un′,un′′)ds,u′′-α2u+f(u,u′,u′′)=0,un+1(t)=u0(t)+∫0t12α(eα(s-t)-eα(t-s))f(un,un′,un′′)ds,u′′′+f(u,u′,u′′,u′′′)=0,un+1(t)=u0(t)-∫0t12(s-t)2f(un,un′,un′′,un′′′)ds,u(4)+f(u,u′,u′′,u′′′,u(4))=0,un+1(t)=u0(t)+∫0t16(s-t)3f(un,un′,un′′,un′′′,un(4))ds,u(k)+f(u,u′,u′′,…,u(k))=0,un+1(t)=u0(t)+(-1)n∫0t1(n-1)!(s-t)n-1f(un,un′,un′′,…,un(k))ds.
The main feature of the method is that the initial solution can be chosen with some unknown parameters in the form of the searched solution. For example, for solitons, we begin with
(5.314)u0=psech2(qξ),ξ=x+ct,
where p and q are the unknown parameters to be further identified after one or few iterations. For a more general form for solitary solutions, we assume that
(5.315)u0(ξ)=∑i=-nmcieiξ∑i=-pqbieiξ,ξ=x+ct,
where ci and bi are constants to be further determined.For discontinuous solitons, we can assume, for example, the following form:(5.316)u(ξ)=pexp(-q|ξ|),
or
(5.317)u0(ξ)=∑i=-nmciei|ξ|∑i=-pqbiei|ξ|,
where p and q are the unknown parameters to be further identified.For compacton-like solution, we assume that the solution has the form(5.318)u0(x,t)=asin2(kx+ωt)b+csin2(kx+ωt),
where a,b,k, and ω are unknown constants further to be determined
(5.319)u0(x,t)=∑iaisin2(kix+ωit)∑j[bj+cjsin2(kjx+ωjt)].As an illustrating example, we consider the following modified KdV equation(5.320)ut+u2ux+uxxx=0.
Its iteration formulation can be constructed as follows:
(5.321)un+1(x,t)=un(x,t)-∫0t{(un)t+un2(un)x+(un)xxx}dt.
To search for its compacton-like solution, we assume that the solution has the form
(5.322)u0(x,t)=asin2(kx+wt)b+csin2(kx+wt),
where a, b, k,and w are unknown constants further to be determined.Substituting (5.322) into (5.321), we can calculate u1 and u2 with ease. In order to identify the constants in the initial solution, we can set
(5.323)∂k∂tkun(x,t)=∂k∂tkun+1(x,t),
setting
(5.324)∂∂tu0(x,t)=∂∂tu1(x,t).
By a simple calculation (see [60] for details), the compacton-like solution is obtained, which reads
(5.325)u(x,t)=±22kcsin2(kx-4k3t)-(3/2)c+csin2(kx-4k3t),
or
(5.326)u(x,t)=±42ksin2(kx-4k3t)-3+2sin2(kx-4k3t).
We can also begin with a more general initial solution in the form
(5.327)u0(x,t)=a+1c+dcos(kx+wt),
where a, b, c, k, and w are unknown constants.Setting∂u0/∂t=∂u1/∂t, we, therefore, obtain the following new compacton-like solution:
(5.328)u(x,t)=a+1-a/3k2±(1/6k2)4a2-6k2cos(kx+(k3-ka2)t).
If we choose k=iK, where K is a constant, then (5.328) becomes
(5.329)u(x,t)=a+1(a/3K2)±(1/6K2)4a2+6K2cosh(Kx+(-K3-Ka2)t)=a+1(a/3K2)±(1/12K2)4a2+6K2[exp(Kx+(-K3-Ka2)t)+exp(-Kx-(-K3-Ka2)t)],
which is a solitary solution. It is interesting that we can convert the compacton-like solution to solitary solution.The initial solution (trial function) can be also constructed in a solitary form. Now we begin with(5.330)u0=a+1b+cexp(kx+wt)+dexp(-kx-wt).
By the same manipulation as illustrated above, we set ∂u0/∂t=∂u1/∂t.So we obtain the following needed solitary solution:(5.331)u(x,t)=a+1a/3k2+cexp(kx+(-k3-ka2)t)+((3k2+2a2)/72k4)(1/c)exp(-kx-(-k3-ka2)t).
Hereby a and c are free parameters.It is also interesting to note that the solitary solution can be converted into a compacton-like solution if we choosek=iK, where K is a constant. If k=iK, (5.331) becomes
(5.332)u(x,t)=a+1-a/3K2+(c+𝒜)cos(Kx+(K3-Ka2)t)+(c-𝒜)isin(Kx+(K3-Ka2)t),
where  𝒜 donates (-3K2+2a2)/72K4c. In the above derivation, we use the relations
(5.333)exp(ikx+iwt)=cos(kx+wt)+isin(kx+wt),exp(-ikx-iwt)=cos(kx+wt)-isin(kx+wt).
In order to convert (5.332) into a compact form, the last term in denominator must be vanished, which requires
(5.334)c--3K2+2a272K4c=0.Solvingc from (5.334) results in
(5.335)c=±112-6K2+4a2K2.Substituting (5.335) into (5.332) yields a compacton-like solution
(5.336)u(x,t)=a+1-a/3K2±(1/6K2)4a2-6K2cos(Kx+(K3-ka2)t).


## 5.7. Homotopy Perturbation Method

### 5.7.1. Homotopy Technology

Before proceeding the method, we give an interesting application of the homotopy technology to an asymptotic match of Rayleigh grain size distribution and Hillert grain size distribution. Grain growth is a well-known phenomenon in the evolution of crystalline microstructures. A stochastic continuity or Fokker-Planck continuity equation was proposed to accurately predict properties of grain growth [66]. The equation can be expressed in the form [66]
(5.337)εd2fdx2+(α+εx-α+2x)dfdx+(6-α+εx2)f=0.
Let us consider two limiting cases where the driving force is either due to diffusion (Rayleigh) or to the drift velocity (Hillert). In the first case, the Rayleigh grain size distribution occurs when α=0 and ε≠0 [66]:
(5.338)fr(x)=π2xexp(-π4x2).
The other limiting case, α=8 and ε=0, leads to Hillert grain size distribution [66]:
(5.339)fh(x)=8exp(2x/(x-2))(x-2)2.
The actual solution to (5.337) falls between (5.338) and (5.339); in the work of Pande and Cooper [66], a very good but approximate solution was given (see (18) of Pande and Cooper’s publication [66]). Hereby we will suggest a simple matching technology to bridge the limiting cases.An asymptotic match of (5.338) and (5.339) can be expressed in the form [67]
(5.340)f(x,α,ε)=C(α,ε)fr(x)+D(α,ε)fh(x)=C(α,ε)π2xexp(-π4x2)+D(α,ε)8exp(2x/(x-2))(x-2)2,
where C and D are matching parameters.Equation (5.340) should turn out to be exactly (5.338) and (5.339), respectively, for the above two limiting cases. This requires
(5.341)C(0,ε)=1,D(0,ε)=0,C(8,0)=0,D(8,0)=1.
We can freely choose C and D satisfying (5.341) and the following normalization condition
(5.342)C+D=1.
Hereby we give simple expressions for C and D in the forms
(5.343)C(α,ε)=1-18α(1-ε),D(α,ε)=18α(1-ε).
We, therefore, obtain the following approximate solution to (5.337) [67]:
(5.344)f(x)=π2(1-18α(1-ε))xexp(-π4x2)+18α(1-ε)8exp(2x/(x-2))(x-2)2.
This equation is simpler than that given by Pande and Cooper [66]. There are many alternative ways for determination of C and D satisfying (5.341) and (5.342).In this short paper we suggest an asymptotic match to bridge the two limited cases, and the obtained result is valid for the whole case. The asymptotic match is a simple and useful mathematical tool in engineering for reliable treatment of a nonlinear problem whose analytical solution can be easily obtained for two limited cases (e.g.,ε→0 and ε→∞, resp.) by some analytical methods.Consider another example of the relativistic oscillator(5.345)u′′+u1+u2=0,
with initial conditions u(0)=A,u′(0)=0.It is easy to obtain the following approximate frequency [68]:
(5.346)ω=22π(1+4π2·A2)-1/4,forA≫1,ω=(1+12A2)-1/4,forA≪1.In order to match both the casesA→0 and A→∞, we construct the following homotopy [68]:
(5.347)ω~=e-αAf(A)+(1-e-αA)g(A),
where f(A)=(22/π)(1+(4/π2)·A2)-1/4, g(A)=(1+(1/2)A2)-1/4, and α is a free parameter. Now considering the case when A=1, we have exact frequency, which is ω~=0.8736. From this relationship, we can identify α as follows:
(5.348)α=0.4962.
Finally, we obtain the following result:
(5.349)ω~=e-0.4962A·22π(1+4π2·A2)-1/4+(1-e-0.496A)·(1+12A2)-1/4.


### 5.7.2. Homotopy Perturbation Method with an Auxiliary Term [69]

The two most important steps in application of the homotopy perturbation method [69–72] are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be constructed such that, when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach for construction of the homotopy equation with an auxiliary term Duffing equation is used as example to illustrate the solution procedure.Consider a general nonlinear equation(5.350)Lu+Nu=0,
where L and N are, respectively, the linear operator and nonlinear operator.The first step for the method is to construct a homotopy equation in the form(5.351)L~u+p(Lu-L~u+Nu)=0,
where L~ is a linear operator with a possible unknown constant, and L~u=0 can best describe the solution property. The embedding parameter p monotonically increases from zero to unit as the trivial problem, L~u=0, is continuously deformed to the original one.For example, consider a nonlinear oscillator [9]
(5.352)u′′+εu3=0,u(0)=A,u′(0)=0.
For an oscillator, we can use sine or cosine function. We assume that the approximate solution of (5.352) is
(5.353)u(t)=Acosωt,
where ω is the frequency to be determined later. We, therefore, can choose
(5.354)L~u=u¨+ω2u.
Accordingly, we can construct a homotopy equation in the form
(5.355)u¨+ω2u+p(u3-ω2u)=0.
When p=0, we have
(5.356)u¨+ω2u=0,u(0)=A,u′(0)=0,
which describes the basic solution property of the original nonlinear equation (5.352).Whenp=1, (5.355) becomes the original one. So the solution procedure is to deform from the initial solution, (5.353), to the real one. Due to one unknown parameter in the initial solution, only one iteration is enough. For detailed solution procedure, refer to [9].If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form(5.357)u¨+0·u+pu3=0.
We expand the solution and the coefficient, zero, of the linear term into a series in p:
(5.358)u=u0+pu1+p2u2+⋯,(5.359)0=ω2+pa1+p2a2+⋯,
where the unknown constant, ai, is determined in the (i+1)th iteration. The solution procedure is given in [9].In this paper, we suggest an alternative approach for construction of homotopy equation, which is(5.360)L~u+p(Lu-L~u+Nu)+αp(1-p)u=0,
where α is an auxiliary parameter. When α=0, (5.360) turns out to be that of the classical one, expressed in (5.351). The auxiliary term, αp(1-p)u, vanishes completely when p=0 or p=1, so the auxiliary term will affect neither the initial solution (p=0), nor the real solution (p=1).To illustrate the solution procedure, we consider a nonlinear oscillator in the form(5.361)d2udt2+bu+cu3=0,u(0)=A,u′(0)=0,
where b and c are positive constants.Equation (5.361) admits a periodic solution, and the linearized equation of (5.361) is
(5.362)u′′+ω2u=0,u(0)=A,u′(0)=0,
where ω is the frequency of (5.361).We construct the following homotopy equation with an auxiliary term [69]:
(5.363)u′′+ω2u+p[(b-ω2)u+cu3]+αp(1-p)u=0.
Assume that the solution can be expressed in a power series in p as shown in (5.358). Substituting (5.358) into (5.363), and processing as the standard perturbation method, we have
(5.364)u0′′+ω2u0=0,u0(0)=A,u0′(0)=0,(5.365)u1′′+ω2u1+(b-ω2)u0+cu03+αu0=0,u1(0)=0,u1′(0)=0,(5.366)u2′′+ω2u2+(b-ω2)u1+3cu02u1+α(u1-u0)=0,
with initial conditions
(5.367)∑i=0ui(0)=A,∑i=0ui′(0)=0.
Solving (5.364), we have
(5.368)u0=Acosωt.Substitutingu0 into (5.365) results in
(5.369)u1′′+ω2u1+A(α+b-ω2+(34)cA2)cosωt+14cA3cos3ωt=0.Eliminating the secular term needs(5.370)α+b-ω2+34cA2=0.
A special solution of (5.369) is
(5.371)u1=-cA332ω2cos3ωt.
If only a first-order approximate solution is enough, we just set α=0, and this results in
(5.372)ω=b+34cA2.
The accuracy reaches 7.6% even for the case cA2→∞.The solution procedure continues by submittingu1 into (5.366), and after some simple calculation, we obtain
(5.373)u2′′+ω2u2-(αA+3c2A5128ω2)cosωt-(cA3(b-ω2)32ω2+3c2A564ω2+αcA332ω2)cos3ωt-3c2A5128ω2cos5ω=0.No secular term inu2 requires
(5.374)αA+3c2A5128ω2=0.Solving (5.370) and (5.374) simultaneously, we obtain
(5.375)ω=b+(3/4)cA2+(b+(3/4)cA2)2+(3/32)c2A42,
and the approximate solution is u(t)=Acosωt, where ω is given in (5.375).In order to compare with the perturbation solution and the exact solution, we setb=1. In case c≪1, (5.375) agrees with that obtained by the classical perturbation method; when c→∞, we have
(5.376)limc→∞ω=3/4+(3/4)2+3/322cA2=0.8832cA2.
The exact period reads
(5.377)Tex=41+cA2∫0π/2dx1-ksin2x,
where k=cA2/2(1+cA2).In casec→∞, we have
(5.378)limc→∞Tex=6.743cA2,(5.379)ωex≈2π6.743cA2.
Comparing between (5.376) and (5.378), we find that the accuracy reaches 5.5%, while accuracy of the first-order approximate frequency is 7.6%.If a higher-order approximate solution is needed, we rewrite the homotopy equation in the form(5.380)u′′+ω2u+p[(b-ω2)u+cu3]+1·p(1-p)u=0.
The coefficient, 1, in the auxiliary term is also expanded in a series in p in the form
(5.381)1=α0+pα1+p2α2+⋯,
where αi is identified in the (i+2)th iteration. The solution procedure is similar to that illustrated above.Generally, the homotopy equation can be constructed in the form(5.382)L~u+p(Lu-L~u+Nu)+1·f(p)g(p)h(u,u′,u′′,…)=0,
where f and g are functions of p, satisfying f(0)=0 and g(1)=0, and h can be generally expressed in the form
(5.383)h=u+β1u′+β2u′′+⋯.


### 5.7.3. Homotopy Perturbation Method for Solitary Solutions

Homotopy perturbation method [69–72] provides a simple mathematical tool for searching for soliton solutions without any small perturbation. Considering the following nonlinear equation:
(5.384)∂u∂t+au∂u∂x+b∂3u∂x3+N(u)=0,a>0,b>0,
we can construct a homotopy in the form
(5.385)(1-p){∂u∂t+6u∂u∂x+∂3u∂x3}+p{∂u∂t+au∂u∂x+b∂3u∂x3+N(u)}=0.
When p=0, we have
(5.386)∂u∂t+6u∂u∂x+∂3u∂x3=0,
a well-known KdV equation, whose soliton solution is known. When p=1, (5.385) turns out to be the original equation. According to the homotopy perturbation method, we assume that
(5.387)u=u0+pu1+p2u2+⋯.
Substituting (5.387) into (5.385), and proceeding the same process as the traditional perturbation method does, we can easily solve u0, u1, and other components. The solution can be expressed finally in the form
(5.388)u=u0+u1+u2+⋯.
The homotopy perturbation method always stops before the second iteration, so the solution can be expressed as
(5.389)u=u0+u1,
for most cases.For traveling wave solutions, we can use the following transformation:(5.390)u(t,x)=u(ξ),ξ=x-Dt.
Equation (5.384) becomes
(5.391)-Duξ+auuξ+buξξξ+N(u)=0.
Integrating the equation with respect to ξ, we have
(5.392)-Du+12au2+buξξ+F(u)=0,
where ∂F/∂u=N(u)+C, and C is an integral constant.We begin with a soliton in the form(5.393)u0=1e-αξ+eαξ,
which is the solution of the following equation:
(5.394)uξξ-α2u+8α2u3=0.
Accordingly, we construct the following homotopy equation:
(5.395)uξξ-Dbu+0·u3+p[a2bu2+1bF(u)]=0.
The solution and the parameters D/b and 0 are expanded in the forms
(5.396)u=u0+pu1+p2u2+⋯,Db=α2+pα1+p2α2+⋯,0=8α2+pβ1+p2β2+⋯.
Substituting (5.396) into (5.395), and proceeding a similar way as the perturbation method, we have
(5.397)u0ξξ-α2u0+8α2u03=0,u1ξξ-α2u1+α1u0+24α2u02u1+β1u03+a2bu02+1bF(u0)=0.
As an example, we consider the following equation:
(5.398)ut+uux+ηuxxx=0.
By the transformation u(t,x)=u(ξ), ξ=x-Dt, we have
(5.399)-Duξ+uuξ+ηuξξξ=0,
or
(5.400)u′′-Dηu+12ηu2+C=0.
In order to make the solution process simple, we construct a homotopy equation in the form
(5.401)u′′-Dηu+p(12ηu2+C)=0.
Assume that the solution and the parameter D/η can be expanded in the forms
(5.402)u=u0+pu1+p2u2+···,Dη=λ02+pλ1+p2λ2+···.
Substituting (5.402) into (5.401), proceeding as the classical perturbation method does, we have
(5.403)u0′′-λ02u0=0,u1′′-λ02u1-λ1u0+12ηu02+C=0,u2′′-λ02u2-λ2u0-λ1u1+1ηu0u1=0.
The solution of the first equation of (5.403) is
(5.404)u0=Aeλ0ξ+Be-λ0ξ.
Substituting (5.404) into the second equation of (5.403) we have
(5.405)u1′′-λ02u1=λ1(Aeλ0ξ+Be-λ0ξ)-12η(Aeλ0ξ+Be-λ0ξ)2-C=λ1(Aeλ0ξ+Be-λ0ξ)-12η(A2e2λ0ξ+B2e-2λ0ξ)-ABη-C.
We set
(5.406)λ1=0,AB=-ηC
in (5.405) to avoid the secular-like terms ξkeλ0ξ, ξke-λ0ξ or ξk (k≥0), and solving the resultant equation, we have the solution for u1:
(5.407)u1=-16η(A2e2λ0ξ+B2e-2λ0ξ).
Substituting u0 and u1 into the third equation of (5.403) results in
(5.408)u2′′-λ02u2=λ2(Aeλ0ξ+Be-λ0ξ)+16η2(Aeλ0ξ+Be-λ0ξ)(A2e2λ0ξ+B2e-2λ0ξ)=A(λ2+AB6η2)eλ0ξ+B(λ2+AB6η2)e-λ0ξ+16η2(A3e3λ0ξ+B3e-3λ0ξ).
Similarly, we set
(5.409)λ2=-AB6η2.
The solution for u2 is
(5.410)u2=148η2(A3e3λ0ξ+B3e-3λ0ξ).
If the second order approximate solution is enough, then we have
(5.411)u=u0+u1+u2=Aeλ0ξ+Be-λ0ξ-16η(A2e2λ0ξ+B2e-2λ0ξ)+148η2(A3e3λ0ξ+B3e-3λ0ξ).
Solving λ0 from the second equation of (5.402), we have
(5.412)λ0=Dη-(pλ1+p2λ2+···)|p=1,λi=0(i≥3)=Dη+AB6η2.Using in the[1,2] exponential Padé approximant [62], we have
(5.413)u=Aeλ0ξ+Be-λ0ξ1+aeλ0ξ+be-λ0ξ+ce2λ0ξ+de-2λ0ξ,
where the parameters A,B,a,b,c, and d can be identified by the exponential Padé approximant similar to the Padé approximant.The[m,n] exponential Padé approximant is defined as follows.Let(5.414)F=F0+εF1+ε2F2+⋯,ε→0.The [m,n] exponential Padé approximant reads
(5.415)Fmn(ε)=∑i=0mciexpdiε∑i=0naiexpbiε.
The coefficients ai,bi,ci,  and  di, are determined from the following condition: the first (2m+2n) components of the expansion of the rational function Fmn(ε) in a Maclaurin series coincide with the first (2m+2n) components of the series F(ε).


### 5.7.4. Couple Homotopy Perturbation Method with Laplace Transform [73]

Couple of the homotopy perturbation method with the Laplace transform makes the solution procedure much simpler.Consider the following foam drainage equation:(5.416)ut+2u2ux-ux2-12uxxu=0,
with initial conditions
(5.417)u(x,0)=-12+11+ex.
Apply Laplace transformation
(5.418)u(x,s)=1s(-12+11+ex)-1sL(2u2ux-ux2-12uxxu).
Apply inverse Laplace transformation
(5.419)u(x,t)=(-12+11+ex)-L-1(1sL(2u2ux-ux2-12uxxu)).
Apply the convex homotopy
(5.420)u0+pu1+p2u2+⋯=(-12+11+ex)-∫0tL-1(1sL(2(u0+pu1+⋯)2(∂u0∂x+p∂u1∂x+⋯)))dt+∫0tL-1(1sL(u0+pu1+⋯)x2+12(∂2u0∂x2+p∂2u1∂x2+⋯)(u0+pu1+⋯))dt.
Comparing the coefficient of like powers of p,
(5.421)p(0):u0(x,t)=(-12+11+ex),p(1):u1(x,t)=4ext(1+2ex)4+8e2xt(1+2ex)4-5ext(1+2ex)3-2e2xt(1+2ex)3+3ext2(1+2ex)2-ext216(1+2ex)3+e2xt8(1+2ex)3.⋮
The series solution is given by
(5.422)u(x,t)=(-12+11+ex)+4ext(1+2ex)4+8e2xt(1+2ex)4-5ext(1+2ex)3-2e2xt(1+2ex)3+3ext2(1+2ex)2-ext216(1+2ex)3+e2xt8(1+2ex)3⋯.
Applications of the homotopy perturbation method to solitary solutions can be referred to [74–77].


## 5.7.1. Homotopy Technology

Before proceeding the method, we give an interesting application of the homotopy technology to an asymptotic match of Rayleigh grain size distribution and Hillert grain size distribution. Grain growth is a well-known phenomenon in the evolution of crystalline microstructures. A stochastic continuity or Fokker-Planck continuity equation was proposed to accurately predict properties of grain growth [66]. The equation can be expressed in the form [66]
(5.337)εd2fdx2+(α+εx-α+2x)dfdx+(6-α+εx2)f=0.
Let us consider two limiting cases where the driving force is either due to diffusion (Rayleigh) or to the drift velocity (Hillert). In the first case, the Rayleigh grain size distribution occurs when α=0 and ε≠0 [66]:
(5.338)fr(x)=π2xexp(-π4x2).
The other limiting case, α=8 and ε=0, leads to Hillert grain size distribution [66]:
(5.339)fh(x)=8exp(2x/(x-2))(x-2)2.
The actual solution to (5.337) falls between (5.338) and (5.339); in the work of Pande and Cooper [66], a very good but approximate solution was given (see (18) of Pande and Cooper’s publication [66]). Hereby we will suggest a simple matching technology to bridge the limiting cases.An asymptotic match of (5.338) and (5.339) can be expressed in the form [67]
(5.340)f(x,α,ε)=C(α,ε)fr(x)+D(α,ε)fh(x)=C(α,ε)π2xexp(-π4x2)+D(α,ε)8exp(2x/(x-2))(x-2)2,
where C and D are matching parameters.Equation (5.340) should turn out to be exactly (5.338) and (5.339), respectively, for the above two limiting cases. This requires
(5.341)C(0,ε)=1,D(0,ε)=0,C(8,0)=0,D(8,0)=1.
We can freely choose C and D satisfying (5.341) and the following normalization condition
(5.342)C+D=1.
Hereby we give simple expressions for C and D in the forms
(5.343)C(α,ε)=1-18α(1-ε),D(α,ε)=18α(1-ε).
We, therefore, obtain the following approximate solution to (5.337) [67]:
(5.344)f(x)=π2(1-18α(1-ε))xexp(-π4x2)+18α(1-ε)8exp(2x/(x-2))(x-2)2.
This equation is simpler than that given by Pande and Cooper [66]. There are many alternative ways for determination of C and D satisfying (5.341) and (5.342).In this short paper we suggest an asymptotic match to bridge the two limited cases, and the obtained result is valid for the whole case. The asymptotic match is a simple and useful mathematical tool in engineering for reliable treatment of a nonlinear problem whose analytical solution can be easily obtained for two limited cases (e.g.,ε→0 and ε→∞, resp.) by some analytical methods.Consider another example of the relativistic oscillator(5.345)u′′+u1+u2=0,
with initial conditions u(0)=A,u′(0)=0.It is easy to obtain the following approximate frequency [68]:
(5.346)ω=22π(1+4π2·A2)-1/4,forA≫1,ω=(1+12A2)-1/4,forA≪1.In order to match both the casesA→0 and A→∞, we construct the following homotopy [68]:
(5.347)ω~=e-αAf(A)+(1-e-αA)g(A),
where f(A)=(22/π)(1+(4/π2)·A2)-1/4, g(A)=(1+(1/2)A2)-1/4, and α is a free parameter. Now considering the case when A=1, we have exact frequency, which is ω~=0.8736. From this relationship, we can identify α as follows:
(5.348)α=0.4962.
Finally, we obtain the following result:
(5.349)ω~=e-0.4962A·22π(1+4π2·A2)-1/4+(1-e-0.496A)·(1+12A2)-1/4.


## 5.7.2. Homotopy Perturbation Method with an Auxiliary Term [69]

The two most important steps in application of the homotopy perturbation method [69–72] are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be constructed such that, when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach for construction of the homotopy equation with an auxiliary term Duffing equation is used as example to illustrate the solution procedure.Consider a general nonlinear equation(5.350)Lu+Nu=0,
where L and N are, respectively, the linear operator and nonlinear operator.The first step for the method is to construct a homotopy equation in the form(5.351)L~u+p(Lu-L~u+Nu)=0,
where L~ is a linear operator with a possible unknown constant, and L~u=0 can best describe the solution property. The embedding parameter p monotonically increases from zero to unit as the trivial problem, L~u=0, is continuously deformed to the original one.For example, consider a nonlinear oscillator [9]
(5.352)u′′+εu3=0,u(0)=A,u′(0)=0.
For an oscillator, we can use sine or cosine function. We assume that the approximate solution of (5.352) is
(5.353)u(t)=Acosωt,
where ω is the frequency to be determined later. We, therefore, can choose
(5.354)L~u=u¨+ω2u.
Accordingly, we can construct a homotopy equation in the form
(5.355)u¨+ω2u+p(u3-ω2u)=0.
When p=0, we have
(5.356)u¨+ω2u=0,u(0)=A,u′(0)=0,
which describes the basic solution property of the original nonlinear equation (5.352).Whenp=1, (5.355) becomes the original one. So the solution procedure is to deform from the initial solution, (5.353), to the real one. Due to one unknown parameter in the initial solution, only one iteration is enough. For detailed solution procedure, refer to [9].If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form(5.357)u¨+0·u+pu3=0.
We expand the solution and the coefficient, zero, of the linear term into a series in p:
(5.358)u=u0+pu1+p2u2+⋯,(5.359)0=ω2+pa1+p2a2+⋯,
where the unknown constant, ai, is determined in the (i+1)th iteration. The solution procedure is given in [9].In this paper, we suggest an alternative approach for construction of homotopy equation, which is(5.360)L~u+p(Lu-L~u+Nu)+αp(1-p)u=0,
where α is an auxiliary parameter. When α=0, (5.360) turns out to be that of the classical one, expressed in (5.351). The auxiliary term, αp(1-p)u, vanishes completely when p=0 or p=1, so the auxiliary term will affect neither the initial solution (p=0), nor the real solution (p=1).To illustrate the solution procedure, we consider a nonlinear oscillator in the form(5.361)d2udt2+bu+cu3=0,u(0)=A,u′(0)=0,
where b and c are positive constants.Equation (5.361) admits a periodic solution, and the linearized equation of (5.361) is
(5.362)u′′+ω2u=0,u(0)=A,u′(0)=0,
where ω is the frequency of (5.361).We construct the following homotopy equation with an auxiliary term [69]:
(5.363)u′′+ω2u+p[(b-ω2)u+cu3]+αp(1-p)u=0.
Assume that the solution can be expressed in a power series in p as shown in (5.358). Substituting (5.358) into (5.363), and processing as the standard perturbation method, we have
(5.364)u0′′+ω2u0=0,u0(0)=A,u0′(0)=0,(5.365)u1′′+ω2u1+(b-ω2)u0+cu03+αu0=0,u1(0)=0,u1′(0)=0,(5.366)u2′′+ω2u2+(b-ω2)u1+3cu02u1+α(u1-u0)=0,
with initial conditions
(5.367)∑i=0ui(0)=A,∑i=0ui′(0)=0.
Solving (5.364), we have
(5.368)u0=Acosωt.Substitutingu0 into (5.365) results in
(5.369)u1′′+ω2u1+A(α+b-ω2+(34)cA2)cosωt+14cA3cos3ωt=0.Eliminating the secular term needs(5.370)α+b-ω2+34cA2=0.
A special solution of (5.369) is
(5.371)u1=-cA332ω2cos3ωt.
If only a first-order approximate solution is enough, we just set α=0, and this results in
(5.372)ω=b+34cA2.
The accuracy reaches 7.6% even for the case cA2→∞.The solution procedure continues by submittingu1 into (5.366), and after some simple calculation, we obtain
(5.373)u2′′+ω2u2-(αA+3c2A5128ω2)cosωt-(cA3(b-ω2)32ω2+3c2A564ω2+αcA332ω2)cos3ωt-3c2A5128ω2cos5ω=0.No secular term inu2 requires
(5.374)αA+3c2A5128ω2=0.Solving (5.370) and (5.374) simultaneously, we obtain
(5.375)ω=b+(3/4)cA2+(b+(3/4)cA2)2+(3/32)c2A42,
and the approximate solution is u(t)=Acosωt, where ω is given in (5.375).In order to compare with the perturbation solution and the exact solution, we setb=1. In case c≪1, (5.375) agrees with that obtained by the classical perturbation method; when c→∞, we have
(5.376)limc→∞ω=3/4+(3/4)2+3/322cA2=0.8832cA2.
The exact period reads
(5.377)Tex=41+cA2∫0π/2dx1-ksin2x,
where k=cA2/2(1+cA2).In casec→∞, we have
(5.378)limc→∞Tex=6.743cA2,(5.379)ωex≈2π6.743cA2.
Comparing between (5.376) and (5.378), we find that the accuracy reaches 5.5%, while accuracy of the first-order approximate frequency is 7.6%.If a higher-order approximate solution is needed, we rewrite the homotopy equation in the form(5.380)u′′+ω2u+p[(b-ω2)u+cu3]+1·p(1-p)u=0.
The coefficient, 1, in the auxiliary term is also expanded in a series in p in the form
(5.381)1=α0+pα1+p2α2+⋯,
where αi is identified in the (i+2)th iteration. The solution procedure is similar to that illustrated above.Generally, the homotopy equation can be constructed in the form(5.382)L~u+p(Lu-L~u+Nu)+1·f(p)g(p)h(u,u′,u′′,…)=0,
where f and g are functions of p, satisfying f(0)=0 and g(1)=0, and h can be generally expressed in the form
(5.383)h=u+β1u′+β2u′′+⋯.


## 5.7.3. Homotopy Perturbation Method for Solitary Solutions

Homotopy perturbation method [69–72] provides a simple mathematical tool for searching for soliton solutions without any small perturbation. Considering the following nonlinear equation:
(5.384)∂u∂t+au∂u∂x+b∂3u∂x3+N(u)=0,a>0,b>0,
we can construct a homotopy in the form
(5.385)(1-p){∂u∂t+6u∂u∂x+∂3u∂x3}+p{∂u∂t+au∂u∂x+b∂3u∂x3+N(u)}=0.
When p=0, we have
(5.386)∂u∂t+6u∂u∂x+∂3u∂x3=0,
a well-known KdV equation, whose soliton solution is known. When p=1, (5.385) turns out to be the original equation. According to the homotopy perturbation method, we assume that
(5.387)u=u0+pu1+p2u2+⋯.
Substituting (5.387) into (5.385), and proceeding the same process as the traditional perturbation method does, we can easily solve u0, u1, and other components. The solution can be expressed finally in the form
(5.388)u=u0+u1+u2+⋯.
The homotopy perturbation method always stops before the second iteration, so the solution can be expressed as
(5.389)u=u0+u1,
for most cases.For traveling wave solutions, we can use the following transformation:(5.390)u(t,x)=u(ξ),ξ=x-Dt.
Equation (5.384) becomes
(5.391)-Duξ+auuξ+buξξξ+N(u)=0.
Integrating the equation with respect to ξ, we have
(5.392)-Du+12au2+buξξ+F(u)=0,
where ∂F/∂u=N(u)+C, and C is an integral constant.We begin with a soliton in the form(5.393)u0=1e-αξ+eαξ,
which is the solution of the following equation:
(5.394)uξξ-α2u+8α2u3=0.
Accordingly, we construct the following homotopy equation:
(5.395)uξξ-Dbu+0·u3+p[a2bu2+1bF(u)]=0.
The solution and the parameters D/b and 0 are expanded in the forms
(5.396)u=u0+pu1+p2u2+⋯,Db=α2+pα1+p2α2+⋯,0=8α2+pβ1+p2β2+⋯.
Substituting (5.396) into (5.395), and proceeding a similar way as the perturbation method, we have
(5.397)u0ξξ-α2u0+8α2u03=0,u1ξξ-α2u1+α1u0+24α2u02u1+β1u03+a2bu02+1bF(u0)=0.
As an example, we consider the following equation:
(5.398)ut+uux+ηuxxx=0.
By the transformation u(t,x)=u(ξ), ξ=x-Dt, we have
(5.399)-Duξ+uuξ+ηuξξξ=0,
or
(5.400)u′′-Dηu+12ηu2+C=0.
In order to make the solution process simple, we construct a homotopy equation in the form
(5.401)u′′-Dηu+p(12ηu2+C)=0.
Assume that the solution and the parameter D/η can be expanded in the forms
(5.402)u=u0+pu1+p2u2+···,Dη=λ02+pλ1+p2λ2+···.
Substituting (5.402) into (5.401), proceeding as the classical perturbation method does, we have
(5.403)u0′′-λ02u0=0,u1′′-λ02u1-λ1u0+12ηu02+C=0,u2′′-λ02u2-λ2u0-λ1u1+1ηu0u1=0.
The solution of the first equation of (5.403) is
(5.404)u0=Aeλ0ξ+Be-λ0ξ.
Substituting (5.404) into the second equation of (5.403) we have
(5.405)u1′′-λ02u1=λ1(Aeλ0ξ+Be-λ0ξ)-12η(Aeλ0ξ+Be-λ0ξ)2-C=λ1(Aeλ0ξ+Be-λ0ξ)-12η(A2e2λ0ξ+B2e-2λ0ξ)-ABη-C.
We set
(5.406)λ1=0,AB=-ηC
in (5.405) to avoid the secular-like terms ξkeλ0ξ, ξke-λ0ξ or ξk (k≥0), and solving the resultant equation, we have the solution for u1:
(5.407)u1=-16η(A2e2λ0ξ+B2e-2λ0ξ).
Substituting u0 and u1 into the third equation of (5.403) results in
(5.408)u2′′-λ02u2=λ2(Aeλ0ξ+Be-λ0ξ)+16η2(Aeλ0ξ+Be-λ0ξ)(A2e2λ0ξ+B2e-2λ0ξ)=A(λ2+AB6η2)eλ0ξ+B(λ2+AB6η2)e-λ0ξ+16η2(A3e3λ0ξ+B3e-3λ0ξ).
Similarly, we set
(5.409)λ2=-AB6η2.
The solution for u2 is
(5.410)u2=148η2(A3e3λ0ξ+B3e-3λ0ξ).
If the second order approximate solution is enough, then we have
(5.411)u=u0+u1+u2=Aeλ0ξ+Be-λ0ξ-16η(A2e2λ0ξ+B2e-2λ0ξ)+148η2(A3e3λ0ξ+B3e-3λ0ξ).
Solving λ0 from the second equation of (5.402), we have
(5.412)λ0=Dη-(pλ1+p2λ2+···)|p=1,λi=0(i≥3)=Dη+AB6η2.Using in the[1,2] exponential Padé approximant [62], we have
(5.413)u=Aeλ0ξ+Be-λ0ξ1+aeλ0ξ+be-λ0ξ+ce2λ0ξ+de-2λ0ξ,
where the parameters A,B,a,b,c, and d can be identified by the exponential Padé approximant similar to the Padé approximant.The[m,n] exponential Padé approximant is defined as follows.Let(5.414)F=F0+εF1+ε2F2+⋯,ε→0.The [m,n] exponential Padé approximant reads
(5.415)Fmn(ε)=∑i=0mciexpdiε∑i=0naiexpbiε.
The coefficients ai,bi,ci,  and  di, are determined from the following condition: the first (2m+2n) components of the expansion of the rational function Fmn(ε) in a Maclaurin series coincide with the first (2m+2n) components of the series F(ε).


## 5.7.4. Couple Homotopy Perturbation Method with Laplace Transform [73]

Couple of the homotopy perturbation method with the Laplace transform makes the solution procedure much simpler.Consider the following foam drainage equation:(5.416)ut+2u2ux-ux2-12uxxu=0,
with initial conditions
(5.417)u(x,0)=-12+11+ex.
Apply Laplace transformation
(5.418)u(x,s)=1s(-12+11+ex)-1sL(2u2ux-ux2-12uxxu).
Apply inverse Laplace transformation
(5.419)u(x,t)=(-12+11+ex)-L-1(1sL(2u2ux-ux2-12uxxu)).
Apply the convex homotopy
(5.420)u0+pu1+p2u2+⋯=(-12+11+ex)-∫0tL-1(1sL(2(u0+pu1+⋯)2(∂u0∂x+p∂u1∂x+⋯)))dt+∫0tL-1(1sL(u0+pu1+⋯)x2+12(∂2u0∂x2+p∂2u1∂x2+⋯)(u0+pu1+⋯))dt.
Comparing the coefficient of like powers of p,
(5.421)p(0):u0(x,t)=(-12+11+ex),p(1):u1(x,t)=4ext(1+2ex)4+8e2xt(1+2ex)4-5ext(1+2ex)3-2e2xt(1+2ex)3+3ext2(1+2ex)2-ext216(1+2ex)3+e2xt8(1+2ex)3.⋮
The series solution is given by
(5.422)u(x,t)=(-12+11+ex)+4ext(1+2ex)4+8e2xt(1+2ex)4-5ext(1+2ex)3-2e2xt(1+2ex)3+3ext2(1+2ex)2-ext216(1+2ex)3+e2xt8(1+2ex)3⋯.
Applications of the homotopy perturbation method to solitary solutions can be referred to [74–77].


## 5.8. Parameter-Expansion Method

### 5.8.1. An Example

Slab detachment or break-off is appreciated as an important geological process. Ananalytical solution was given in [78] for the nonlinear dynamics of high amplitude necking in a free layer of power-law fluid extended in layer-parallel direction due to buoyancy stress.The power-law flow law for the layer can be written as(5.423)ε˙=Bτn,
where ε˙, rate of deformation parallel to the layer of a plane in the layer, τ is the mean layer-parallel deviatoric stress, and B and n are a material parameter and the stress exponent, respectively.Schmalholz obtained the following detachment time [78]:
(5.424)t=tcn=1nB((1/2)F)n,
where F is the layer-parallel force due to buoyancy at the top of the layer.The power-law fluid assumption, (5.423), is useful because of its simplicity but only approximately describes the behaviour of a real non-Newtonian fluid. We modify (5.423) in the form
(5.425)ε˙=B1τ+B2τn,
where B1 and B2 are material parameters.The governing equation for the dynamics of necking becomes(5.426)-1DdDdt=B1F2D+B2(F2D)n,D(0)=D0,
where D is the local thickness of the layer, and F is the layer-parallel force.Rewrite (5.426) in the form
(5.427)dDdt+A1+A2D1-n=0,D(0)=D0,
where A1=B1F/2, A2=B2(F/2)n.We use the parameter-expansion method (see Section5.8.2) to solve (5.427). We expand the solution in the form
(5.428)D=D-+pD~+p2D~~+⋯,
where D- is a reference thickness, D~ is a perturbation term, D~~ is an even smaller infinitesimal perturbation term, and p is a bookkeeping parameter, p=1 (see Section  5.8.2).The parametersA1 and A2 in (5.427) are expended in a similar way:
(5.429)A1=A-1+pA~1+p2A~~1+⋯,A2=p(A-2+pA~2+p2A~~2+⋯).
Substituting (5.428)-(5.429) into (5.427), collecting terms of the same power of p, and equating coefficients of like powers of  p yield a series of linear equations. We write only the equations for D- and D~, which read
(5.430)dD-dt+A-1=0,D-(0)=D0,(5.431)dD~dt+A~1+A-2D-1-n=0,D~(0)=0.
The solution of (5.430) reads
(5.432)D-=D0-A-1t.
Substituting the result into (5.431), we have
(5.433)dD~dt+A~1+A-2(D0-A-1t)1-n=0,D~(0)=0.
Its solution is
(5.434)D~=-A-2(D0)2-n(2-n)A-1-A~1t+A-2(D0-A-1t)2-n(2-n)A-1.
Note that D~ is a perturbation term. When t is small, it can be approximated as
(5.435)D~=-A-2(D0)2-n(2-n)A-1-A~1t+A-2(2-n)A-1(D02-n-(2-n)D01-nA-1t)=-(A~1-A-2D01-n)t.
We set
(5.436)A~1=A-2D01-n.
If the first-order approximate solution is enough, setting p=1 in (5.428)-(5.429) and equating higher-order perturbation terms to be zero, we have
(5.437)A2=A-2,A~1=A2D01-n,A-1=A1-A2D01-n,D=D-+D~=D0-A-1t+A2(D0)2-n(2-n)A-1-A~1t-A2(D0-A-1t)2-n(2-n)A-1.
For qualitative analysis, we use D- as an approximate solution
(5.438)D=D0-(A1-A2D01-n)t.
The detachment time is
(5.439)t=D0A1-A2D01-n=D0(B1F/2)-B2(F/2)nD01-n.
Equation (5.439) shows that the detachment time depends upon the initial layer thickness and fluid properties and the buoyancy.The nonlinear equation describing the dynamics of necking is of strong nonlinearity, and in order to solve it analytically, oversimple assumption has to be made. The power-law fluid is simple, but it is not a real non-Newtonian fluid. Recent development of analytical methods increases tantalizing possibility of theoretically seeking the approximate solutions to the present problem.


### 5.8.2. Parameter-Expansion Method for Solitary Solutions

Parameter-expansion method includes the modified Lindstedt-Poincare method [79] and bookkeeping parameter method [80]. In the review article [7], it was previously called parameter-expanding method. The method does not require to construct a homotopy. To illustrate its solution procedure, we reconsider (5.384), which is rewritten in the form
(5.440)∂u∂t+au∂u∂x+b∂3u∂x3+1·N(u)=0.
Supposing that the parameters, a, b, and 1, can be expressed in the forms
(5.441)a=a0+pa1+p2a2+⋯,b=b0+pb1+p2b2+⋯,1=pc1+p2c2+⋯,
where p is a bookkeeping parameter, p=1, substituting (5.441) into (5.440), proceeding the same way as the perturbation method, we can easily obtain the needed solution.We consider a simple mathematical model in the form(5.442)u′′+u2=0,u(0)=u(1)=0,u′′+0+1·u2=0.We seek an expansion of the form(5.443)u=u0+pu1+p2u2+⋯,
where the ellipsis dots stand for terms proportional to powers of p greater than 2, p is a bookkeeping parameter, p=1.The constants, 0 and 1, in the left-hand side of (5.442) can be, respectively, expanded in a similar way
(5.444)0=2a+a1p+a2p2+⋯,1=b1p+b2p2+⋯.
Substituting (5.444) to (5.442), we have
(5.445)(u0+pu1+p2u2+⋯)′′+(2a+a1p+a2p2+⋯)+(b1p+b2p2+⋯)·(u0+pu1+p2u2+⋯)2=0,
and equating coefficients of like powers of p we obtain the same equations as illustrated in previous examples:
(5.446)u0′′+2a=0,u0(0)=u0(1)=0,u1′′+b1u02+a1=0,u1(0)=u1(1)=0.
Solving u0 and u1, we have the following first-order approximate solution:
(5.447)u(t)=u0(t)+u1(t)=at(1-t)+at2-a2(130t6-110t5+112t4)-(a-160a2)t.
The discussion on how to determine the constant, a, is given in [69]. Applications of the parameter-expansion method to various nonlinear problems are available in [81–86].


## 5.8.1. An Example

Slab detachment or break-off is appreciated as an important geological process. Ananalytical solution was given in [78] for the nonlinear dynamics of high amplitude necking in a free layer of power-law fluid extended in layer-parallel direction due to buoyancy stress.The power-law flow law for the layer can be written as(5.423)ε˙=Bτn,
where ε˙, rate of deformation parallel to the layer of a plane in the layer, τ is the mean layer-parallel deviatoric stress, and B and n are a material parameter and the stress exponent, respectively.Schmalholz obtained the following detachment time [78]:
(5.424)t=tcn=1nB((1/2)F)n,
where F is the layer-parallel force due to buoyancy at the top of the layer.The power-law fluid assumption, (5.423), is useful because of its simplicity but only approximately describes the behaviour of a real non-Newtonian fluid. We modify (5.423) in the form
(5.425)ε˙=B1τ+B2τn,
where B1 and B2 are material parameters.The governing equation for the dynamics of necking becomes(5.426)-1DdDdt=B1F2D+B2(F2D)n,D(0)=D0,
where D is the local thickness of the layer, and F is the layer-parallel force.Rewrite (5.426) in the form
(5.427)dDdt+A1+A2D1-n=0,D(0)=D0,
where A1=B1F/2, A2=B2(F/2)n.We use the parameter-expansion method (see Section5.8.2) to solve (5.427). We expand the solution in the form
(5.428)D=D-+pD~+p2D~~+⋯,
where D- is a reference thickness, D~ is a perturbation term, D~~ is an even smaller infinitesimal perturbation term, and p is a bookkeeping parameter, p=1 (see Section  5.8.2).The parametersA1 and A2 in (5.427) are expended in a similar way:
(5.429)A1=A-1+pA~1+p2A~~1+⋯,A2=p(A-2+pA~2+p2A~~2+⋯).
Substituting (5.428)-(5.429) into (5.427), collecting terms of the same power of p, and equating coefficients of like powers of  p yield a series of linear equations. We write only the equations for D- and D~, which read
(5.430)dD-dt+A-1=0,D-(0)=D0,(5.431)dD~dt+A~1+A-2D-1-n=0,D~(0)=0.
The solution of (5.430) reads
(5.432)D-=D0-A-1t.
Substituting the result into (5.431), we have
(5.433)dD~dt+A~1+A-2(D0-A-1t)1-n=0,D~(0)=0.
Its solution is
(5.434)D~=-A-2(D0)2-n(2-n)A-1-A~1t+A-2(D0-A-1t)2-n(2-n)A-1.
Note that D~ is a perturbation term. When t is small, it can be approximated as
(5.435)D~=-A-2(D0)2-n(2-n)A-1-A~1t+A-2(2-n)A-1(D02-n-(2-n)D01-nA-1t)=-(A~1-A-2D01-n)t.
We set
(5.436)A~1=A-2D01-n.
If the first-order approximate solution is enough, setting p=1 in (5.428)-(5.429) and equating higher-order perturbation terms to be zero, we have
(5.437)A2=A-2,A~1=A2D01-n,A-1=A1-A2D01-n,D=D-+D~=D0-A-1t+A2(D0)2-n(2-n)A-1-A~1t-A2(D0-A-1t)2-n(2-n)A-1.
For qualitative analysis, we use D- as an approximate solution
(5.438)D=D0-(A1-A2D01-n)t.
The detachment time is
(5.439)t=D0A1-A2D01-n=D0(B1F/2)-B2(F/2)nD01-n.
Equation (5.439) shows that the detachment time depends upon the initial layer thickness and fluid properties and the buoyancy.The nonlinear equation describing the dynamics of necking is of strong nonlinearity, and in order to solve it analytically, oversimple assumption has to be made. The power-law fluid is simple, but it is not a real non-Newtonian fluid. Recent development of analytical methods increases tantalizing possibility of theoretically seeking the approximate solutions to the present problem.


## 5.8.2. Parameter-Expansion Method for Solitary Solutions

Parameter-expansion method includes the modified Lindstedt-Poincare method [79] and bookkeeping parameter method [80]. In the review article [7], it was previously called parameter-expanding method. The method does not require to construct a homotopy. To illustrate its solution procedure, we reconsider (5.384), which is rewritten in the form
(5.440)∂u∂t+au∂u∂x+b∂3u∂x3+1·N(u)=0.
Supposing that the parameters, a, b, and 1, can be expressed in the forms
(5.441)a=a0+pa1+p2a2+⋯,b=b0+pb1+p2b2+⋯,1=pc1+p2c2+⋯,
where p is a bookkeeping parameter, p=1, substituting (5.441) into (5.440), proceeding the same way as the perturbation method, we can easily obtain the needed solution.We consider a simple mathematical model in the form(5.442)u′′+u2=0,u(0)=u(1)=0,u′′+0+1·u2=0.We seek an expansion of the form(5.443)u=u0+pu1+p2u2+⋯,
where the ellipsis dots stand for terms proportional to powers of p greater than 2, p is a bookkeeping parameter, p=1.The constants, 0 and 1, in the left-hand side of (5.442) can be, respectively, expanded in a similar way
(5.444)0=2a+a1p+a2p2+⋯,1=b1p+b2p2+⋯.
Substituting (5.444) to (5.442), we have
(5.445)(u0+pu1+p2u2+⋯)′′+(2a+a1p+a2p2+⋯)+(b1p+b2p2+⋯)·(u0+pu1+p2u2+⋯)2=0,
and equating coefficients of like powers of p we obtain the same equations as illustrated in previous examples:
(5.446)u0′′+2a=0,u0(0)=u0(1)=0,u1′′+b1u02+a1=0,u1(0)=u1(1)=0.
Solving u0 and u1, we have the following first-order approximate solution:
(5.447)u(t)=u0(t)+u1(t)=at(1-t)+at2-a2(130t6-110t5+112t4)-(a-160a2)t.
The discussion on how to determine the constant, a, is given in [69]. Applications of the parameter-expansion method to various nonlinear problems are available in [81–86].


## 5.9. Exp-Function Method

### 5.9.1. Exp-Function Method for Solitary Solutions

Exp-function method [87–89] provides us with a straightforward and concise approach for obtaining generalized solitary solutions and periodic solutions; the solution procedure, by the help of Matlab or Mathematica, is of utter simplicity, see [90–97]. Considering a general nonlinear partial differential equation in the form
(5.448)F(u,ux,uy,uz,ut,uxx,uyy,uzz,utt,uxy,uxt,uyt,…)=0.
The exp-function method assumes that the solitary solution can be expressed in the form
(5.449)u(x,y,z,t)=∑n=-klpnexp(anx+bny+cnz+dnt)∑m=-ijqmexp(amx+bmy+cmz+dmt).
We can also introduce a transformation
(5.450)ξ=ax+by+cz+dt.
We can rewrite (5.448) in the following nonlinear ordinary differential equation:
(5.451)F(u,u′,u′′,u′′′,…)=0,
where the prime denotes the derivation with respect to ξ.According to the exp-function method, the traveling wave solutions can be expressed in the form(5.452)u(ξ)=∑n=-klanexp(nξ)∑m=-ijbmexp(mξ),
where i, j, k, and l are positive integers which could be freely chosen; an and bm are unknown constants to be determined.Consider a fractional Benjamin-Bona-Mahony (BBM) equation [97]
(5.453)∂u∂t-u∂u∂x-∂∂t∂2u∂x2=0,
using a transformation
(5.454)u(x,t)=U(ξ),ξ=kx+wt,
(5.453) becomes an ordinary differential equation,
(5.455)wU′-kUU′-k2wU′′′=0.
By the exp-function method, we assume that the solution of (5.460) can be expressed as
(5.456)U(ξ)=a1exp(ξ)+a0+a-1exp(-ξ)b1exp(ξ)+b0+b-1exp(-ξ),
where ai,bi(i=1,0,-1) are unknown parameters.Substituting (5.456) into (5.455), collecting terms of the same term of exp(iξ), we have
(5.457)1A[C7exp(7ξ)+C6exp(6ξ)+C5exp(5ξ)+C4exp(4ξ)+C3exp(3ξ)+C2exp(2ξ)+C1exp(ξ)]=0.Equating coefficients,Ci(i=1~7), to zero yields a series of linear equations
(5.458)Ci=0(i=1~7).
Solving the system of algebraic equation, we can identify parameters in (5.456). Finally, we obtain [97]
(5.459)u(x,t)=-w(k2-1)k+6k2b0b1exp[(kx+wt)]+b0+(b02/4b1)exp[-(kx+wt)],
where b0 and b1 are free parameters.Byk=iK, w=iW, the obtained solitary solution can be converted into periodic solution.Other expressions are listed as follows:(5.460)u(x,t)=∑nanexp(qnx+pnt)∑mbmexp(αmx+βmt),
where ai,biαi,βiqi,  and  pi are unknown constants to be further determined.For a two-wave solution, we can use the double exp-function method [8], which requires
(5.461)u=a-1exp(-ξ)+b1exp(ξ)+a0+b-1exp(-η)+b1exp(η)c-1exp(-ξ)+c1exp(ξ)+b0+d-1exp(-η)+d1exp(η),
where ξ=a1x+b1y+c1z-d1t and η=a2x+b2y+c2z-d2t.


### 5.9.2. The Exp-Function Method for Periodic Solutions

The French mathematician J. Fourier (1768–1830) showed that any periodic motion can be represented by a series of sines and cosines that are harmonically related(5.462)u(t)=12a0+a1cosω1t+a2cosω2t+⋯+b1sinω1t+b2sinω2t+⋯,
where ωn=nω1, and an and bn are, respectively, expressed as
(5.463)an=2T∫-T/2T/2u(t)cosωntdt,bn=2T∫-T/2T/2u(t)sinωntdt.
Using Euler’s formula, we have
(5.464)sint=12(eit+e-it),cost=-12i(eit-e-it).
We, therefore, can rewrite the Fourier series, (5.462), in the form
(5.465)u(t)=∑n=-∞∞cneiωnt,
where c0=(1/2)a0 and cn=(1/2)(an-ibn).For periodic solution, we can assume that the solution can be expressed in the form(5.466)u(t)=∑n=-cdpnexp(iωnt)∑m=-fgqmexp(iωmt),
or
(5.467)u(t)=∑0Ncneiωnt.


## 5.9.1. Exp-Function Method for Solitary Solutions

Exp-function method [87–89] provides us with a straightforward and concise approach for obtaining generalized solitary solutions and periodic solutions; the solution procedure, by the help of Matlab or Mathematica, is of utter simplicity, see [90–97]. Considering a general nonlinear partial differential equation in the form
(5.448)F(u,ux,uy,uz,ut,uxx,uyy,uzz,utt,uxy,uxt,uyt,…)=0.
The exp-function method assumes that the solitary solution can be expressed in the form
(5.449)u(x,y,z,t)=∑n=-klpnexp(anx+bny+cnz+dnt)∑m=-ijqmexp(amx+bmy+cmz+dmt).
We can also introduce a transformation
(5.450)ξ=ax+by+cz+dt.
We can rewrite (5.448) in the following nonlinear ordinary differential equation:
(5.451)F(u,u′,u′′,u′′′,…)=0,
where the prime denotes the derivation with respect to ξ.According to the exp-function method, the traveling wave solutions can be expressed in the form(5.452)u(ξ)=∑n=-klanexp(nξ)∑m=-ijbmexp(mξ),
where i, j, k, and l are positive integers which could be freely chosen; an and bm are unknown constants to be determined.Consider a fractional Benjamin-Bona-Mahony (BBM) equation [97]
(5.453)∂u∂t-u∂u∂x-∂∂t∂2u∂x2=0,
using a transformation
(5.454)u(x,t)=U(ξ),ξ=kx+wt,
(5.453) becomes an ordinary differential equation,
(5.455)wU′-kUU′-k2wU′′′=0.
By the exp-function method, we assume that the solution of (5.460) can be expressed as
(5.456)U(ξ)=a1exp(ξ)+a0+a-1exp(-ξ)b1exp(ξ)+b0+b-1exp(-ξ),
where ai,bi(i=1,0,-1) are unknown parameters.Substituting (5.456) into (5.455), collecting terms of the same term of exp(iξ), we have
(5.457)1A[C7exp(7ξ)+C6exp(6ξ)+C5exp(5ξ)+C4exp(4ξ)+C3exp(3ξ)+C2exp(2ξ)+C1exp(ξ)]=0.Equating coefficients,Ci(i=1~7), to zero yields a series of linear equations
(5.458)Ci=0(i=1~7).
Solving the system of algebraic equation, we can identify parameters in (5.456). Finally, we obtain [97]
(5.459)u(x,t)=-w(k2-1)k+6k2b0b1exp[(kx+wt)]+b0+(b02/4b1)exp[-(kx+wt)],
where b0 and b1 are free parameters.Byk=iK, w=iW, the obtained solitary solution can be converted into periodic solution.Other expressions are listed as follows:(5.460)u(x,t)=∑nanexp(qnx+pnt)∑mbmexp(αmx+βmt),
where ai,biαi,βiqi,  and  pi are unknown constants to be further determined.For a two-wave solution, we can use the double exp-function method [8], which requires
(5.461)u=a-1exp(-ξ)+b1exp(ξ)+a0+b-1exp(-η)+b1exp(η)c-1exp(-ξ)+c1exp(ξ)+b0+d-1exp(-η)+d1exp(η),
where ξ=a1x+b1y+c1z-d1t and η=a2x+b2y+c2z-d2t.


## 5.9.2. The Exp-Function Method for Periodic Solutions

The French mathematician J. Fourier (1768–1830) showed that any periodic motion can be represented by a series of sines and cosines that are harmonically related(5.462)u(t)=12a0+a1cosω1t+a2cosω2t+⋯+b1sinω1t+b2sinω2t+⋯,
where ωn=nω1, and an and bn are, respectively, expressed as
(5.463)an=2T∫-T/2T/2u(t)cosωntdt,bn=2T∫-T/2T/2u(t)sinωntdt.
Using Euler’s formula, we have
(5.464)sint=12(eit+e-it),cost=-12i(eit-e-it).
We, therefore, can rewrite the Fourier series, (5.462), in the form
(5.465)u(t)=∑n=-∞∞cneiωnt,
where c0=(1/2)a0 and cn=(1/2)(an-ibn).For periodic solution, we can assume that the solution can be expressed in the form(5.466)u(t)=∑n=-cdpnexp(iωnt)∑m=-fgqmexp(iωmt),
or
(5.467)u(t)=∑0Ncneiωnt.


## 5.10. Subequation Method

There are many subequation methods in the open literature for solitary solution [98–100]. We consider a general nonlinear equation in the form
(5.468)F(u,ut,ux,uy,uz,uxx,uyy,uzz,…)=0.
The sub-equation method follows the following steps.Step 1.
Letu=u(x,y,z,t)=u(ξ), where ξ=ax+by+cz+dt+ξ0. Equation (5.468) turns out to be an ordinary differential equation
(5.469)F(u,uξ,uξξ,uξξξ,…)=0.Step 2.
Assume that the solution can be expressed in the form(5.470)u=∑nanfn(ξ),
or in a more general form,
(5.471)u=U(f(ξ)),
where f is the solution of an auxiliary equation
(5.472)f'2=b0+b1f+b2f2+b3f3+b4f4+⋯.
The exact solution of (5.472) for fixed bi(i=0,1,2,3,…) is known.Step 3.
Use some mathematical software to determinean in (5.470).The auxiliary equation can be constructed using the Hamiltonian invariant as discussed in Section5.3.5. For (5.128), and the following auxiliary equation can be constructed:
(5.473)u′=cu2+2u3.
For (5.227), the following auxiliary equation should be in the form:
(5.474)u′=cu2-12(n+1)(n+2)un+2.
In such case, the solution of the auxiliary equation is the special solution of the problem.


## 5.11. Ancient Chinese Mathematics

### 5.11.1. Introduction to the Ancient Chinese Mathematics

Can ancient Chinese mathematics still work for nonlinear science? Yes, it can. It is simple but effective.To illustrate the basic idea of the ancient Chinese method, we consider an algebraic equation:(5.475)f(x)=0.Letx1 and x2 be the approximate solutions of the equation, which lead to the remainders f(x1) and f(x2), respectively; the ancient Chinese algorithm leads to the result
(5.476)x=x2f(x1)-x1f(x2)f(x1)-f(x2).Now we consider a generalized nonlinear oscillator in the form(5.477)u′′+f(u)=0,u(0)=A,u′(0)=0.
We use two trial functions u1(t)=Acosω1t and u2=Acosω2t, which are, respectively, the solutions of the following linear oscillator equations:
(5.478)u′′+ω12u=0,u′′+ω22u=0.
The residuals are
(5.479)R1(ω1t)=-ω12cosω1t+f(Acosω1t),R2(ω2t)=-ω22cosω2t+f(Acosω2t).
The residuals depend upon t, and in our previous applications [7], we just use R1(0) and R2(0) for simplicity when we use the ancient Chinese method, resulting in relatively low accuracy [7]
(5.480)ω2=ω12R2(0)-ω22R1(0)R2(0)-R1(0).
Equation (5.480) is called He’s frequency formulation or He’s frequency-amplitude formulation in the literature [101–106].In order to improve its accuracy, we introduce two new residual variablesR~1 and R~2 defined as [105, 106]
(5.481)R~1=4T1∫0T1/4R1(t)cos(2πT1t)dt,(5.482)R~2=4T2∫0T2/4R2(t)cos(2πT2t)dt.
According to the ancient Chinese method, we can approximately determine ω2 in the form
(5.483)ω2=ω12R~2-ω22R~1R~2-R~1.
In this letter, we will use the well-known Duffing equation.Using trial functionsu1(t)=Acost and u2=Acosω2t, respectively, for (5.481), we obtain the following residuals:
(5.484)R1(t)=εA3cos3t,R2(t)=A(1-ω22)cosω2t+εA3cos3ω2t,
where ω2>0 and ω2≠1.By simple calculation, we obtain(5.485)R~1=4T1∫0T1/4R1(t)cos(2πT1t)dt=2π∫0π/2εA3cos4tdt=34π,R~2=4T2∫0T2/4R2(t)cos(2πT2t)dt=2ωπ∫0T2/4{A(1-ω22)cos2ω2t+εA3cos4ω2t}dt=2π∫0π/2{A(1-ω22)cos2s+εA3cos4s}ds=A(1-ω22)1π+34πεA3.
Applying (5.483), we have
(5.486)ω2=ω12R~2-ω22R~1R~2-R~1=A(1-ω22)(1/π)+(3/4π)εA3-(3/4π)εA3ω22A(1-ω22)(1/π)+(3/4π)εA3-(3/4π)εA3=(1-ω22)+(3/4)εA3-(3/4)εA3ω22(1-ω22)=1+34εA2.


### 5.11.2. Ancient Chinese Mathematics for Optimization [107]

To elucidate the basic idea of application of the ancient Chinese mathematics to optimal problems, we consider first the following simple example:(5.487)f=xy→max,(5.488)s.t.g(x,y)=x+y-1=0.To make the Lagrange functionf  maximal, it requires
(5.489)df=xdy+ydx=0,
where dx and dy are not independent, and according to the constraint equation, (5.490), we have
(5.490)dx+dy=0.
Considering the relationship between dx and dy, (5.492), (5.491) becomes
(5.491)df=(x-y)dy=0.
Due to the arbitrary dy in (5.491), we have the following Euler equation:
(5.492)x-y=0.
Solving (5.488) and (5.492) simultaneously, we find the optimal values
(5.493)x*=y*=12.A general solution procedure is given in [8], and the stationary condition (Euler equation) can be simply obtained using He’s brackets [8, 108]:
(5.494)〈f,g〉x,y=fxgy-fygx=0,
where fx=∂f/∂x=y, fy=∂f/∂y=x, gx=∂g/∂x=1, and gy=∂g/∂y=1.Equation (5.494) is exactly the same with (5.492).According to the ancient Chinese mathematics, to solve a problem, we should choose double trial solutions,x1 and x2. We can choose x1 arbitrarily, for example, x1=2/7, according to the constraint equation (5.488), and we have y1=5/7.From (5.487), we obtain
(5.495)f1=x1y1=1049.
The condition (5.489) can be approximately expressed as
(5.496)Δf=xΔy+yΔx=0,
where Δx=x2-x1, Δy=y2-y1, and Δf=f2-f1.Equation (5.496) implies that
(5.497)f2=f1.
This means that
(5.498)f2=x2y2=1049.
Considering the constraint equation, (5.488), again, we have
(5.499)x2+y2=1.
From (5.498) and (5.499), we can obtain immediately x2=5/7 and y2=2/7.Substitutingx1, x2, y1, and y2 into (5.498), we have
(5.500)Δf=x(y2-y1)+y(x2-x1)=x(27-57)+y(57-27)=0.
Solving (5.488) and (5.500) simultaneously, we find the optimal values given in (5.493).Example 5.1.
Condiser the following [109]:
(5.501)f=x2+y2→min,(5.502)s.t.g(x,y)=y-2-x2=0.
To make (5.501) minimal, it requires that
(5.503)Δf=2xΔx+2yΔy2x2+y2=xΔx+yΔyx2+y2=0.
Choosing x1=1, we have y1=3 and f1=10. We set f2=10, which means that
(5.504)x22+y22=10,(5.505)y2=2+x22.
Solving (5.504) and (5.505) simultaneously, we have x2=-1 and y2=3.
Substitutingx1, x2, y1, and y2 into (5.503), we obtain
(5.506)Δf=x(x2-x1)+y(y2-y1)x2+y2=-2xx2+y2=0.
We, therefore, obtain the optimal valuesx*=0 and y*=2.Example 5.2.
Consider the following problem [109]:
(5.507)f(x,y)=x2+y2→min,(5.508)s.t.y2-(x-1)3=0.
The Lagrange multiplier method becomes invalid for this example [109]. Using the basic idea of the ancient Chinese mathematics, we choose x1=2, y1=1 and x2=2, y2=-1, and both cases lead to f1=f2=5.
Making (5.507) stationary requires
(5.509)Δf=xΔx+yΔyx2+y2=x(x2-x1)+y(y2-y1)x2+y2=-2yx2+y2=0,
which leads to immediately y*=0 and x*=1 if the constraint equation, (5.508), is considered.


### 5.11.3. Ancient Chinese Mathematics for Nonlinear Wave Equations [110]

Now we consider the nonlinear Schrödinger equation(5.510)iut+uxx+2|u|2u=0,u(x,0)=eikx.
We use the trial functions
(5.511)u1(x,t)=ei(kx+t),u2(x,t)=ei(kx+ωt).
The residuals are
(5.512)R1(x,t)=(1-k2)ei(kx+t),R2(x,t)=(2-ω-k2)ei(kx+ωt).
According to frequency-amplitude formulation, (5.480), we have
(5.513)ω2=ω12R2(x,0)-ω22R1(x,0)R2(x,0)-R1(x,0)=-(ω+ωk2-2ω+k2-2).We get the equation aboutω, and the solution is
(5.514)ω=2-k2,u(x,t)=ei{kx+(2-k2)ωt}.
Now we consider the Schrödinger equation with power law nonlinearity
(5.515)iut+uxx-2|u|2ru=0,u(x,0)=(2(r+1)sech2(2rx))1/2r,r≥1.
Following the analysis presented above, we choose the trial functions
(5.516)u1(x,t)=(2(r+1)sech2(2rx))1/2reit,u2(x,t)=(2(r+1)sech2(2rx))1/2reiωt.
The residuals are
(5.517)R1(x,t)=-(2(r+1)(sech(2rx))2)1/2reit+4(2(r+1)(sech(2rx))2)1/2r(tanh(2rx))2eit-4(2(r+1)(sech(2rx))2)1/2r(1-(tanh(2rx))2)reit+2(2(r+1)(sech(2rx))2)1/2r-1eit,R2(x,t)=-(2(r+1)(sech(2rx))2)1/2reiωtω+4(2(r+1)(sech(2rx))2)1/2r(tanh(2rx))2eiωt-4(2(r+1)(sech(2rx))2)1/2r(1-(tanh(2rx))2)reiωt+2(2(r+1)(sech(2rx))2)1/2reiωt.
According to the frequency-amplitude formulation, (5.480), setting x=0 and t=0,  we have
(5.518)ω2=-4r+2ω|(21/2r(r+1)1/2r)2r|-ω-4ωr+|(21/2r(r+1)1/2r)2r|.
Solving ω from the above equation yields
(5.519)ω=-4r+2|(21/2r(r+1)1/2r)2r|.
We, therefore, obtain the solution
(5.520)u2(x,t)=(2(r+1)sech2(2rx))1/2reiωt.


### 5.11.4. Solitary-Solution Formulation for Nonlinear Equations

Now consider a general nonlinear equation(5.521)L(u)+N(u)=0,
where L is a linear operator, and N is a nonlinear operator.We choose two trial equations(5.522)L(u)=0,L~(u)=0,
where L~ is a linearized linear operator or a simple nonlinear operator, and its solution L~(u)=0 is known. For example, we consider a nonlinear oscillator: u′′+u3=0, and we can choose L(u)=u′′+u=0 and L~(u)=u′′+ω2u=0, where ω is the frequency to be searched for.Substituting the solutions of (5.522) with same boundary/initial conditions as (5.521) into (5.521), we obtain, respectively, the residuals  R1 and R2.For traveling wave solutions, we can make a transformation(5.523)ξ=ax+by+cz+ωt.
The solitary-solution formulation can be expressed as
(5.524)ω2=R1ω12-R2ω22R1-R2,
or
(5.525)ωn=R1ω1n-R2ω2nR1-R2.The KdV equation is a nonlinear, dispersive partial differential equation for a function of two real variables, spacex and time t:
(5.526)ut+6uux+uxxx=0.
According to the solitary-solution formulation (5.524), we construct two trial functions in the following forms:
(5.527)u1(x,t)=Aexp(kx+t+D)+exp[-(kx+t+D)]+B,u2(x,t)=Aexp(kx+ωt+D)+exp[-(kx+ωt+D)]+B,
where A, B, D are arbitrary constants.Substitutingu1 and u2 into (5.526), respectively, we obtain the following residuals:
(5.528)R1(x,t)=-A{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}2-6A2k{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}3-Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}2-6Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}3{exp(kx+t+D)+exp[-(kx+t+D)]+B}4+6Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}3,R2(x,t)=-Aω{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}2-6A2k{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}3-Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}2-6Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}3{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}4+6Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}3.
Using the solitary-solution formulation (5.480), and choosing the location point (0,0), we obtain
(5.529)ω2=ω12R2(0,0)-ω22R1(0,0)R2(0,0)-R1(0,0).
Simplifying (5.529) results in
(5.530)ω2+ω+(ω+1)k3+[2Bω2+2ωB+(ω+1)(6kA-4k3B)]exp(-D)+[2ω2+B2ω2+2ω+ωB2+(ω+1)(6kAB-22k3+k3B2)]exp(-2D)+[2Bω2+2ωB+(ω+1)(6kA-4k3B)]exp(-3D)+[ω2+ω+(ω+1)k3]exp(-4D)=0.
Setting the coefficients of exp(-nD) (n=0,1,2,3,4) to be zero, we obtain
(5.531)ω2+ω+(ω+1)k3=0,(5.532)2Bω2+2ωB+(ω+1)(6kA-4k3B)=0,(5.533)2ω2+B2ω2+2ω+ωB2+(ω+1)(6kAB-22k3+k3B2)=0.
Solving (5.531)~(5.533) simultaneously, we obtain the following relationships:(1)
the relationship between solitary wavenumber and solitary frequency:(5.534)ω=-k3,(2)
the relationship between solitary amplitude and solitary wavenumber:(5.535)A=k2B,(3)
the morphofactor:(5.536)B2=4, so the solitary wave solution can be readily obtained, which reads
(5.537)u(x,t)=2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]+2,
or
(5.538)u(x,t)=-2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]-2,
which are, by chance, exact solutions.


### 5.11.5. Ancient Chinese Inequality and Application

In an ancient history book, it is write (see [111]):He Chengtian uses 26/49 as the strong, and 9/17 as the weak. Among the strong and the weak, Chengtian tries to find a more accurate denominator of the fractional day of the Moon. Chengtian obtains 752 as the denominator by using the 15 and 1 as weighting factors, respectively, for the strong and the weak. No other calendar can reach such a high accuracy after Chengtian, who uses heuristically the strong and weak weighting factors.The statement is rather cryptic, and in modern mathematical term, the statement can be explained as follows.According to the observation data, He Chengtian (369?~447AD) finds that(5.539)292649days>1Moon>29917days.
Using the weighting factors (15 and 1), He Chengtian obtains
(5.540)Thefractionalday=26×15+9×149×15+17×1=399752,
so
(5.541)1Moon=29399752days.
He Chengtian actually uses the following inequality.If(5.542)ab<x<dc,
where a,b,c, and d are real numbers, then
(5.543)ab<ma+ndmb+nc<dc,
and x is approximated by
(5.544)x=ma+ndmb+nc,
where m and n are weighting factors.The above inequity was developed by the max-min approach to nonlinear oscillators [112]. Hereby we use the inequity for a chemical problem.A mass balance on a differential volume element of porous medium for a spherical catalyst pellet gives(5.545)∂c′∂t=∇·De∇c′-RA,
where c′ is the chemical reactant concentration, -RA is the rate of reaction per unit volume, and De  is the effective diffusion coefficient for reactant.Assuming the coupled process of diffusion and reaction is steady, we reduce (5.545) to the form
(5.546)De(∂2c′∂r2+2r∂c′∂r)=RA.
The boundary conditions are ∂c′/∂r=0,r=0 (center of catalyst), and c′=cs,r=r0 (surface of catalyst).Introducing dimensionless variablesC=c′/cs and R=r/r0, we have the following nonlinear equation for nth-order and irreversible reaction at isothermal condition:
(5.547)∂2C∂R2+2R∂C∂R=ϕ2Cn,∂C∂R(0)=0,C(1)=1,
where ϕ is the Thiele modulus, defined as ϕ=r02kνcsn-1/De, where  kν is the reaction constant.Equation (5.547) can also be written in Cartesian geometry, and it is easy to transform Cartesian geometry to cylindrical or spherical geometry.To investigate the nature of the solution of (5.547), consider the analogous linear equation
(5.548)∂2C∂R2+2R∂C∂R=k,∂C∂R(0)=0,C(1)=1.MultiplyingR2 on both sides of (5.548), we have
(5.549)∂∂R(R2∂C∂R)=kR2,∂C∂R(0)=0,C(1)=1.Its solution is(5.550)C=1+16k(R2-1).Now we rewrite (5.544) in the form
(5.551)∂2C∂R2+2R∂C∂R=g(C),
where g(C)=ϕ2Cn.The left side of (5.551) is analogous to k in (5.548).Observe that0<C<1, so C is never less than the solution of the following initial value problem:
(5.552)∂2C∂R2+2R∂C∂R=ϕ2Cmaxn=ϕ2.The solution of (5.552) reads
(5.553)C=1+16ϕ2(R2-1).
So it follows that (see Figure 4)
(5.554)6+ϕ2(R2-1)6<C<1.
According to He Chengtian’s interpolation, we have
(5.555)C=6p+pϕ2(R2-1)+q6p+q,
or
(5.556)C=1+ξϕ2(R2-1),
where p and q are weighting factors, ξ=p/(6p+q). The free parameter, ξ, in (5.556) can be identified via various methods. Hereby we illustrate a simple approach by the residual method. Substituting (5.556) into (5.544), we obtain the following residual equation:
(5.557)Π(R,ξ)=∂2C∂R2+2R∂C∂R-ϕ2Cn=2ξϕ2+4ξϕ2-ϕ2[1+ξϕ2(R2-1)]n.Figure 4
Qualitative sketch for the diffusion reaction in spherical porous catalyst. The dashed line presents an initial guessC=1+ϕ2(R2-1)/6, and the continued line presents the real solution.Chemists and technologists always want to have an accurate halfway through the change. Assume that the above residual vanishes completely atC=1/2, that is, 1+ξϕ2(R2-1)=1/2, then we have
(5.558)Π(R,ξ)|R=R*=6ξϕ2-12nϕ2,
where R=R* is the root of the equation 1+ξϕ2(R2-1)=1/2. From (5.558), the unknown ξ can be identified as
(5.559)ξ=2-n6.Finally, we obtain the following design formulation:(5.560)C(R)=1+2-n6ϕ2(R2-1).Comparison with exact solution is illustrated in Figure5.Figure 5
Comparison between exact solution and the approximate solution (n=1, ϕ=1). The dashed line presents an approximate solution, and the continued line presents the exact solution. The maximal error is about 7.8%.In order to identify the weighting factors,p and q, in (5.555), we proceed the same manipulation as illustrated above. Substituting (5.555) into (5.544), we obtain a residual equation:
(5.561)Π(R,p,q)=∂2C∂R2+2R∂C∂R-ϕ2Cn=2pϕ26p+q+4pϕ26p+q-ϕ2(6p+pϕ2(R2-1)+q6p+q)n=0.
Locating at R=0 and R=1/2, we obtain
(5.562)Π(0,p,q)=2pϕ26p+q+4pϕ26p+q-ϕ2(6p-pϕ2+q6p+q)n=0,(5.563)Π(12,p,q)=2pϕ26p+q+4pϕ26p+q-ϕ2(6p-(3/4)pϕ2+q6p+q)n=0.
Solving (5.562) and (5.563) simultaneously, the values of p and q can be readily determined.The accuracy of vanishing residuals depends upon the trial functions, (5.555) and (5.556), which are derived from an analogous linear equation, so the obtained solutions are valid for the whole solution domain, keeping maximal accuracy at the location points.The preceding analysis has the virtue of utter simplicity, and the illustrating example shows that the suggested method is very effective and convenient in solving nonlinear equations. The suggested method can be readily applied to the search for analytical solutions for various nonlinear equations, which should have the character of monotonic increase or monotonic decrease. Many nonlinear problems arising in chemical engineering can be described by, for example, exponential decay/rise functions, so the ancient Chinese method—the He Chengtian interpolation—might be a powerful mathematical tool in chemical engineering.


## 5.11.1. Introduction to the Ancient Chinese Mathematics

Can ancient Chinese mathematics still work for nonlinear science? Yes, it can. It is simple but effective.To illustrate the basic idea of the ancient Chinese method, we consider an algebraic equation:(5.475)f(x)=0.Letx1 and x2 be the approximate solutions of the equation, which lead to the remainders f(x1) and f(x2), respectively; the ancient Chinese algorithm leads to the result
(5.476)x=x2f(x1)-x1f(x2)f(x1)-f(x2).Now we consider a generalized nonlinear oscillator in the form(5.477)u′′+f(u)=0,u(0)=A,u′(0)=0.
We use two trial functions u1(t)=Acosω1t and u2=Acosω2t, which are, respectively, the solutions of the following linear oscillator equations:
(5.478)u′′+ω12u=0,u′′+ω22u=0.
The residuals are
(5.479)R1(ω1t)=-ω12cosω1t+f(Acosω1t),R2(ω2t)=-ω22cosω2t+f(Acosω2t).
The residuals depend upon t, and in our previous applications [7], we just use R1(0) and R2(0) for simplicity when we use the ancient Chinese method, resulting in relatively low accuracy [7]
(5.480)ω2=ω12R2(0)-ω22R1(0)R2(0)-R1(0).
Equation (5.480) is called He’s frequency formulation or He’s frequency-amplitude formulation in the literature [101–106].In order to improve its accuracy, we introduce two new residual variablesR~1 and R~2 defined as [105, 106]
(5.481)R~1=4T1∫0T1/4R1(t)cos(2πT1t)dt,(5.482)R~2=4T2∫0T2/4R2(t)cos(2πT2t)dt.
According to the ancient Chinese method, we can approximately determine ω2 in the form
(5.483)ω2=ω12R~2-ω22R~1R~2-R~1.
In this letter, we will use the well-known Duffing equation.Using trial functionsu1(t)=Acost and u2=Acosω2t, respectively, for (5.481), we obtain the following residuals:
(5.484)R1(t)=εA3cos3t,R2(t)=A(1-ω22)cosω2t+εA3cos3ω2t,
where ω2>0 and ω2≠1.By simple calculation, we obtain(5.485)R~1=4T1∫0T1/4R1(t)cos(2πT1t)dt=2π∫0π/2εA3cos4tdt=34π,R~2=4T2∫0T2/4R2(t)cos(2πT2t)dt=2ωπ∫0T2/4{A(1-ω22)cos2ω2t+εA3cos4ω2t}dt=2π∫0π/2{A(1-ω22)cos2s+εA3cos4s}ds=A(1-ω22)1π+34πεA3.
Applying (5.483), we have
(5.486)ω2=ω12R~2-ω22R~1R~2-R~1=A(1-ω22)(1/π)+(3/4π)εA3-(3/4π)εA3ω22A(1-ω22)(1/π)+(3/4π)εA3-(3/4π)εA3=(1-ω22)+(3/4)εA3-(3/4)εA3ω22(1-ω22)=1+34εA2.


## 5.11.2. Ancient Chinese Mathematics for Optimization [107]

To elucidate the basic idea of application of the ancient Chinese mathematics to optimal problems, we consider first the following simple example:(5.487)f=xy→max,(5.488)s.t.g(x,y)=x+y-1=0.To make the Lagrange functionf  maximal, it requires
(5.489)df=xdy+ydx=0,
where dx and dy are not independent, and according to the constraint equation, (5.490), we have
(5.490)dx+dy=0.
Considering the relationship between dx and dy, (5.492), (5.491) becomes
(5.491)df=(x-y)dy=0.
Due to the arbitrary dy in (5.491), we have the following Euler equation:
(5.492)x-y=0.
Solving (5.488) and (5.492) simultaneously, we find the optimal values
(5.493)x*=y*=12.A general solution procedure is given in [8], and the stationary condition (Euler equation) can be simply obtained using He’s brackets [8, 108]:
(5.494)〈f,g〉x,y=fxgy-fygx=0,
where fx=∂f/∂x=y, fy=∂f/∂y=x, gx=∂g/∂x=1, and gy=∂g/∂y=1.Equation (5.494) is exactly the same with (5.492).According to the ancient Chinese mathematics, to solve a problem, we should choose double trial solutions,x1 and x2. We can choose x1 arbitrarily, for example, x1=2/7, according to the constraint equation (5.488), and we have y1=5/7.From (5.487), we obtain
(5.495)f1=x1y1=1049.
The condition (5.489) can be approximately expressed as
(5.496)Δf=xΔy+yΔx=0,
where Δx=x2-x1, Δy=y2-y1, and Δf=f2-f1.Equation (5.496) implies that
(5.497)f2=f1.
This means that
(5.498)f2=x2y2=1049.
Considering the constraint equation, (5.488), again, we have
(5.499)x2+y2=1.
From (5.498) and (5.499), we can obtain immediately x2=5/7 and y2=2/7.Substitutingx1, x2, y1, and y2 into (5.498), we have
(5.500)Δf=x(y2-y1)+y(x2-x1)=x(27-57)+y(57-27)=0.
Solving (5.488) and (5.500) simultaneously, we find the optimal values given in (5.493).Example 5.1.
Condiser the following [109]:
(5.501)f=x2+y2→min,(5.502)s.t.g(x,y)=y-2-x2=0.
To make (5.501) minimal, it requires that
(5.503)Δf=2xΔx+2yΔy2x2+y2=xΔx+yΔyx2+y2=0.
Choosing x1=1, we have y1=3 and f1=10. We set f2=10, which means that
(5.504)x22+y22=10,(5.505)y2=2+x22.
Solving (5.504) and (5.505) simultaneously, we have x2=-1 and y2=3.
Substitutingx1, x2, y1, and y2 into (5.503), we obtain
(5.506)Δf=x(x2-x1)+y(y2-y1)x2+y2=-2xx2+y2=0.
We, therefore, obtain the optimal valuesx*=0 and y*=2.Example 5.2.
Consider the following problem [109]:
(5.507)f(x,y)=x2+y2→min,(5.508)s.t.y2-(x-1)3=0.
The Lagrange multiplier method becomes invalid for this example [109]. Using the basic idea of the ancient Chinese mathematics, we choose x1=2, y1=1 and x2=2, y2=-1, and both cases lead to f1=f2=5.
Making (5.507) stationary requires
(5.509)Δf=xΔx+yΔyx2+y2=x(x2-x1)+y(y2-y1)x2+y2=-2yx2+y2=0,
which leads to immediately y*=0 and x*=1 if the constraint equation, (5.508), is considered.


## 5.11.3. Ancient Chinese Mathematics for Nonlinear Wave Equations [110]

Now we consider the nonlinear Schrödinger equation(5.510)iut+uxx+2|u|2u=0,u(x,0)=eikx.
We use the trial functions
(5.511)u1(x,t)=ei(kx+t),u2(x,t)=ei(kx+ωt).
The residuals are
(5.512)R1(x,t)=(1-k2)ei(kx+t),R2(x,t)=(2-ω-k2)ei(kx+ωt).
According to frequency-amplitude formulation, (5.480), we have
(5.513)ω2=ω12R2(x,0)-ω22R1(x,0)R2(x,0)-R1(x,0)=-(ω+ωk2-2ω+k2-2).We get the equation aboutω, and the solution is
(5.514)ω=2-k2,u(x,t)=ei{kx+(2-k2)ωt}.
Now we consider the Schrödinger equation with power law nonlinearity
(5.515)iut+uxx-2|u|2ru=0,u(x,0)=(2(r+1)sech2(2rx))1/2r,r≥1.
Following the analysis presented above, we choose the trial functions
(5.516)u1(x,t)=(2(r+1)sech2(2rx))1/2reit,u2(x,t)=(2(r+1)sech2(2rx))1/2reiωt.
The residuals are
(5.517)R1(x,t)=-(2(r+1)(sech(2rx))2)1/2reit+4(2(r+1)(sech(2rx))2)1/2r(tanh(2rx))2eit-4(2(r+1)(sech(2rx))2)1/2r(1-(tanh(2rx))2)reit+2(2(r+1)(sech(2rx))2)1/2r-1eit,R2(x,t)=-(2(r+1)(sech(2rx))2)1/2reiωtω+4(2(r+1)(sech(2rx))2)1/2r(tanh(2rx))2eiωt-4(2(r+1)(sech(2rx))2)1/2r(1-(tanh(2rx))2)reiωt+2(2(r+1)(sech(2rx))2)1/2reiωt.
According to the frequency-amplitude formulation, (5.480), setting x=0 and t=0,  we have
(5.518)ω2=-4r+2ω|(21/2r(r+1)1/2r)2r|-ω-4ωr+|(21/2r(r+1)1/2r)2r|.
Solving ω from the above equation yields
(5.519)ω=-4r+2|(21/2r(r+1)1/2r)2r|.
We, therefore, obtain the solution
(5.520)u2(x,t)=(2(r+1)sech2(2rx))1/2reiωt.


## 5.11.4. Solitary-Solution Formulation for Nonlinear Equations

Now consider a general nonlinear equation(5.521)L(u)+N(u)=0,
where L is a linear operator, and N is a nonlinear operator.We choose two trial equations(5.522)L(u)=0,L~(u)=0,
where L~ is a linearized linear operator or a simple nonlinear operator, and its solution L~(u)=0 is known. For example, we consider a nonlinear oscillator: u′′+u3=0, and we can choose L(u)=u′′+u=0 and L~(u)=u′′+ω2u=0, where ω is the frequency to be searched for.Substituting the solutions of (5.522) with same boundary/initial conditions as (5.521) into (5.521), we obtain, respectively, the residuals  R1 and R2.For traveling wave solutions, we can make a transformation(5.523)ξ=ax+by+cz+ωt.
The solitary-solution formulation can be expressed as
(5.524)ω2=R1ω12-R2ω22R1-R2,
or
(5.525)ωn=R1ω1n-R2ω2nR1-R2.The KdV equation is a nonlinear, dispersive partial differential equation for a function of two real variables, spacex and time t:
(5.526)ut+6uux+uxxx=0.
According to the solitary-solution formulation (5.524), we construct two trial functions in the following forms:
(5.527)u1(x,t)=Aexp(kx+t+D)+exp[-(kx+t+D)]+B,u2(x,t)=Aexp(kx+ωt+D)+exp[-(kx+ωt+D)]+B,
where A, B, D are arbitrary constants.Substitutingu1 and u2 into (5.526), respectively, we obtain the following residuals:
(5.528)R1(x,t)=-A{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}2-6A2k{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}3-Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}2-6Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}3{exp(kx+t+D)+exp[-(kx+t+D)]+B}4+6Ak3{exp(kx+t+D)-exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]}{exp(kx+t+D)+exp[-(kx+t+D)]+B}3,R2(x,t)=-Aω{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}2-6A2k{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}3-Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}2-6Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}3{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}4+6Ak3{exp(kx+ωt+D)-exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]}{exp(kx+ωt+D)+exp[-(kx+ωt+D)]+B}3.
Using the solitary-solution formulation (5.480), and choosing the location point (0,0), we obtain
(5.529)ω2=ω12R2(0,0)-ω22R1(0,0)R2(0,0)-R1(0,0).
Simplifying (5.529) results in
(5.530)ω2+ω+(ω+1)k3+[2Bω2+2ωB+(ω+1)(6kA-4k3B)]exp(-D)+[2ω2+B2ω2+2ω+ωB2+(ω+1)(6kAB-22k3+k3B2)]exp(-2D)+[2Bω2+2ωB+(ω+1)(6kA-4k3B)]exp(-3D)+[ω2+ω+(ω+1)k3]exp(-4D)=0.
Setting the coefficients of exp(-nD) (n=0,1,2,3,4) to be zero, we obtain
(5.531)ω2+ω+(ω+1)k3=0,(5.532)2Bω2+2ωB+(ω+1)(6kA-4k3B)=0,(5.533)2ω2+B2ω2+2ω+ωB2+(ω+1)(6kAB-22k3+k3B2)=0.
Solving (5.531)~(5.533) simultaneously, we obtain the following relationships:(1)
the relationship between solitary wavenumber and solitary frequency:(5.534)ω=-k3,(2)
the relationship between solitary amplitude and solitary wavenumber:(5.535)A=k2B,(3)
the morphofactor:(5.536)B2=4, so the solitary wave solution can be readily obtained, which reads
(5.537)u(x,t)=2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]+2,
or
(5.538)u(x,t)=-2k2exp(kx-k3t+D)+exp[-(kx-k3t+D)]-2,
which are, by chance, exact solutions.


## 5.11.5. Ancient Chinese Inequality and Application

In an ancient history book, it is write (see [111]):He Chengtian uses 26/49 as the strong, and 9/17 as the weak. Among the strong and the weak, Chengtian tries to find a more accurate denominator of the fractional day of the Moon. Chengtian obtains 752 as the denominator by using the 15 and 1 as weighting factors, respectively, for the strong and the weak. No other calendar can reach such a high accuracy after Chengtian, who uses heuristically the strong and weak weighting factors.The statement is rather cryptic, and in modern mathematical term, the statement can be explained as follows.According to the observation data, He Chengtian (369?~447AD) finds that(5.539)292649days>1Moon>29917days.
Using the weighting factors (15 and 1), He Chengtian obtains
(5.540)Thefractionalday=26×15+9×149×15+17×1=399752,
so
(5.541)1Moon=29399752days.
He Chengtian actually uses the following inequality.If(5.542)ab<x<dc,
where a,b,c, and d are real numbers, then
(5.543)ab<ma+ndmb+nc<dc,
and x is approximated by
(5.544)x=ma+ndmb+nc,
where m and n are weighting factors.The above inequity was developed by the max-min approach to nonlinear oscillators [112]. Hereby we use the inequity for a chemical problem.A mass balance on a differential volume element of porous medium for a spherical catalyst pellet gives(5.545)∂c′∂t=∇·De∇c′-RA,
where c′ is the chemical reactant concentration, -RA is the rate of reaction per unit volume, and De  is the effective diffusion coefficient for reactant.Assuming the coupled process of diffusion and reaction is steady, we reduce (5.545) to the form
(5.546)De(∂2c′∂r2+2r∂c′∂r)=RA.
The boundary conditions are ∂c′/∂r=0,r=0 (center of catalyst), and c′=cs,r=r0 (surface of catalyst).Introducing dimensionless variablesC=c′/cs and R=r/r0, we have the following nonlinear equation for nth-order and irreversible reaction at isothermal condition:
(5.547)∂2C∂R2+2R∂C∂R=ϕ2Cn,∂C∂R(0)=0,C(1)=1,
where ϕ is the Thiele modulus, defined as ϕ=r02kνcsn-1/De, where  kν is the reaction constant.Equation (5.547) can also be written in Cartesian geometry, and it is easy to transform Cartesian geometry to cylindrical or spherical geometry.To investigate the nature of the solution of (5.547), consider the analogous linear equation
(5.548)∂2C∂R2+2R∂C∂R=k,∂C∂R(0)=0,C(1)=1.MultiplyingR2 on both sides of (5.548), we have
(5.549)∂∂R(R2∂C∂R)=kR2,∂C∂R(0)=0,C(1)=1.Its solution is(5.550)C=1+16k(R2-1).Now we rewrite (5.544) in the form
(5.551)∂2C∂R2+2R∂C∂R=g(C),
where g(C)=ϕ2Cn.The left side of (5.551) is analogous to k in (5.548).Observe that0<C<1, so C is never less than the solution of the following initial value problem:
(5.552)∂2C∂R2+2R∂C∂R=ϕ2Cmaxn=ϕ2.The solution of (5.552) reads
(5.553)C=1+16ϕ2(R2-1).
So it follows that (see Figure 4)
(5.554)6+ϕ2(R2-1)6<C<1.
According to He Chengtian’s interpolation, we have
(5.555)C=6p+pϕ2(R2-1)+q6p+q,
or
(5.556)C=1+ξϕ2(R2-1),
where p and q are weighting factors, ξ=p/(6p+q). The free parameter, ξ, in (5.556) can be identified via various methods. Hereby we illustrate a simple approach by the residual method. Substituting (5.556) into (5.544), we obtain the following residual equation:
(5.557)Π(R,ξ)=∂2C∂R2+2R∂C∂R-ϕ2Cn=2ξϕ2+4ξϕ2-ϕ2[1+ξϕ2(R2-1)]n.Figure 4
Qualitative sketch for the diffusion reaction in spherical porous catalyst. The dashed line presents an initial guessC=1+ϕ2(R2-1)/6, and the continued line presents the real solution.Chemists and technologists always want to have an accurate halfway through the change. Assume that the above residual vanishes completely atC=1/2, that is, 1+ξϕ2(R2-1)=1/2, then we have
(5.558)Π(R,ξ)|R=R*=6ξϕ2-12nϕ2,
where R=R* is the root of the equation 1+ξϕ2(R2-1)=1/2. From (5.558), the unknown ξ can be identified as
(5.559)ξ=2-n6.Finally, we obtain the following design formulation:(5.560)C(R)=1+2-n6ϕ2(R2-1).Comparison with exact solution is illustrated in Figure5.Figure 5
Comparison between exact solution and the approximate solution (n=1, ϕ=1). The dashed line presents an approximate solution, and the continued line presents the exact solution. The maximal error is about 7.8%.In order to identify the weighting factors,p and q, in (5.555), we proceed the same manipulation as illustrated above. Substituting (5.555) into (5.544), we obtain a residual equation:
(5.561)Π(R,p,q)=∂2C∂R2+2R∂C∂R-ϕ2Cn=2pϕ26p+q+4pϕ26p+q-ϕ2(6p+pϕ2(R2-1)+q6p+q)n=0.
Locating at R=0 and R=1/2, we obtain
(5.562)Π(0,p,q)=2pϕ26p+q+4pϕ26p+q-ϕ2(6p-pϕ2+q6p+q)n=0,(5.563)Π(12,p,q)=2pϕ26p+q+4pϕ26p+q-ϕ2(6p-(3/4)pϕ2+q6p+q)n=0.
Solving (5.562) and (5.563) simultaneously, the values of p and q can be readily determined.The accuracy of vanishing residuals depends upon the trial functions, (5.555) and (5.556), which are derived from an analogous linear equation, so the obtained solutions are valid for the whole solution domain, keeping maximal accuracy at the location points.The preceding analysis has the virtue of utter simplicity, and the illustrating example shows that the suggested method is very effective and convenient in solving nonlinear equations. The suggested method can be readily applied to the search for analytical solutions for various nonlinear equations, which should have the character of monotonic increase or monotonic decrease. Many nonlinear problems arising in chemical engineering can be described by, for example, exponential decay/rise functions, so the ancient Chinese method—the He Chengtian interpolation—might be a powerful mathematical tool in chemical engineering.


## 6. Solitons in Lattice Systems

### 6.1. Physical Understanding of Differential-Difference Equations

Governing equations for continuum media are well established by using the Gauss’ divergence theorems:(6.1)∭V∇ϕdV=∯∂VnϕdS,(6.2)∭V∇·AdV=∯∂Vn·AdS,(6.3)∭V∇×AdV=∯∂Vn×AdS.
Mathematically, the above Gauss’ divergence theorems are invalid for fractal media, such as porous media, weaves. So the governing equations for noncontinuum media cannot be expressed by differential equations in space but can be done by difference equations. Considering the continuous time, such problems can be described by differential-difference equations:
(6.4)∂un∂t=F(un-1,un,un+1),∂un∂t=F(∂un-1∂x,∂un∂x,∂un-1∂x,un-1,un,un+1),∂un∂t=F(∂un∂x,∂un∂y,∂un∂z,un-1,un,un+1).Unlike difference equations which are fully discretized, DDEs are semidiscretized with some of their spatial variables discretized, while time is usually kept continuous.Solitons and compactons for difference-differential equations have caught much attention due to the fact that discrete spacetime may be the most radical and logical viewpoint of reality [113].A better physical understanding of differential-difference equations can be obtained by considering the flow through a lattice where the conservation of mass requires [114]
(6.5)dρidt+ρi+1ui+1-ρi-1ui-1=0,
with  ρi and ui being, respectively, the gas density and velocity at the ith lattice point (see Figure 6).Figure 6
Conservation of mass.


### 6.2. Exp-Function Method

Consider a hybrid-lattice system [115]
(6.6)∂un∂t=(1+αun+βun2)(un-1-un+1),
where α and  β are constants.Letun=un(ξ),ξ=dn+c1x+c2t+ξ0, then (6.6) becomes
(6.7)c2un′=(1+αun+βun2)(un-1-un+1).
According to the exp-function method, we assume that
(6.8)un(ξ)=a1exp(ξ)+a0+a-1exp(-ξ)b1exp(ξ)+b0+exp(-ξ),un-1(ξ)=a1exp(ξ-d)+a0+a-1exp(-ξ+d)b1exp(ξ-d)+b0+exp(-ξ+d),un+1(ξ)=a1exp(ξ+d)+a0+a-1exp(-ξ-d)b1exp(ξ+d)+b0+exp(-ξ-d).
Substituting (6.8) into (6.7), and by the help of Maple, clearing the denominator, and setting the coefficients of power terms like exp(jξ),j=1,2,… to zero yield a system of algebraic equations, and we obtain some generalized exact solutions, one of which is expressed in the form [115]
(6.9)un=((2αβa02e2d)/(α2-4β)(e2d-1)2)exp(ξ)+a0-(α/2β)exp(-ξ)-((4αβ2a02e2d)/(α2-4β)(e2d-1)2)exp(ξ)+exp(-ξ),
where c2=(α2-4β)(e2d-1)/(4βed), ξ=dn+c1x+c2t+ξ0, and c1 and a0 are free parameters.The application of the exp-function method to the discrete mKdV lattice and the discrete Schrödinger equation was discussed in [116, 117].


### 6.3. Variational Iteration Method

Consider the Volterra lattice equation [118]
(6.10)∂un∂t=un(un+1-un-1),un(0)=n.
Using the variational iteration method, an iteration formulation can be constructed as follows:
(6.11)un,m=un,m-1-∫0tun,m-1{un+1,m-1-un-1,m-1}dt.
Begining with (un)0=n, we can obtain the following series solution [118]:
(6.12)un(t)=n{1+2t+4t2+8t3+16t4+32t5+64t6+O(t7)},
which converges to the exact solution un(t)=n/(1-2t).


### 6.4. Homotopy Perturbation Method

Consider the discretized mKdV lattice equation [119]
(6.13)dundt=un2(un+1-un-1),un(0)=1-1n2.
Construct a homotopy
(6.14)(1-p)(dun(t)dt-dun,0(t)dt)+p(dun(t)dt-un2(t)·(un+1(t)-un-1(t)))=0.
Suppose the solution of (6.13) to be in the following form:
(6.15)un(t)=un,0(t)+pun,1(t)+p2un,2+⋯.
By a simple calculation, beginning with un,0(t)=1-1/n2, we have [119]
(6.16)un(t)=1-1n2+4tn3-12t2n4+32t3n5-80t4n6+⋯,
which converges to the exact solution un(t)=1-1/(n+2t)2.


### 6.5. Parameter-Expansion Method

We rewrite (6.13) in the form
(6.17)dundt=1·un2(un+1-un-1).
We expand, respectively, the solution and the coefficient, 1, in the forms
(6.18)un(t)=un,0(t)+pun,1(t)+p2un,2+⋯,p=1,1=pa1+p2a2+⋯.
The solution procedure is given in Section 5.8.


### 6.6. Parameterized Perturbation Method

Parameterized perturbation method was proposed in 1999 [120]; it is useful to obtain exact and approximate solutions of nonlinear differential equations. The method requires no linearization or discretization; large computational work and round-off errors are avoided. It has been used to solve effectively, easily, and accurately a large class of nonlinear problems with approximations. These approximations converge rapidly to accurate solutions [120, 121].We consider the following general nonlinear DDEs given in the form [122]:
(6.19)dun(t)dt=f(un-1,un,un+1),
with the initial condition un(0)=A(n), where a, b, and c are constants, and f is a nonlinear function.According to the PPM, an expanding parameter is introduced by a linear transformation(6.20)un=ενn,
where ε is the introduced perturbation parameter.Substituting (6.20) into (6.19) results in
(6.21)dνn(t)dt=f(ε;νn-1,νn,νn+1),νn(0)=A(n)ε,
supposing that the solution of (6.21) can be expressed in the form
(6.22)νn=νn,0+ενn,1+ε2νn,2+…,
and processing in a traditional fashion of perturbation technique.In the following examples, we will illustrate the usefulness and effectiveness of the proposed technique.Consider the following Volterra equation:(6.23)dundt=un(un+1-un-1),
with the initial condition
(6.24)un(0)=n.Using the transformation (6.20), the original equation (6.23) becomes
(6.25)dνndt=ενn(νn+1-νn-1),νn(0)=nε.
Substituting (6.22) into (6.25) and equating coefficients of like powers of ε yield the following equations:
(6.26)dνn,0dt=0,νn,0(0)=νn(0)=nε,dνn,idt=∑k=0i-1νn,k(νn+1,i-k-1-νn-1,i-k-1),νn,i(0)=0,i=1,2,….
We can start with νn,0(0)=n/ε, and we obtain the following successive approximations:
(6.27)νn,k(t)=2knεk+1tk,k=0,1,2,….
Hence, the solution series in general gives
(6.28)νn=nε(1+2t+4t2+8t3+16t4+⋯),
then
(6.29)un=ενn=n(1+2t+4t2+8t3+16t4+⋯).
The closed form of the series is
(6.30)un(t)=n1-2t,
which gives the exact solution of the problem.


### 6.7. Solitary-Solution Formulation [123]

Suppose that the differential-difference equation we discuss in this paper is in the following nonlinear polynomial form:(6.31)dun(t)dt=f(un-1,un,un+1),
where un=u(n,t) is a dependent variable; t is a continuous variable; n,pi∈Z.Using the basic idea of the ancient Chinese algorithm, we choose two trial functions in the forms(6.32)un,1(n,t)=f(ξn+ω1t),un,2(n,t)=g(ξn+ω2t),
where ξn=nd+ξ0, ξ0 is arbitrary, and f and g are known functions. If a periodic solution is searched for, f and g must be periodic functions; if a solitary solution is solved, f and g must be of solitary structures. In this paper, a bell solitary solution of a differential-difference equation is considered, and trial functions are chosen as follows:
(6.33)un,1(n,t)=Aeξn+ω1t+e-(ξn+ω1t)+B,un,2(n,t)=Aeξn+ωt+e-(ξn+ωt)+B.
For un, un-1 and un+1 should be compatible, then
(6.34)un-1,1(n,t)=Aeξn-d+t+e-(ξn-d+t)+B,(6.35)un+1,1(n,t)=Aeξn+d+t+e-(ξn+d+t)+B,(6.36)un-1,2(n,t)=Aeξn-d+ωt+e-(ξn-d+ωt)+B,(6.37)un+1,2(n,t)=Aeξn+d+ωt+e-(ξn+d+ωt)+B.


### 6.8. Solution Procedure

Step 1.
Define the residual function(6.38)R~(t)=dun(t)dt-f(un-1,un,un+1).
Substituting (6.40)~(6.43) into (6.37), we can obtain, respectively, the residual functions R~1 and R~2.Step 2.
SubstituteR~1 and R~2 into the following equation:
(6.39)ω2=ω12R~2(0)-ω22R~1(0)R~2(0)-R~1(0),
where ω2=ω.Step 3.
Combining the coefficients ofeξn in (6.45), and setting them to be zero, we can solve the algebraic equations to find the values of ω, A, and B. Finally, an explicit solution is obtained.
The famous mKdV lattice equation reads(6.40)dundt=(α-un2)(un-1-un+1),
where α≠0.Substituting (6.40)~(6.43) into (6.46), we obtain, respectively, the following residual functions R~1 and R~2, and using the solitary-solution formulation, (6.45), we have
(6.41)e3ξn(ω3+ω2αe-d-ω2αed-ω-αe-d+αed)-e-3ξn(ω3+ω2αe-d-ω2αed-ω-αe-d+αed)+e2ξnB(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)-e-2ξnB(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)+eξn(ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d)-e-ξn(ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d)=0.
Setting the coefficients of eiξn to be zero, we have
(6.42)ω3+ω2αe-d-ω2αed-ω-αe-d+αed=0,B(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)=0,ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d=0.
Solving (6.48)–(6.49) simultaneously, we have
(6.43)ω=2αsinh(d),(6.44)A=2-αsinh(d),(6.45)B=0.
We, therefore, obtain the following needed solitary solution:
(6.46)un=2-αsinh(d)eξn+2αsinh(d)t+e-(ξn+2αsinh(d)t)=-αsinh(d)sech{2αsinh(d)t+nd+ξ0}.
We can also use the following assumptions for solitary solutions:
(6.47)un,1(n,t)=daeξn+ω1t+be-(ξn+ω1t)+c,(6.48)un,2(n,t)=∑n=-cdanexp(ξn+ωt)∑m=-fgbmexp(ξn+ωt).
For compact-like solutions, we can assume that the solution has the form
(6.49)un,1(n,t)=da+cos2(ξn+ω1t),(6.50)u0(x,t)=asin2(kx+wt)b+csin2(kx+wt).


## 6.1. Physical Understanding of Differential-Difference Equations

Governing equations for continuum media are well established by using the Gauss’ divergence theorems:(6.1)∭V∇ϕdV=∯∂VnϕdS,(6.2)∭V∇·AdV=∯∂Vn·AdS,(6.3)∭V∇×AdV=∯∂Vn×AdS.
Mathematically, the above Gauss’ divergence theorems are invalid for fractal media, such as porous media, weaves. So the governing equations for noncontinuum media cannot be expressed by differential equations in space but can be done by difference equations. Considering the continuous time, such problems can be described by differential-difference equations:
(6.4)∂un∂t=F(un-1,un,un+1),∂un∂t=F(∂un-1∂x,∂un∂x,∂un-1∂x,un-1,un,un+1),∂un∂t=F(∂un∂x,∂un∂y,∂un∂z,un-1,un,un+1).Unlike difference equations which are fully discretized, DDEs are semidiscretized with some of their spatial variables discretized, while time is usually kept continuous.Solitons and compactons for difference-differential equations have caught much attention due to the fact that discrete spacetime may be the most radical and logical viewpoint of reality [113].A better physical understanding of differential-difference equations can be obtained by considering the flow through a lattice where the conservation of mass requires [114]
(6.5)dρidt+ρi+1ui+1-ρi-1ui-1=0,
with  ρi and ui being, respectively, the gas density and velocity at the ith lattice point (see Figure 6).Figure 6
Conservation of mass.


## 6.2. Exp-Function Method

Consider a hybrid-lattice system [115]
(6.6)∂un∂t=(1+αun+βun2)(un-1-un+1),
where α and  β are constants.Letun=un(ξ),ξ=dn+c1x+c2t+ξ0, then (6.6) becomes
(6.7)c2un′=(1+αun+βun2)(un-1-un+1).
According to the exp-function method, we assume that
(6.8)un(ξ)=a1exp(ξ)+a0+a-1exp(-ξ)b1exp(ξ)+b0+exp(-ξ),un-1(ξ)=a1exp(ξ-d)+a0+a-1exp(-ξ+d)b1exp(ξ-d)+b0+exp(-ξ+d),un+1(ξ)=a1exp(ξ+d)+a0+a-1exp(-ξ-d)b1exp(ξ+d)+b0+exp(-ξ-d).
Substituting (6.8) into (6.7), and by the help of Maple, clearing the denominator, and setting the coefficients of power terms like exp(jξ),j=1,2,… to zero yield a system of algebraic equations, and we obtain some generalized exact solutions, one of which is expressed in the form [115]
(6.9)un=((2αβa02e2d)/(α2-4β)(e2d-1)2)exp(ξ)+a0-(α/2β)exp(-ξ)-((4αβ2a02e2d)/(α2-4β)(e2d-1)2)exp(ξ)+exp(-ξ),
where c2=(α2-4β)(e2d-1)/(4βed), ξ=dn+c1x+c2t+ξ0, and c1 and a0 are free parameters.The application of the exp-function method to the discrete mKdV lattice and the discrete Schrödinger equation was discussed in [116, 117].


## 6.3. Variational Iteration Method

Consider the Volterra lattice equation [118]
(6.10)∂un∂t=un(un+1-un-1),un(0)=n.
Using the variational iteration method, an iteration formulation can be constructed as follows:
(6.11)un,m=un,m-1-∫0tun,m-1{un+1,m-1-un-1,m-1}dt.
Begining with (un)0=n, we can obtain the following series solution [118]:
(6.12)un(t)=n{1+2t+4t2+8t3+16t4+32t5+64t6+O(t7)},
which converges to the exact solution un(t)=n/(1-2t).


## 6.4. Homotopy Perturbation Method

Consider the discretized mKdV lattice equation [119]
(6.13)dundt=un2(un+1-un-1),un(0)=1-1n2.
Construct a homotopy
(6.14)(1-p)(dun(t)dt-dun,0(t)dt)+p(dun(t)dt-un2(t)·(un+1(t)-un-1(t)))=0.
Suppose the solution of (6.13) to be in the following form:
(6.15)un(t)=un,0(t)+pun,1(t)+p2un,2+⋯.
By a simple calculation, beginning with un,0(t)=1-1/n2, we have [119]
(6.16)un(t)=1-1n2+4tn3-12t2n4+32t3n5-80t4n6+⋯,
which converges to the exact solution un(t)=1-1/(n+2t)2.


## 6.5. Parameter-Expansion Method

We rewrite (6.13) in the form
(6.17)dundt=1·un2(un+1-un-1).
We expand, respectively, the solution and the coefficient, 1, in the forms
(6.18)un(t)=un,0(t)+pun,1(t)+p2un,2+⋯,p=1,1=pa1+p2a2+⋯.
The solution procedure is given in Section 5.8.


## 6.6. Parameterized Perturbation Method

Parameterized perturbation method was proposed in 1999 [120]; it is useful to obtain exact and approximate solutions of nonlinear differential equations. The method requires no linearization or discretization; large computational work and round-off errors are avoided. It has been used to solve effectively, easily, and accurately a large class of nonlinear problems with approximations. These approximations converge rapidly to accurate solutions [120, 121].We consider the following general nonlinear DDEs given in the form [122]:
(6.19)dun(t)dt=f(un-1,un,un+1),
with the initial condition un(0)=A(n), where a, b, and c are constants, and f is a nonlinear function.According to the PPM, an expanding parameter is introduced by a linear transformation(6.20)un=ενn,
where ε is the introduced perturbation parameter.Substituting (6.20) into (6.19) results in
(6.21)dνn(t)dt=f(ε;νn-1,νn,νn+1),νn(0)=A(n)ε,
supposing that the solution of (6.21) can be expressed in the form
(6.22)νn=νn,0+ενn,1+ε2νn,2+…,
and processing in a traditional fashion of perturbation technique.In the following examples, we will illustrate the usefulness and effectiveness of the proposed technique.Consider the following Volterra equation:(6.23)dundt=un(un+1-un-1),
with the initial condition
(6.24)un(0)=n.Using the transformation (6.20), the original equation (6.23) becomes
(6.25)dνndt=ενn(νn+1-νn-1),νn(0)=nε.
Substituting (6.22) into (6.25) and equating coefficients of like powers of ε yield the following equations:
(6.26)dνn,0dt=0,νn,0(0)=νn(0)=nε,dνn,idt=∑k=0i-1νn,k(νn+1,i-k-1-νn-1,i-k-1),νn,i(0)=0,i=1,2,….
We can start with νn,0(0)=n/ε, and we obtain the following successive approximations:
(6.27)νn,k(t)=2knεk+1tk,k=0,1,2,….
Hence, the solution series in general gives
(6.28)νn=nε(1+2t+4t2+8t3+16t4+⋯),
then
(6.29)un=ενn=n(1+2t+4t2+8t3+16t4+⋯).
The closed form of the series is
(6.30)un(t)=n1-2t,
which gives the exact solution of the problem.


## 6.7. Solitary-Solution Formulation [123]

Suppose that the differential-difference equation we discuss in this paper is in the following nonlinear polynomial form:(6.31)dun(t)dt=f(un-1,un,un+1),
where un=u(n,t) is a dependent variable; t is a continuous variable; n,pi∈Z.Using the basic idea of the ancient Chinese algorithm, we choose two trial functions in the forms(6.32)un,1(n,t)=f(ξn+ω1t),un,2(n,t)=g(ξn+ω2t),
where ξn=nd+ξ0, ξ0 is arbitrary, and f and g are known functions. If a periodic solution is searched for, f and g must be periodic functions; if a solitary solution is solved, f and g must be of solitary structures. In this paper, a bell solitary solution of a differential-difference equation is considered, and trial functions are chosen as follows:
(6.33)un,1(n,t)=Aeξn+ω1t+e-(ξn+ω1t)+B,un,2(n,t)=Aeξn+ωt+e-(ξn+ωt)+B.
For un, un-1 and un+1 should be compatible, then
(6.34)un-1,1(n,t)=Aeξn-d+t+e-(ξn-d+t)+B,(6.35)un+1,1(n,t)=Aeξn+d+t+e-(ξn+d+t)+B,(6.36)un-1,2(n,t)=Aeξn-d+ωt+e-(ξn-d+ωt)+B,(6.37)un+1,2(n,t)=Aeξn+d+ωt+e-(ξn+d+ωt)+B.


## 6.8. Solution Procedure

Step 1.
Define the residual function(6.38)R~(t)=dun(t)dt-f(un-1,un,un+1).
Substituting (6.40)~(6.43) into (6.37), we can obtain, respectively, the residual functions R~1 and R~2.Step 2.
SubstituteR~1 and R~2 into the following equation:
(6.39)ω2=ω12R~2(0)-ω22R~1(0)R~2(0)-R~1(0),
where ω2=ω.Step 3.
Combining the coefficients ofeξn in (6.45), and setting them to be zero, we can solve the algebraic equations to find the values of ω, A, and B. Finally, an explicit solution is obtained.
The famous mKdV lattice equation reads(6.40)dundt=(α-un2)(un-1-un+1),
where α≠0.Substituting (6.40)~(6.43) into (6.46), we obtain, respectively, the following residual functions R~1 and R~2, and using the solitary-solution formulation, (6.45), we have
(6.41)e3ξn(ω3+ω2αe-d-ω2αed-ω-αe-d+αed)-e-3ξn(ω3+ω2αe-d-ω2αed-ω-αe-d+αed)+e2ξnB(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)-e-2ξnB(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)+eξn(ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d)-e-ξn(ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d)=0.
Setting the coefficients of eiξn to be zero, we have
(6.42)ω3+ω2αe-d-ω2αed-ω-αe-d+αed=0,B(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)=0,ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d=0.
Solving (6.48)–(6.49) simultaneously, we have
(6.43)ω=2αsinh(d),(6.44)A=2-αsinh(d),(6.45)B=0.
We, therefore, obtain the following needed solitary solution:
(6.46)un=2-αsinh(d)eξn+2αsinh(d)t+e-(ξn+2αsinh(d)t)=-αsinh(d)sech{2αsinh(d)t+nd+ξ0}.
We can also use the following assumptions for solitary solutions:
(6.47)un,1(n,t)=daeξn+ω1t+be-(ξn+ω1t)+c,(6.48)un,2(n,t)=∑n=-cdanexp(ξn+ωt)∑m=-fgbmexp(ξn+ωt).
For compact-like solutions, we can assume that the solution has the form
(6.49)un,1(n,t)=da+cos2(ξn+ω1t),(6.50)u0(x,t)=asin2(kx+wt)b+csin2(kx+wt).


## 7. Fractional Differential Equations

### 7.1. Physical Understanding of Fractional Differential Equations

Fractional differential equations have caught much attention recently due to the exact description of nonlinear phenomena. No analytical method was available before 1998 for such equations even for linear fractional differential equations.In 1998, the variational iteration method was first proposed to solve fractional differential equations with greatest success, see [124]. Following the above idea, Draganescu, Momani, and Odibat applied the variational iteration method to more complex fractional differential equations, showing effectiveness and accuracy of the used method, see [125, 126]. In 2002, the Adomian method was suggested to solve fractional differential equations [127]. But many researchers found that it is very difficult to calculate the Adomian polynomial, see comments in [128]. Ghorbani and Saberi-Nadjafi suggested a very simple method for calculating the Adomian polynomial using the homotopy perturbation method [129], and He polynomial should be used instead of Adomian polynomial [130].In 2007, Momani and Odibat [131] applied the homotopy perturbation method to fractional differential equations and revealed that the homotopy perturbation method is an alternative analytical method for fractional differential equations.Although the fractional calculus was invented by Newton and Leibnitz over three centuries ago, it only became a hot topic recently owing to the development of the computer and its exact description of many real-life problems. To give a physical interpretation of the fractional calculus, we begin with a simple function(7.1)y=x2.
When x=x1 and x=x2, we have
(7.2)y1=x12,y2=x22.
We therefore have the difference
(7.3)Δy=y1-y2=x12-x22=(x1-x2)(x1+x2)=(x1+x2)Δx.
In case x1→x2 or Δx→0, we have the differential
(7.4)dy=2xdx.
The above derivation, however, is only valid for continuous functions. To show its applications, we consider the action in String theory [132]
(7.5)S=mc∫ds,
where m is the mass of a particle, c is the speed of light, and ds is the relativistic metric that can be expressed in the form
(7.6)ds=c2dt2-dx2-dy2-dz2=c1-u2c2dt.
Substitution of (7.6) into (7.5) leads to
(7.7)S=∫mc21-u2c2dt,
from which the equation of free motion is obtained in the usual way
(7.8)ddt(mu1-u2/c2)=0.The above derivation assumes that space and time are both continuous. The distance between two points, for example, cannot be expressed in the form of (7.6) in a discontinuous world (e.g., in the Julia set).Now consider a plane with fractal structure (see Figure7). The shortest path between two points is not a line, and we have
(7.9)dsE=kdsD,
where dsE is the actual distance between two points (discontinuous line in Figure 7), ds is the line distance between two points (continuous line in Figure 7), D is the fractal dimension, and k is a constant.Figure 7
The distance between two points in a discontinuous spacetime.The action in a discontinuous spacetime can therefore be written in the following form using fractional calculus:(7.10)S=mc∫dsE=∫mc1+Dk(1-u2c2)D/2dtD.
Fractional calculus is therefore valid for discontinuous problems. Now we consider a well-known predator-prey model (the Lotka-Volterra equation) [133]
(7.11)dxdt=x(a-by),dydt=-y(c-dy),
where y is the number of predators (e.g., wolves), x is the number of its prey (e.g., rabbits), and a,b,c, and d are parameters representing the interaction of the two species.In general, the growth of the two populations is discontinuous, and a simple modification of the predator-prey model is to replacedy/dt  and  dx/dt  by fractional derivatives
(7.12)DαxDtα=x(a-by),DβyDtβ=-y(c-dy),
where the populations of the predator and prey may be greatly affected by the fractional orders, α and β.


### 7.2. Variational Iteration Method

Consider the following general fractional differential equation:(7.13a)DαuDtα+f=0.
In the case 0<α<1, we rewrite (7.13a) in the form
(7.13b)dudt+DαuDtα-dudt+f=0,and the variational iteration algorithms are given as follows:
(7.14)un+1(t)=un(t)-∫0t(DαunDtα+fn)ds,un+1(t)=u0(t)-∫0t(DαunDtα-dundt+fn)ds,un+1(t)=u0(t)-∫0t{(DαunDtα-dundt+fn)-(Dαun-1Dtα-dun-1dt+fn-1)}ds.
In the case 1<α<2, the above iteration formulas are also valid. We can rewrite (7.13a) in the form
(7.15)d2udt2+DαuDtα-d2udt2+f=0,
and the following iteration formulae are suggested:
(7.16)un+1(t)=un(t)+∫0t(s-t)(DαunDtα+fn)ds,un+1(t)=u0(t)+∫0t(s-t)(DαunDtα-d2undt2+fn)ds,un+1(t)=un(t)+∫0t(s-t){(DαunDtα-d2undt2+fn)-(Dαun-1Dtα-d2un-1dt2+fn-1)}ds.
When α is close to 1, (7.14) is better, while (7.16) is recommended for α approaching 2.For the caseN<α<N+1, where N is a natural number, the iteration formulas
(7.17)un+1(t)=un(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα+fn)ds,un+1(t)=u0(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα-dNundtN+fn)ds,un+1(t)=un(t)+(-1)N×∫0t1(N-1)!(s-t)N-1×{(DαunDtα-dNundtN+fn)-(Dαun-1Dtα-dNun-1dtN+fn-1)}ds,
or
(7.18)un+1(t)=un(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα+fn)ds,un+1(t)=u0(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα-dN+1undtN+1+fn)ds,un+1(t)=un(t)+(-1)N+1×∫0t1N!(s-t)N{(DαunDtα-dN+1undtN+1+fn)-(Dαun-1Dtα-dN+1un-1dtN+1+fn-1)}ds
can be used. When α is close to N, (7.17) is recommended, and (7.18) works more effectively for α approaching N+1.We consider the fractional calculus version of the standard vibration equation in one dimension as [133]
(7.19)∂2u∂r2+1r∂u∂r=1c2∂βu∂tβ,r≥0,t≥0,1<β≤2,
with initial conditions
(7.20)u(r,0)=r2,∂∂tu(r,0)=cr.
Equation (7.19) can be written as
(7.21)∂2u∂t2=c2∂2-β∂t2-β[∂2u∂r2+1r∂u∂r].
According to the variational iteration method, we consider the correction functional in t-direction in the following form:
(7.22)un+1(r,t)=un(r,t)+∫0t(ξ-t){∂2un(r,ξ)∂ξ2-c2∂2-β∂t2-β(∂2u~n(r,ξ)∂r2+1r∂u~n(r,ξ)∂r)}dξ.
Beginning with an initial approximation
(7.23)u0(r,t)=u(r,0)+tut(r,0)=r2+ctr,
we obtain [133]
(7.24)u(r,t)=limn→∞un(r,t)=r2+ctr+4c2tβΓ(β+1)+c3tβ+1rΓ(β+2)+c5t2β+1r3Γ(2β+2)+9c7t3β+1r5Γ(3β+2)+⋯=r2+4c2tβΓ(β+1)+ctrEβ,2(c2r2ktβ),
where kn=[1×3×5×…×(2n-3)]2 and Eβ,b(t)=∑n=0∞tn/Γ(nβ+b) are the generalized Mittag-Leffler function.Consider another nonlinear fractional differential equation [134]
(7.25)D*αu(x)+u′(x)+u3(x)=0,x>0,
where 1<α≤2, subject to the initial condition
(7.26)u(0)=1,u′(0)=0.
We can construct a correction functional in the form
(7.27)un+1(x)=un(x)+∫0xλ(ξ)(D(2)un(ξ)+Mu~n(ξ)-D(2)u~n(ξ)+Dαu~n(ξ))dξ.
After identification of the Lagrange multiplier, we obtain the following iteration formulation:
(7.28)un+1(x)=un(x)+∫0x(ξ-x)[D*αun(ξ)+un′(ξ)+u3(ξ)]dξ.
Consequently, beginning with u0=1, we find the following approximations:
(7.29)u0(x)=1,u1(x)=1-x22,u2(x)=1-x2+x36+x48-x640+x8448-Γ(4-α)x4-αΓ(3-α)Γ(5-α)+x4-αΓ(4-α).


### 7.3. Homotopy Perturbation Method

The homotopy perturbation method becomes an effective tool to fractional differential equations [134–137]. Consider the nonlinear time-fractional RLW equation [134]
(7.30)∂αu∂tα-uxxt+(u22)x=0,t>0,
where 0<α≤1, subject to the initial condition
(7.31)u(x,0)=x.
We can construct the following homotopy:
(7.32)∂u∂t=p[∂u∂t+uxxt-(u22)x-Dtαu].
Processing the same solution procedure as the homotopy perturbation method, we obtain
(7.33)u0=x,u1=-xt,u2=x(-t+t2+t2-αΓ(3-α)),u3=x(-t+2t2-t3+2t2-αΓ(3-α)-4t3-αΓ(4-α)-t3-2αΓ(4-2α)),
and the solution can be approximately expressed as
(7.34)u=u0+u1+u2+u3+….


### 7.4. Fractional Complex Transform

The fractional complex transform [138, 139] was originally suggested to convert a fractional differential equation with Jumarie’s modification of Riemann-Liouville derivative into its classical differential partner.Consider the following general fractional differential equation:(7.35)f(u,ut(α),ux(β),uy(γ),uz(λ),ut(2α),ux(2β),uy(2γ),uz(2λ),…)=0,0<α≤1,0<β≤1,0<γ≤1,0<λ≤1,
where ut(α)=Dtαu=Dαu/Dtα denotes Jumarie’s fractional derivation, which is a modified Riemann-Liouville derivative defined as [140]
(7.36)Dtαu(t,x,y,z)=1Γ(1-α)ddt∫0t‍(t-ξ)-α(u(ξ,x,y,z)-u(0,x,y,z))dξ,
where u is a continuous (but not necessarily differentiable) function
(7.37)∂αc∂tα=0,(7.38)∂α∂tα[cu]=c∂αu∂tα,(7.39)∂αtβ∂tα=Γ(1+β)Γ(1+β-α)tβ-α,β≥α>0.
In our previous publications [138], there were some flaws. The solution procedure should be followed as follows.Using the following transforms:(7.40)s=tα,X=xβ,Y=yγ,Z=zλ,
we have
(7.41)∂αu∂tα=∂u∂s·∂βs∂xβ=σs∂u∂s,∂βu∂xβ=∂u∂X·∂βX∂xβ=σX∂u∂X,∂γu∂yγ=∂u∂Y·∂γY∂yγ=σY∂u∂Y,∂λu∂zλ=∂u∂Z·∂λZ∂zλ=σZ∂u∂Z,
where σs, σX, σY, and σZ are fractal indexes [139]. We can, therefore, easily convert fractional differential equations into partial differential equations, so that everyone familiar with advanced calculus can deal with fractional calculus without any difficulty.To determineσs, we consider a special case s=tα and  u=sm, and we have
(7.42)∂αu∂tα=Γ(1+mα)·tmα-αΓ(1+mα-α)=σ∂u∂s=σmtmα-α.
We, therefore, can determine σs as follows:
(7.43)σs=Γ(1+mα)mΓ(1+mα-α).
Other fractal indexes (σX, σY, σZ) can be determined in a similar way.As an example, consider the fractional differential equation(7.44)∂αu∂tα+Bu=0,0<α<1,u(0)=1.
Assume that the solution can be expressed in a series in the form
(7.45)u=∑m=0∞amsm,
where am(m=0,1,2,3,…) are constants to be further determined.Using the transforms=tα, and submitting (7.45) into (7.44), we have
(7.46)∂∂s∑m=0∞σsmamsm+B∑m=0∞amsm=0,
or
(7.47)∑m=0∞mσsmamsm-1+B∑m=0∞amsm=0.
According to (7.43), the fractal index σsm can be determined as follows:
(7.48)σsm=Γ(1+mα)mΓ(1+mα-α).
From (7.47) and (7.48), we obtain
(7.49)Γ(1+mα)Γ(1+mα-α)am+Bam-1=0.
Generally, we begin with u0=u(0)=1. After a simple calculation, we have
(7.50)am=(-B)mΓ(1+mα).
We, therefore, obtain the following solution:
(7.51)u(s)=∑m=0∞(-B)mΓ(1+mα)sm,
or
(7.52)u(t)=∑m=0∞(-B)mΓ(1+mα)tmα=Eα(-Btα),
where Eα is a Mittag-Leffler function defined as
(7.53)Eα(t)=∑m=0∞tmΓ(1+mα).
Equation (7.53) is the exact solution of the example.


### 7.5. Yang-Laplace Transform [141]

The local fractional functional analysis was first proposed by Yang [141]. The Yang-Laplace transform of f(x) is defined as(7.54)Lα{f(x)}=fsL,α(s)≔1Γ(1+α)∫0∞Eα(-sαxα)f(x)(dx)α,0<α≤1,
and the inverse formula is
(7.55)Lα-1(fsL,α(s))=f(t)=:1(2π)α∫β-i∞β+i∞Eα(sαxα)fsL,α(s)(ds)α,
where sα=βα+iα∞α and Re(sα)=βα>0α.Table of the Yang-Laplace Transform of Elementary Functions(7.56)(1)Lα{δα(x)}=1.(2)Lα{δα(x-a)}=Eα(-aαsα).(3)Lα{Hα(x)}=1sα.(4)Lα{Hα(x)xα}=1s2α.(5)Lα{δα(nα)(x)}=snα.(6)Lα{1}=1sα.(7)Lα{Eα(aαxα)}=1sα-aα.(8)Lα{xαEα(aαxα)}=1(s-a)2α.(9)Lα{sinα(ax)α}=aαs2α-a2α.(10)Lα{cosα(ax)α}=sαs2α+a2α.(11)Lα{Eα(-bαxα)sinα(ax)α}=aα(s+b)2α+a2α.(12)Lα{Eα(-bαxα)cosα(ax)α}=aα+bα(s+b)2α+a2α.(13)Lα{xkα}=Γ(1+kα)s(k+1)α.(14)Lα{xkαEα(aαxα)}=Γ(1+kα)(s-a)(k+1)α.(15)Lα{Eα(aαxα)-Eα(bαxα)}=aα-bα(s-a)α(s-b)α.
The Yang-Laplace transform is an effective mathematical tool for solving local fractional differential equations. Consider the following local fractional differential equation:
(7.57)y(α)(t)+2y(t)=Eα(-tα),0<α≤1,
with initial condition y(t)|t=0=0.By the Yang-Laplace transform, (7.55) becomes
(7.58)sαysL,α(s)+2ysL,α(s)=11+s.
Solving ysL,α from (7.58), we have
(7.59)ysL,α(ω)=11+sα-12+sα.
The inverse Yang-Laplace transform of (7.59) gives
(7.60)y(t)=Eα(-tα)-Eα(-2tα).
Consider another local fractional differential equation given by
(7.61)∂αu∂yα+kαu=0,y>0,0<α≤1,u(0)=1.
Taking the Yang-Laplace transform, we have
(7.62)sαu~(s)+kαu~(s)-u(0)=0.
It is obvious that
(7.63)u~(s)=1sα+kα.
The inverse Yang-Laplace transform results in the following solution:
(7.64)u(y)=Eα(kαyα).


### 7.6. Yang-Fourier Transform

Yang-Fourier transform is also an effective method for local fractional differential equations. The Yang-Fourier transform off(x) is given by
(7.65)Fα{f(x)}=fωF,α(ω)∶=1Γ(1+α)∫-∞∞Eα(-iαωαxα)f(x)(dx)α,
and the inverse formula of the Yang-Fourier transform is
(7.66)f(x)=Fα-1(fωF,α(ω))∶=1(2π)α∫-∞∞Eα(iαωαxα)fωF,α(ω)(dω)α.
The basic properties of the Yang-Fourier transform are as follows:
(7.67)Fα{af(x)+bg(x)}=aFα{f(x)}+bFα{g(x)},Fα{f(α)(x)}=iαωαFα{f(x)}.Table of Local Fractional Fourier Transforms(7.68)(1)Fα{δα(x)}=1.(2)Fα{1}=(2π)αΓ(1+α)δα(ω).(3)Fα{Eα(iαω0αxα)}=(2π)αΓ(1+α)δα(ω-ω0).(4)Fα{δα(x-x0)f(x)}=Eα(-iαωαx0α)f(x0).(5)Fα{δα(x-x0)}=Eα(-iαωαx0α).(6)Fα{cosα(ax)α}=πα[δα(ω+a)+δα(ω-a)].(7)Fα{sinα(ax)α}=παiα[δα(ω+a)-δα(ω-a)].(8)Fα{Hα(x)Eα(iαω0αxα)}=1iα(ω-a)α+(2π)α2Γ(1+α)δα(ω-a).(9)Fα{|x|α}=-2ω2α.(10)Fα{δα(n)(x)}=jnαωnα.(11)Fα{xnα}=(2π)αΓ(1+α)δα(nα)(ω).(12)Fα{Hα(x)xnα}=Γ(1+nα)inαωnα+(2π)α2Γ(1+α)inαδα(nα)(ω).
Consider a local fractional differential equation
(7.69)y(α)(t)+2y(t)=Eα(-tα),0<α≤1,
with initial condition y(0)=0.Taking Yang-Fourier transform for (7.69),
(7.70)iαωαyωF,α(ω)+2yωF,α(ω)=11+iαωα,we have(7.71)yωF,α(ω)=11+iαωα-12+iαωα.The inverse Yang-Fourier transform gives(7.72)f(t)=Eα(-tα)-Eα(-2tα).Consider another local fractional differential equation(7.73)∂αu∂yα+kαu=0,y∈ℝ,0<α≤1,u(0)=1.Taking the Yang-Fourier transform yields(7.74)iαωαu~(ω)+kαu~(ω)-1=0,
or
(7.75)u~(ω)=1iαωα+kα.
The inverse Yang-Fourier transform gives the solution
(7.76)u(y)=Eα(kαyα).


### 7.7. Fractal Derivative andq-Derivative

The fractal derivative is defined by transforming the standard integer-dimensional space time(x,t) into a fractal space time [142]
(7.77)du(t)dtD=lims→tu(t)-u(s)tD-sD,
where D is the order of the fractal derivative. This definition is much simpler but lacks physical understanding.As an example, consider the fractal derivative relaxation equation [142]
(7.78)dudtD+Bu=0,0<D<1,u(0)=1,
whose analytical solution is [142]
(7.79)u(t)=exp(-BtD).
We write down a general fractal differential equation of the form
(7.80)dudtD+f=0,
and use the transformation tD=x to convert (7.80) into an ordinary differential equation
(7.81)dudx+f=0,
so that the variational iteration algorithms above can be directly applied:
(7.82)un+1(x)=un(x)-∫0x(dundx+fn)dx,un+1(x)=u0(x)-∫0xfndx,un+1(x)=u0(x)-∫0x{fn-fn-1}dx,
or
(7.83)un+1(tD)=un(tD)-∫0tD(dundtD+fn)DtD-1dt,un+1(tD)=u0(tD)-∫0tDfnDtD-1dt,un+1(tD)=u0(tD)-∫0tD{fn-fn-1}DtD-1dx.
As another example, consider the Nth-order fractal differential equation
(7.84)dNudtD+f=0,
which can be converted by the transformation
(7.85)tD=xN
into the ordinary differential equation
(7.86)dNudxN+f=0,
and the variational iteration algorithms are
(7.87)un+1(x)=un(x)+(-1)N∫0x1(N-1)!(s-x)N-1(un(N)+fn)ds,un+1(x)=u0(x)+(-1)N∫0t1(N-1)!(s-x)N-1fnds,un+1(x)=un(x)+(-1)N∫0t1(N-1)!(s-t)N-1(fn-fn-1)ds.
The fractal derivative is simpler than its fractional counterpart in many applications, and it is also valid for discontinuous cases. We rewrite (7.77) in the form
(7.88)du(x)dxD=1klimA→Bu(A)-u(B)x~A-x~B,
where k is a constant, and A and B are arbitrary points in discontinuous space or spacetime (as shown in Figure 8). (x~A, x~B) are called the fractal coordinates and are defined by
(7.89)x~A=k(xA-0)D=k(xA)D,(7.90)x~B=k(xB-0)D=k(xB)D,
where (xA,xB) are the coordinates, and D is the fractional dimension in x-direction. Substituting (7.88) and (7.89) into (7.88), we obtain
(7.91)du(x)dxD=limA→Bu(A)-u(B)(xA)D-(xB)D.
The fractal differential model is particularly suitable for describing discontinuous matter and is the preferred model for describing flow or heat conduction through porous media. For instance, the principle of mass conservation can be written in the form
(7.92)∂ρ∂t+k1∂(ρu)∂xD1+k2∂(ρv)∂yD2+k3∂(ρw)∂zD3=0,
where D1, D2, and D3 are the fractal dimensions of porosity in the x, y, and z directions, respectively, and ki(i=1,2,3) are constants that are related to the fractal dimensions. In particular, we have ki=1, when Di=1. Similarly, the momentum equation for one-dimensional porous flow can be written in the form
(7.93)∂u∂t+ku∂u∂xD=-kρ∂P∂xD+k∂∂xD(μk∂u∂xD),
when D=1 and k=1,  and (7.93) turns out to be the classical one.Figure 8
A schematic diagram of distance betweenO and A (i.e., the fractal coordinate of A) in a fractal space, and D=ln2/ln3, while the fractional dimensions for the plane are ln8/ln3.The one-dimensional heat conduction equation in porous media can be expressed as(7.94)∂T∂t+k∂∂xD(μk∂T∂xD)=0,
where μ is the conduction coefficient and k=1 when D=1.An oscillator swinging in a porous medium can also be described by fractal differential equations. For example, the Duffing equation with fractal damp can be expressed as(7.95)d2udt2+u+kμdudtD+εu3=0.
It is easy to establish fractal differential equations for discontinuous media by replacing ∂/∂x in the classical approach by k∂/∂xD.Now we consider a fractal media illustrated in Figure9, and assume that the smallest measure is L0, any discontinuity less than L0 is ignored, then the distance between two points of A and B in Figure 1 can be expressed using fractal geometry. Hereby we introduce a new fractal derivative for engineering application:
(7.96)Du(t)Dxα=limΔx→L0u(A)-u(B)Thedistancebetweentwopoints=duds=limΔx→L0u(A)-u(B)kL0α,
where k is a constant, α is the fractal dimension, and the distance between two points in a discontinuous space can be expressed as ds=kL0α.Figure 9
The distance between two points in a discontinuous spacetime.Please note in the above definition thatΔx does tend to zero, but to the smallest measure size, L0.Example 7.1.
As a simple application, we consider the Fourier’s law heat conduction, which reads(7.97)∂qh∂t=-cdTdn,
where T is the thermal potential, and qh is heat flow.
In the discontinuous media, the Fourier’s law can be simply modified as(7.98)∂qh∂t=-cDTDnα,
where DT/Dnα is a fractal derivative defined in (7.96).The suggested fractal derivative is easy to be used for any discontinuous problems, and equations with fractal derivative can be easily solved using classical calculus.Letf(x;y;…) be a multivariable real continuous function. The q-derivative and the partial q-derivative are defined by [143]:
(7.99)Dqxf(x)=f(qx)-f(x)(q-1)x,∂qxf(x;y;…)=f(qx;y;…)-f(x;y;…)(q-1)x,
and ∂qxf(x;y;…)|x=0=limn→∞(f(xqn;y;…)-f(0;y;…))/xqn,
(7.100)Dqtu+f(u)=0,0<q<1.
In the case 0<α<1, we rewrite (7.13a) in the form
(7.101)dudt+{Dqtu+f(u)-dudt}=0,
and the variational iteration algorithms are given as follows:
(7.102)un+1(t)=un(t)-∫0t(Dqtun+f(un))ds,un+1(t)=u0(t)-∫0t(Dqtun-dundt+f(un))ds,un+1(t)=u0(t)-∫0t{(Dqtun-dundt+f(un))-(Dαun-1Dtα-dun-1dt+f(un-1))}ds.
In the case 1<α<2, the above iteration formulas are also valid. We can rewrite (7.100) in the form
(7.103)d2udt2+DαuDtα-d2udt2+f=0
and the following iteration formulae are suggested
(7.104)un+1(t)=un(t)+∫0t(s-t)(DαunDtα+fn)ds,un+1(t)=u0(t)+∫0t(s-t)(DαunDtα-d2undt2+fn)ds,un+1(t)=un(t)+∫0t(s-t){(DαunDtα-d2undt2+fn)}-(Dαun-1Dtα-d2un-1dt2+fn-1)ds,
when α is close to 1, (7.102) is better, while (7.104) is recommended for α approaching 2.For the caseN<α<N+1, where N is a natural number, the iteration formulas are as follows
(7.105)un+1(t)=un(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα+fn)dsun+1(t)=u0(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα-dNundtN+fn)dsun+1(t)=un(t)+(-1)N+∫0t1(N-1)!(s-t)N-1×{(DαunDtα-dNundtN+fn)-(Dαun-1Dtα-dNun-1dtN+fn-1)}ds,or(7.106)un+1(t)=un(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα+fn)ds,un+1(t)=u0(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα-dN+1undtN+1+fn)ds,un+1(t)=un(t)+(-1)N+1×∫0t1N!(s-t)N{(DαunDtα-dN+1undtN+1+fn)-(Dαun-1Dtα-dN+1un-1dtN+1+fn-1)}ds.


## 7.1. Physical Understanding of Fractional Differential Equations

Fractional differential equations have caught much attention recently due to the exact description of nonlinear phenomena. No analytical method was available before 1998 for such equations even for linear fractional differential equations.In 1998, the variational iteration method was first proposed to solve fractional differential equations with greatest success, see [124]. Following the above idea, Draganescu, Momani, and Odibat applied the variational iteration method to more complex fractional differential equations, showing effectiveness and accuracy of the used method, see [125, 126]. In 2002, the Adomian method was suggested to solve fractional differential equations [127]. But many researchers found that it is very difficult to calculate the Adomian polynomial, see comments in [128]. Ghorbani and Saberi-Nadjafi suggested a very simple method for calculating the Adomian polynomial using the homotopy perturbation method [129], and He polynomial should be used instead of Adomian polynomial [130].In 2007, Momani and Odibat [131] applied the homotopy perturbation method to fractional differential equations and revealed that the homotopy perturbation method is an alternative analytical method for fractional differential equations.Although the fractional calculus was invented by Newton and Leibnitz over three centuries ago, it only became a hot topic recently owing to the development of the computer and its exact description of many real-life problems. To give a physical interpretation of the fractional calculus, we begin with a simple function(7.1)y=x2.
When x=x1 and x=x2, we have
(7.2)y1=x12,y2=x22.
We therefore have the difference
(7.3)Δy=y1-y2=x12-x22=(x1-x2)(x1+x2)=(x1+x2)Δx.
In case x1→x2 or Δx→0, we have the differential
(7.4)dy=2xdx.
The above derivation, however, is only valid for continuous functions. To show its applications, we consider the action in String theory [132]
(7.5)S=mc∫ds,
where m is the mass of a particle, c is the speed of light, and ds is the relativistic metric that can be expressed in the form
(7.6)ds=c2dt2-dx2-dy2-dz2=c1-u2c2dt.
Substitution of (7.6) into (7.5) leads to
(7.7)S=∫mc21-u2c2dt,
from which the equation of free motion is obtained in the usual way
(7.8)ddt(mu1-u2/c2)=0.The above derivation assumes that space and time are both continuous. The distance between two points, for example, cannot be expressed in the form of (7.6) in a discontinuous world (e.g., in the Julia set).Now consider a plane with fractal structure (see Figure7). The shortest path between two points is not a line, and we have
(7.9)dsE=kdsD,
where dsE is the actual distance between two points (discontinuous line in Figure 7), ds is the line distance between two points (continuous line in Figure 7), D is the fractal dimension, and k is a constant.Figure 7
The distance between two points in a discontinuous spacetime.The action in a discontinuous spacetime can therefore be written in the following form using fractional calculus:(7.10)S=mc∫dsE=∫mc1+Dk(1-u2c2)D/2dtD.
Fractional calculus is therefore valid for discontinuous problems. Now we consider a well-known predator-prey model (the Lotka-Volterra equation) [133]
(7.11)dxdt=x(a-by),dydt=-y(c-dy),
where y is the number of predators (e.g., wolves), x is the number of its prey (e.g., rabbits), and a,b,c, and d are parameters representing the interaction of the two species.In general, the growth of the two populations is discontinuous, and a simple modification of the predator-prey model is to replacedy/dt  and  dx/dt  by fractional derivatives
(7.12)DαxDtα=x(a-by),DβyDtβ=-y(c-dy),
where the populations of the predator and prey may be greatly affected by the fractional orders, α and β.


## 7.2. Variational Iteration Method

Consider the following general fractional differential equation:(7.13a)DαuDtα+f=0.
In the case 0<α<1, we rewrite (7.13a) in the form
(7.13b)dudt+DαuDtα-dudt+f=0,and the variational iteration algorithms are given as follows:
(7.14)un+1(t)=un(t)-∫0t(DαunDtα+fn)ds,un+1(t)=u0(t)-∫0t(DαunDtα-dundt+fn)ds,un+1(t)=u0(t)-∫0t{(DαunDtα-dundt+fn)-(Dαun-1Dtα-dun-1dt+fn-1)}ds.
In the case 1<α<2, the above iteration formulas are also valid. We can rewrite (7.13a) in the form
(7.15)d2udt2+DαuDtα-d2udt2+f=0,
and the following iteration formulae are suggested:
(7.16)un+1(t)=un(t)+∫0t(s-t)(DαunDtα+fn)ds,un+1(t)=u0(t)+∫0t(s-t)(DαunDtα-d2undt2+fn)ds,un+1(t)=un(t)+∫0t(s-t){(DαunDtα-d2undt2+fn)-(Dαun-1Dtα-d2un-1dt2+fn-1)}ds.
When α is close to 1, (7.14) is better, while (7.16) is recommended for α approaching 2.For the caseN<α<N+1, where N is a natural number, the iteration formulas
(7.17)un+1(t)=un(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα+fn)ds,un+1(t)=u0(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα-dNundtN+fn)ds,un+1(t)=un(t)+(-1)N×∫0t1(N-1)!(s-t)N-1×{(DαunDtα-dNundtN+fn)-(Dαun-1Dtα-dNun-1dtN+fn-1)}ds,
or
(7.18)un+1(t)=un(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα+fn)ds,un+1(t)=u0(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα-dN+1undtN+1+fn)ds,un+1(t)=un(t)+(-1)N+1×∫0t1N!(s-t)N{(DαunDtα-dN+1undtN+1+fn)-(Dαun-1Dtα-dN+1un-1dtN+1+fn-1)}ds
can be used. When α is close to N, (7.17) is recommended, and (7.18) works more effectively for α approaching N+1.We consider the fractional calculus version of the standard vibration equation in one dimension as [133]
(7.19)∂2u∂r2+1r∂u∂r=1c2∂βu∂tβ,r≥0,t≥0,1<β≤2,
with initial conditions
(7.20)u(r,0)=r2,∂∂tu(r,0)=cr.
Equation (7.19) can be written as
(7.21)∂2u∂t2=c2∂2-β∂t2-β[∂2u∂r2+1r∂u∂r].
According to the variational iteration method, we consider the correction functional in t-direction in the following form:
(7.22)un+1(r,t)=un(r,t)+∫0t(ξ-t){∂2un(r,ξ)∂ξ2-c2∂2-β∂t2-β(∂2u~n(r,ξ)∂r2+1r∂u~n(r,ξ)∂r)}dξ.
Beginning with an initial approximation
(7.23)u0(r,t)=u(r,0)+tut(r,0)=r2+ctr,
we obtain [133]
(7.24)u(r,t)=limn→∞un(r,t)=r2+ctr+4c2tβΓ(β+1)+c3tβ+1rΓ(β+2)+c5t2β+1r3Γ(2β+2)+9c7t3β+1r5Γ(3β+2)+⋯=r2+4c2tβΓ(β+1)+ctrEβ,2(c2r2ktβ),
where kn=[1×3×5×…×(2n-3)]2 and Eβ,b(t)=∑n=0∞tn/Γ(nβ+b) are the generalized Mittag-Leffler function.Consider another nonlinear fractional differential equation [134]
(7.25)D*αu(x)+u′(x)+u3(x)=0,x>0,
where 1<α≤2, subject to the initial condition
(7.26)u(0)=1,u′(0)=0.
We can construct a correction functional in the form
(7.27)un+1(x)=un(x)+∫0xλ(ξ)(D(2)un(ξ)+Mu~n(ξ)-D(2)u~n(ξ)+Dαu~n(ξ))dξ.
After identification of the Lagrange multiplier, we obtain the following iteration formulation:
(7.28)un+1(x)=un(x)+∫0x(ξ-x)[D*αun(ξ)+un′(ξ)+u3(ξ)]dξ.
Consequently, beginning with u0=1, we find the following approximations:
(7.29)u0(x)=1,u1(x)=1-x22,u2(x)=1-x2+x36+x48-x640+x8448-Γ(4-α)x4-αΓ(3-α)Γ(5-α)+x4-αΓ(4-α).


## 7.3. Homotopy Perturbation Method

The homotopy perturbation method becomes an effective tool to fractional differential equations [134–137]. Consider the nonlinear time-fractional RLW equation [134]
(7.30)∂αu∂tα-uxxt+(u22)x=0,t>0,
where 0<α≤1, subject to the initial condition
(7.31)u(x,0)=x.
We can construct the following homotopy:
(7.32)∂u∂t=p[∂u∂t+uxxt-(u22)x-Dtαu].
Processing the same solution procedure as the homotopy perturbation method, we obtain
(7.33)u0=x,u1=-xt,u2=x(-t+t2+t2-αΓ(3-α)),u3=x(-t+2t2-t3+2t2-αΓ(3-α)-4t3-αΓ(4-α)-t3-2αΓ(4-2α)),
and the solution can be approximately expressed as
(7.34)u=u0+u1+u2+u3+….


## 7.4. Fractional Complex Transform

The fractional complex transform [138, 139] was originally suggested to convert a fractional differential equation with Jumarie’s modification of Riemann-Liouville derivative into its classical differential partner.Consider the following general fractional differential equation:(7.35)f(u,ut(α),ux(β),uy(γ),uz(λ),ut(2α),ux(2β),uy(2γ),uz(2λ),…)=0,0<α≤1,0<β≤1,0<γ≤1,0<λ≤1,
where ut(α)=Dtαu=Dαu/Dtα denotes Jumarie’s fractional derivation, which is a modified Riemann-Liouville derivative defined as [140]
(7.36)Dtαu(t,x,y,z)=1Γ(1-α)ddt∫0t‍(t-ξ)-α(u(ξ,x,y,z)-u(0,x,y,z))dξ,
where u is a continuous (but not necessarily differentiable) function
(7.37)∂αc∂tα=0,(7.38)∂α∂tα[cu]=c∂αu∂tα,(7.39)∂αtβ∂tα=Γ(1+β)Γ(1+β-α)tβ-α,β≥α>0.
In our previous publications [138], there were some flaws. The solution procedure should be followed as follows.Using the following transforms:(7.40)s=tα,X=xβ,Y=yγ,Z=zλ,
we have
(7.41)∂αu∂tα=∂u∂s·∂βs∂xβ=σs∂u∂s,∂βu∂xβ=∂u∂X·∂βX∂xβ=σX∂u∂X,∂γu∂yγ=∂u∂Y·∂γY∂yγ=σY∂u∂Y,∂λu∂zλ=∂u∂Z·∂λZ∂zλ=σZ∂u∂Z,
where σs, σX, σY, and σZ are fractal indexes [139]. We can, therefore, easily convert fractional differential equations into partial differential equations, so that everyone familiar with advanced calculus can deal with fractional calculus without any difficulty.To determineσs, we consider a special case s=tα and  u=sm, and we have
(7.42)∂αu∂tα=Γ(1+mα)·tmα-αΓ(1+mα-α)=σ∂u∂s=σmtmα-α.
We, therefore, can determine σs as follows:
(7.43)σs=Γ(1+mα)mΓ(1+mα-α).
Other fractal indexes (σX, σY, σZ) can be determined in a similar way.As an example, consider the fractional differential equation(7.44)∂αu∂tα+Bu=0,0<α<1,u(0)=1.
Assume that the solution can be expressed in a series in the form
(7.45)u=∑m=0∞amsm,
where am(m=0,1,2,3,…) are constants to be further determined.Using the transforms=tα, and submitting (7.45) into (7.44), we have
(7.46)∂∂s∑m=0∞σsmamsm+B∑m=0∞amsm=0,
or
(7.47)∑m=0∞mσsmamsm-1+B∑m=0∞amsm=0.
According to (7.43), the fractal index σsm can be determined as follows:
(7.48)σsm=Γ(1+mα)mΓ(1+mα-α).
From (7.47) and (7.48), we obtain
(7.49)Γ(1+mα)Γ(1+mα-α)am+Bam-1=0.
Generally, we begin with u0=u(0)=1. After a simple calculation, we have
(7.50)am=(-B)mΓ(1+mα).
We, therefore, obtain the following solution:
(7.51)u(s)=∑m=0∞(-B)mΓ(1+mα)sm,
or
(7.52)u(t)=∑m=0∞(-B)mΓ(1+mα)tmα=Eα(-Btα),
where Eα is a Mittag-Leffler function defined as
(7.53)Eα(t)=∑m=0∞tmΓ(1+mα).
Equation (7.53) is the exact solution of the example.


## 7.5. Yang-Laplace Transform [141]

The local fractional functional analysis was first proposed by Yang [141]. The Yang-Laplace transform of f(x) is defined as(7.54)Lα{f(x)}=fsL,α(s)≔1Γ(1+α)∫0∞Eα(-sαxα)f(x)(dx)α,0<α≤1,
and the inverse formula is
(7.55)Lα-1(fsL,α(s))=f(t)=:1(2π)α∫β-i∞β+i∞Eα(sαxα)fsL,α(s)(ds)α,
where sα=βα+iα∞α and Re(sα)=βα>0α.Table of the Yang-Laplace Transform of Elementary Functions(7.56)(1)Lα{δα(x)}=1.(2)Lα{δα(x-a)}=Eα(-aαsα).(3)Lα{Hα(x)}=1sα.(4)Lα{Hα(x)xα}=1s2α.(5)Lα{δα(nα)(x)}=snα.(6)Lα{1}=1sα.(7)Lα{Eα(aαxα)}=1sα-aα.(8)Lα{xαEα(aαxα)}=1(s-a)2α.(9)Lα{sinα(ax)α}=aαs2α-a2α.(10)Lα{cosα(ax)α}=sαs2α+a2α.(11)Lα{Eα(-bαxα)sinα(ax)α}=aα(s+b)2α+a2α.(12)Lα{Eα(-bαxα)cosα(ax)α}=aα+bα(s+b)2α+a2α.(13)Lα{xkα}=Γ(1+kα)s(k+1)α.(14)Lα{xkαEα(aαxα)}=Γ(1+kα)(s-a)(k+1)α.(15)Lα{Eα(aαxα)-Eα(bαxα)}=aα-bα(s-a)α(s-b)α.
The Yang-Laplace transform is an effective mathematical tool for solving local fractional differential equations. Consider the following local fractional differential equation:
(7.57)y(α)(t)+2y(t)=Eα(-tα),0<α≤1,
with initial condition y(t)|t=0=0.By the Yang-Laplace transform, (7.55) becomes
(7.58)sαysL,α(s)+2ysL,α(s)=11+s.
Solving ysL,α from (7.58), we have
(7.59)ysL,α(ω)=11+sα-12+sα.
The inverse Yang-Laplace transform of (7.59) gives
(7.60)y(t)=Eα(-tα)-Eα(-2tα).
Consider another local fractional differential equation given by
(7.61)∂αu∂yα+kαu=0,y>0,0<α≤1,u(0)=1.
Taking the Yang-Laplace transform, we have
(7.62)sαu~(s)+kαu~(s)-u(0)=0.
It is obvious that
(7.63)u~(s)=1sα+kα.
The inverse Yang-Laplace transform results in the following solution:
(7.64)u(y)=Eα(kαyα).


## 7.6. Yang-Fourier Transform

Yang-Fourier transform is also an effective method for local fractional differential equations. The Yang-Fourier transform off(x) is given by
(7.65)Fα{f(x)}=fωF,α(ω)∶=1Γ(1+α)∫-∞∞Eα(-iαωαxα)f(x)(dx)α,
and the inverse formula of the Yang-Fourier transform is
(7.66)f(x)=Fα-1(fωF,α(ω))∶=1(2π)α∫-∞∞Eα(iαωαxα)fωF,α(ω)(dω)α.
The basic properties of the Yang-Fourier transform are as follows:
(7.67)Fα{af(x)+bg(x)}=aFα{f(x)}+bFα{g(x)},Fα{f(α)(x)}=iαωαFα{f(x)}.Table of Local Fractional Fourier Transforms(7.68)(1)Fα{δα(x)}=1.(2)Fα{1}=(2π)αΓ(1+α)δα(ω).(3)Fα{Eα(iαω0αxα)}=(2π)αΓ(1+α)δα(ω-ω0).(4)Fα{δα(x-x0)f(x)}=Eα(-iαωαx0α)f(x0).(5)Fα{δα(x-x0)}=Eα(-iαωαx0α).(6)Fα{cosα(ax)α}=πα[δα(ω+a)+δα(ω-a)].(7)Fα{sinα(ax)α}=παiα[δα(ω+a)-δα(ω-a)].(8)Fα{Hα(x)Eα(iαω0αxα)}=1iα(ω-a)α+(2π)α2Γ(1+α)δα(ω-a).(9)Fα{|x|α}=-2ω2α.(10)Fα{δα(n)(x)}=jnαωnα.(11)Fα{xnα}=(2π)αΓ(1+α)δα(nα)(ω).(12)Fα{Hα(x)xnα}=Γ(1+nα)inαωnα+(2π)α2Γ(1+α)inαδα(nα)(ω).
Consider a local fractional differential equation
(7.69)y(α)(t)+2y(t)=Eα(-tα),0<α≤1,
with initial condition y(0)=0.Taking Yang-Fourier transform for (7.69),
(7.70)iαωαyωF,α(ω)+2yωF,α(ω)=11+iαωα,we have(7.71)yωF,α(ω)=11+iαωα-12+iαωα.The inverse Yang-Fourier transform gives(7.72)f(t)=Eα(-tα)-Eα(-2tα).Consider another local fractional differential equation(7.73)∂αu∂yα+kαu=0,y∈ℝ,0<α≤1,u(0)=1.Taking the Yang-Fourier transform yields(7.74)iαωαu~(ω)+kαu~(ω)-1=0,
or
(7.75)u~(ω)=1iαωα+kα.
The inverse Yang-Fourier transform gives the solution
(7.76)u(y)=Eα(kαyα).


## 7.7. Fractal Derivative andq-Derivative

The fractal derivative is defined by transforming the standard integer-dimensional space time(x,t) into a fractal space time [142]
(7.77)du(t)dtD=lims→tu(t)-u(s)tD-sD,
where D is the order of the fractal derivative. This definition is much simpler but lacks physical understanding.As an example, consider the fractal derivative relaxation equation [142]
(7.78)dudtD+Bu=0,0<D<1,u(0)=1,
whose analytical solution is [142]
(7.79)u(t)=exp(-BtD).
We write down a general fractal differential equation of the form
(7.80)dudtD+f=0,
and use the transformation tD=x to convert (7.80) into an ordinary differential equation
(7.81)dudx+f=0,
so that the variational iteration algorithms above can be directly applied:
(7.82)un+1(x)=un(x)-∫0x(dundx+fn)dx,un+1(x)=u0(x)-∫0xfndx,un+1(x)=u0(x)-∫0x{fn-fn-1}dx,
or
(7.83)un+1(tD)=un(tD)-∫0tD(dundtD+fn)DtD-1dt,un+1(tD)=u0(tD)-∫0tDfnDtD-1dt,un+1(tD)=u0(tD)-∫0tD{fn-fn-1}DtD-1dx.
As another example, consider the Nth-order fractal differential equation
(7.84)dNudtD+f=0,
which can be converted by the transformation
(7.85)tD=xN
into the ordinary differential equation
(7.86)dNudxN+f=0,
and the variational iteration algorithms are
(7.87)un+1(x)=un(x)+(-1)N∫0x1(N-1)!(s-x)N-1(un(N)+fn)ds,un+1(x)=u0(x)+(-1)N∫0t1(N-1)!(s-x)N-1fnds,un+1(x)=un(x)+(-1)N∫0t1(N-1)!(s-t)N-1(fn-fn-1)ds.
The fractal derivative is simpler than its fractional counterpart in many applications, and it is also valid for discontinuous cases. We rewrite (7.77) in the form
(7.88)du(x)dxD=1klimA→Bu(A)-u(B)x~A-x~B,
where k is a constant, and A and B are arbitrary points in discontinuous space or spacetime (as shown in Figure 8). (x~A, x~B) are called the fractal coordinates and are defined by
(7.89)x~A=k(xA-0)D=k(xA)D,(7.90)x~B=k(xB-0)D=k(xB)D,
where (xA,xB) are the coordinates, and D is the fractional dimension in x-direction. Substituting (7.88) and (7.89) into (7.88), we obtain
(7.91)du(x)dxD=limA→Bu(A)-u(B)(xA)D-(xB)D.
The fractal differential model is particularly suitable for describing discontinuous matter and is the preferred model for describing flow or heat conduction through porous media. For instance, the principle of mass conservation can be written in the form
(7.92)∂ρ∂t+k1∂(ρu)∂xD1+k2∂(ρv)∂yD2+k3∂(ρw)∂zD3=0,
where D1, D2, and D3 are the fractal dimensions of porosity in the x, y, and z directions, respectively, and ki(i=1,2,3) are constants that are related to the fractal dimensions. In particular, we have ki=1, when Di=1. Similarly, the momentum equation for one-dimensional porous flow can be written in the form
(7.93)∂u∂t+ku∂u∂xD=-kρ∂P∂xD+k∂∂xD(μk∂u∂xD),
when D=1 and k=1,  and (7.93) turns out to be the classical one.Figure 8
A schematic diagram of distance betweenO and A (i.e., the fractal coordinate of A) in a fractal space, and D=ln2/ln3, while the fractional dimensions for the plane are ln8/ln3.The one-dimensional heat conduction equation in porous media can be expressed as(7.94)∂T∂t+k∂∂xD(μk∂T∂xD)=0,
where μ is the conduction coefficient and k=1 when D=1.An oscillator swinging in a porous medium can also be described by fractal differential equations. For example, the Duffing equation with fractal damp can be expressed as(7.95)d2udt2+u+kμdudtD+εu3=0.
It is easy to establish fractal differential equations for discontinuous media by replacing ∂/∂x in the classical approach by k∂/∂xD.Now we consider a fractal media illustrated in Figure9, and assume that the smallest measure is L0, any discontinuity less than L0 is ignored, then the distance between two points of A and B in Figure 1 can be expressed using fractal geometry. Hereby we introduce a new fractal derivative for engineering application:
(7.96)Du(t)Dxα=limΔx→L0u(A)-u(B)Thedistancebetweentwopoints=duds=limΔx→L0u(A)-u(B)kL0α,
where k is a constant, α is the fractal dimension, and the distance between two points in a discontinuous space can be expressed as ds=kL0α.Figure 9
The distance between two points in a discontinuous spacetime.Please note in the above definition thatΔx does tend to zero, but to the smallest measure size, L0.Example 7.1.
As a simple application, we consider the Fourier’s law heat conduction, which reads(7.97)∂qh∂t=-cdTdn,
where T is the thermal potential, and qh is heat flow.
In the discontinuous media, the Fourier’s law can be simply modified as(7.98)∂qh∂t=-cDTDnα,
where DT/Dnα is a fractal derivative defined in (7.96).The suggested fractal derivative is easy to be used for any discontinuous problems, and equations with fractal derivative can be easily solved using classical calculus.Letf(x;y;…) be a multivariable real continuous function. The q-derivative and the partial q-derivative are defined by [143]:
(7.99)Dqxf(x)=f(qx)-f(x)(q-1)x,∂qxf(x;y;…)=f(qx;y;…)-f(x;y;…)(q-1)x,
and ∂qxf(x;y;…)|x=0=limn→∞(f(xqn;y;…)-f(0;y;…))/xqn,
(7.100)Dqtu+f(u)=0,0<q<1.
In the case 0<α<1, we rewrite (7.13a) in the form
(7.101)dudt+{Dqtu+f(u)-dudt}=0,
and the variational iteration algorithms are given as follows:
(7.102)un+1(t)=un(t)-∫0t(Dqtun+f(un))ds,un+1(t)=u0(t)-∫0t(Dqtun-dundt+f(un))ds,un+1(t)=u0(t)-∫0t{(Dqtun-dundt+f(un))-(Dαun-1Dtα-dun-1dt+f(un-1))}ds.
In the case 1<α<2, the above iteration formulas are also valid. We can rewrite (7.100) in the form
(7.103)d2udt2+DαuDtα-d2udt2+f=0
and the following iteration formulae are suggested
(7.104)un+1(t)=un(t)+∫0t(s-t)(DαunDtα+fn)ds,un+1(t)=u0(t)+∫0t(s-t)(DαunDtα-d2undt2+fn)ds,un+1(t)=un(t)+∫0t(s-t){(DαunDtα-d2undt2+fn)}-(Dαun-1Dtα-d2un-1dt2+fn-1)ds,
when α is close to 1, (7.102) is better, while (7.104) is recommended for α approaching 2.For the caseN<α<N+1, where N is a natural number, the iteration formulas are as follows
(7.105)un+1(t)=un(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα+fn)dsun+1(t)=u0(t)+(-1)N∫0t1(N-1)!(s-t)N-1(DαunDtα-dNundtN+fn)dsun+1(t)=un(t)+(-1)N+∫0t1(N-1)!(s-t)N-1×{(DαunDtα-dNundtN+fn)-(Dαun-1Dtα-dNun-1dtN+fn-1)}ds,or(7.106)un+1(t)=un(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα+fn)ds,un+1(t)=u0(t)+(-1)N+1∫0t1N!(s-t)N(DαunDtα-dN+1undtN+1+fn)ds,un+1(t)=un(t)+(-1)N+1×∫0t1N!(s-t)N{(DαunDtα-dN+1undtN+1+fn)-(Dαun-1Dtα-dN+1un-1dtN+1+fn-1)}ds.


## 8. Nonlinear Wave Transform for Nonlinear Wave Equations

The linear wave transform is widely used in nonlinear community to search for solitary solutions. Take the well-known KdV equation as an example(8.1)∂u∂t-6u∂u∂x+∂3u∂x3=0.
The general solution approach is to introduce a wave transform
(8.2)ξ=x-Vt
to convert (8.1) into an ordinary differential equation. In the linear wave transform, V has a definite physical understanding, and it is the wave speed.The nonlinear wave transform introduces a complex variableξ defined as
(8.3)ξ=a(t)x+b(t)y+c(t)z+d(t),
where a,b,c, and d are unknown functions of time.In order to elucidate the solution procedure, we consider the following equation:(8.4)ut+t2ux+stuux+rtuxxx=0,
where r and s are constants.By a nonlinear wave transform,(8.5)ξ=x-a1t-a2t2-a3t3,
where a1, a2, and a3 are constants, we can convert (8.4) into an ordinary differential equation,
(8.6)(-a1-2a2t-3a3t2)u′+t2u′+stuu′+rtu′′′=0,
where prime denotes the derivation with respect to ξ.In order to eliminate the time-dependent coefficients in (8.6), we set
(8.7)a1=0,a3=13.
Equation (8.6) becomes
(8.8)-2a2u′+suu′+ru′′′=0.We rewrite (8.8) in the form
(8.9)u′-s2a2uu′-r2a2u′′′=0.
Equation (8.9) can be effectively solved by the exp-function method (see Section 5.9). Hereby we consider a special case
(8.10)s2a2=6,r2a2=-1.
Equation (8.9) becomes the standard KdV equation, which admits the following solitary solution:
(8.11)u=-A2sech2A4ξ=-A2sech2A4(x-a2t2-13t3).



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*Source: 916793-2012-11-29.xml*