# Solution of Singular and Nonsingular Initial and Boundary Value Problems by Modified Variational Iteration Method

Authors: Muhammad Aslam Noor; Syed Tauseef Mohyud-Din
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

We apply the modified variational iteration method (MVIM) for solving the singular and nonsingular initial and boundary value problems in this paper. The proposed modification is made by introducing Adomian's polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discretization, linearization, perturbation, or restrictive assumptions. Several examples are given to verify the efficiency and reliability of the suggested algorithm.


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## Body

## 1. Introduction

Many problems in applied sciences can be modeled by
singular and nonsingular boundary value problems. The application of these
problems involve astrophysics, experimental and mathematical physics, nuclear
charge in heavy atoms, thermal behavior of a spherical cloud of gas,
thermodynamics, population models, chemical kinetics, and fluid mechanics, see
[1–46] and the references therein. Several techniques including
decomposition, variational iteration, finite difference, polynomial spline, and
homotopy perturbation have been developed for solving such problems, see [1–51]
and the references therein. Most of these methods have their inbuilt
deficiencies coupled with the major drawback of huge computational work. These facts have motivated to develop other
methods for solving these problems. Adomian’s decomposition method [42–44, 51]
was employed for finding solution of linear and nonlinear boundary value
problems. He [12–18] developed the variational
iteration method (VIM) for solving linear, nonlinear, initial, and boundary
value problems. It is worth mentioning that the origin of variational iteration
method can be traced back to Inokuti et al. [19]. In these methods,
the solution is given in an infinite series usually converging to an accurate
solution, see [1–5, 12–19, 21, 23, 31–44, 46, 48, 51] and the references
therein. In this paper, we apply the modified variational iteration method
(MVIM), which is obtained by the elegant coupling of variational iteration
method and the Adomian’s polynomials for solving singular and nonsingular
initial and boundary value problems. This idea has been used by Abbasbandy [1, 2] implicitly, and by Noor and Mohyud-Din [36, 38, 40] for the solution of
nonlinear boundary value problems. The basic motivation of this paper is to
apply this modified variational iteration method (MVIM) for finding the solution of singular and nonsingular initial
and boundary value problems. It is shown that the MVIM provides the
solution in a rapid convergent series with easily computable components. We
write the correct functional for the boundary value problem and calculate the
Lagrange multiplier optimally. The Adomian’s polynomials are introduced in the
correct functional and evaluated by using the specific algorithm [42–44] and
the references therein. Finally, the approximants are calculated by employing
the Lagrange multipliers and the Adomian’s polynomial scheme simultaneously.
The use of Lagrange multiplier reduces the successive application of the
integral operator and minimizes the computational work. Moreover, the selection
of the initial value is done by exploiting the concept of modified
decomposition method. In the present study, we apply this technique to solve
boundary layer problem, unsteady flow of gas, singularly perturbed sixth-order
Boussinesq, third-order dispersive, and fourth-order parabolic equations. To
make the work more concise and to get a better understanding of the solution
behavior, in case of boundary layer problem and the unsteady flow of gas, we
replace the series solutions by the powerful Pade approximants [22, 28, 34, 43, 44, 47]. The use of Pade approximants shows real promise in solving boundary
value problems in an infinite domain. The proposed MVIM solves effectively,
easily, and accurately a large class of linear, nonlinear, partial,
deterministic, or stochastic differential equations with approximate solutions
which converge very rapidly to accurate solutions. Our results can be viewed as
important and significant improvement of the previously known results.


## 2. Variational Iteration Method

To illustrate the basic concept of the technique, we consider the following general differential equation:(2.1)Lu+Nu=g(x),where L is a
linear operator, N a nonlinear operator, and g(x) is the inhomogeneous term.
According to variational iteration method [1–5, 13–19, 21, 23, 31–41, 46, 48],
we can construct a correct functional as follows:
(2.2)un+1(x)=un(x)+∫0xλ(s)(Lun(s)+Nu˜n(s)−g(s))ds,where λ(s) is a Lagrange multiplier [13–18], which can be
identified optimally via variational iteration method. The subscripts n denote
the nth approximation, u˜n is considered as a restricted variation, that is, δu˜n=0. Relational  (2.2) is called as a correct functional. The solution of the linear
problems can be solved in a single iteration step due to the exact
identification of the Lagrange multiplier. The principles of variational
iteration method and its applicability for various kinds of differential
equations are given in [13–18]. In this method, it is required first to
determine the Lagrange multiplier λ optimally. The successive approximation un+1,n≥0 of the solution u will be readily obtained
upon using the determined Lagrange multiplier and any selective function. Consequently,
the solution is given by u=limn→∞un. For the convergence criteria and error
estimates of variational iteration method, see Ramos [41].


## 3. Adomian’s Decomposition Method

To convey an
idea of the technique, we consider the differential equation [42–44] of the form
(3.1)Lu+Ru+Nu=g,
where L is the
highest-order derivative which is assumed to be invertible, R is a linear
differential operator of order lesser than
L, Nu represents the nonlinear terms, and g is the
source term. Applying the inverse operator L−1 to both sides of (3.1) and using the given conditions, we obtain
(3.2)u=f−L−1(Ru)−L−1(Nu),where the function f represents the terms arising from integrating
the source term g and by using the given conditions. Adomian’s decomposition
method [42–44] defines the solution by the series
(3.3)u(x)=∑n=0∞un(x),where the components un(x) are usually determined recurrently by using
the relation
(3.4)u0=f,uk+1=L−1(Ruk)−L−1(Nuk),k≥0.
The nonlinear
operator N(u) can be decomposed into an infinite series of
polynomials given by
(3.5)N(u)=∑n=0∞An,where An are the so-called Adomian’s polynomials that can
be generated for various classes of nonlinearities according to the specific algorithm
developed in [42–44] which yields
(3.6)An=(1n!)(dndλn)N(∑i=0n(λiui))λ=0,n=0,1,2,….


## 4. Modified Variational Iteration Method (MVIM)

To illustrate
the basic concept of the variational decomposition method, we consider the
following general differential (4.1), we have
(4.1)Lu+Nu=g(x),where L is a linear operator, N is a nonlinear operator, and g(x) is the forcing term.According to variational iteration method [1–5, 13–19, 21, 23, 31–41, 46, 48], we can construct a correct functional as follows:
(4.2)un+1(x)=un(x)+∫0xλ(Lun(s)+Nu˜n(s)−g(s))ds,where λ is a Lagrange multiplier [13–18], which can be
identified optimally via variational iteration method. The subscripts n denote
the nth approximation, u˜n is considered as a restricted variation, that is, δu˜n=0 (4.2) is called as a correct functional. We define the solution u(x) by the series
(4.3)u(x)=∑i=0∞ui(x),and the nonlinear term
(4.4)N(u)=∑n=0∞An(u0,u1,u2,…,ui),where An are the so-called Adomian’s polynomials and
can be generated for all types
of nonlinearities according to the algorithm developed, in [42–44] which yields
the following:
(4.5)An=(1n!)(dndλn)N(u(λ)).Hence, we obtain
the following iterative scheme for finding the approximate solution
(4.6)un+1(x)=un(x)+∫0tλ(Lun(x)+∑n=0∞An−g(x))dx,which is called the
modified variational iteration method (MVIM) and is formulated by the elegant coupling
of variational iteration method and the Adomian’s polynomials.


## 5. Pade Approximants

A Pade
approximant is the ratio of two polynomials constructed from the coefficients
of the Taylor series expansion of a
functionu(x). The [L/M] Pade
approximants to a function y(x) are given by [22, 28, 34, 43, 44, 47]
(5.1)[LM]=PL(x)QM(x),where PL(x) is polynomial of degree at most L and QM(x) is a polynomial of degree at most M. The formal
power series
(5.2)y(x)=∑i=1∞aixi,y(x)−PL(x)QM(x)=O(xL+M+1),
determine the coefficients of PL(x)andQM(x) by the equation. Since we can clearly multiply
the numerator and denominator by a constant and leave [L/M] unchanged, we
imposed the normalization condition
(5.3)QM(0)=1.0.
Finally, we require that PL(x)andQM(x) have non common factors. If we write the
coefficient of PL(x)andQM(x) as
(5.4)PL(x)=p0+p1x+p2x2+⋯+pLxL,QM(x)=q0+q1x+q2x2+⋯+qMxM.Then by (5.3) and (5.4), we may
multiply (5.5) by QM(x),
which linearizes the coefficient equations. We can write out (5.5) in more
details as
(5.5)aL+1+aLq1+⋯+aL−MqM=0,qL+2+qL+1q1+⋯+aL−M+2qM=0,⋮aL+M+aL+M−1q1+⋯+aLqM=0,(5.6)a0=p0,a0+a0q1+⋯+=p1,⋮aL+aL−1q1+⋯+a0qL=pL.To solve these
equations, we start with (5.5), which is a set of linear equations for all the
unknown q's.
Once the q's are known, then (5.6) gives an explicit formula for the unknown p's,
which complete the solution. If (5.5) and (5.6) are nonsingular, then we can
solve them directly and obtain (5.7) [22], where (5.7) holds, and if the lower
index on a sum exceeds the upper, the sum is replaced by zero:
(5.7)[LM]=det[aL−M+1aL−M+2⋯aL+1⋮⋮⋱⋮aLaL+1⋯aL+M∑j=MLaj−Mxj∑j=M−1Laj−M+1xj⋯∑j=0Lajxj]det[aL−M+1aL−M+2⋯aL+1⋮⋮⋱aLaL+1⋯aL+MxMxM−1⋯1].To obtain diagonal Pade
approximants of different order such as [2/2], [4/4], or [6/6], we can use the
symbolic calculus software Maple.


## 6. Numerical Applications

In this section,
we apply the modified variational iteration method (MVIM) for solving the singular
and nonsingular boundary value problems. We write the correct functional for
the boundary value problem and carefully select the initial value because the
approximants are heavily dependant on the initial value. The Adomian’s
polynomials are introduced in the correct functional for the nonlinear terms.
The results are very encouraging indicating the reliability and efficiency of
the proposed method. We apply the MVIM for
solving the boundary layer problem; unsteady flow of gas through a porous
medium; Boussinesq equations, third-order dispersive, and fourth-order
parabolic singular partial differential equations. The powerful Pade
approximants are applied in case of boundary-layer problem and unsteady flow in
order to make the work more concise and for better understanding of the
solution behavior.Example 6.1  (see [43]).
Consider the following nonlinear third-order boundary layer problem which
appears mostly in the mathematical modeling of physical phenomena in fluid
mechanics [43, 45]:
(6.1)f′′′(x)+(k−1)f(x)f′′(x)−2n(f′(x))2=0,n>0,with boundary conditions
(6.2)f(0)=0,f′(0)=1,f′(∞)=0,k>0.
The correct functional is given as
(6.3)fn+1(x)=fn(x)+∫0xλ(s)(fn′′′(s)+(k−1)f˜n(x)f˜′′n(s)−2n(f˜n′(s))2)ds=0,k>0.Making the
correct functional stationary, the Lagrange multipliers can be identified as λ(s)=−(1/2!)(s−x)2, consequently, we have
(6.4)fn+1(x)=fn(x)−∫0x12!(s−x)2(fn′′′(s)+(k−1)f˜n(s)f˜′′n(x)−2n(f˜n′(s))2)ds=0,k>0,where f′′(0)=α<0. Applying the modified variational iteration
method, we have
(6.5)fn+1(x)=fn(x)+∫0x12!(s−x)2(fn′′′(s)+(k−1)∑n=0∞An−2n∑n=0∞Bn)ds=0,where An and Bn are the so-called Adomian’s polynomials and
can be generated for all types
of nonlinearities according to the algorithm defined in [42–44]. Consequently
the following approximants are made:
(6.6)f0(x)=x,f1(x)=x+12αx2+13x3,f2(x)=x+12αx2+13x3+124α(3n+1)x4+130n(n+1)x5,f3(x)=x+12αx2+13x3+124α(3n+1)x4+130n(n+1)x5+1120α2(3n+1)x5+1720α(19n2+18n+3)x6+1315n(2n2+2n+1)x7,f4(x)=x+12αx2+13x3+124α(3n+1)x4+130n(n+1)x5+1120α2(3n+1)x5+1720α(19n2+18n+3)x6+1315n(2n2+2n+1)x7,15040α2(27n2+42n+11)x7+140320α(167n3+297n2+161n+15)x8+122680n(13n3+38n2+23n+6)x9,⋮The series solution is given as
(6.7)f(x)=x+αx22+nx33+(18nα+124α)x4+(130n2+140nα2+1120α2+130n)x5+(19720n2α+1240α+140nα)x6+(1120nα2+1315n+2315n3+115040α2+3560n2α2+2315n2)x7+(1140320α3+334480n2α+34480α3n2+235760nα+12688α+16740320n3α+1960α3n)x8+(13780n+527362880n3α2+1911340n3+709362880nα2+238064n2α2+2322680n2+1322680n4+43120960α2)x9+⋯.Example 6.2  (see [34, 44]).
Consider the following
nonlinear differential equation which
governs the unsteady flow of gas through a porous medium:(6.8)y′′(x)+2x1−αyy′(x)=0,0<α<1.With the following typical boundary
conditions imposed by the physical properties [34, 44],
(6.9)y(0)=1,limx→∞y(x)=0.The correct
functional is given as
(6.10)yn+1(x)=yn(x)+∫0xλ(s)(y′′(s)+2x1−αyy′(s))ds,0<α<1.Making the
correct functional stationary, using λ=x−s, as the Lagrange multiplier, we get the
following iterative formula:
(6.11)yn+1(x)=yn(x)+∫0x(s−x)(y′′(s)+2x1−αyy′(s))ds,0<α<1,where
(6.12)A=y′(0).Applying the modified
variational iteration method, we have
(6.13)yn+1(x)=yn(x)+∫0x(s−x)(y′′(s)+2x∑n=0∞An)ds,0<α<1,where An are the so-called Adomian’s polynomials and
can be generated for all types
of nonlinearities according to the algorithm defined in [42–44]. First few
Adomian’s polynomials are as under:
(6.14)A0=(1−αy0)−1/2y0′,A1=(1−αy0)−1/2y1′+α2(1−αy0)−3/2y0′y1,A2=(1−αy0)−1/2y2′+α2(1−αy0)−3/2y1′y1+α2(1−αy0)−3/2y0′y2+38α2(1−αy0)−5/2y0′y12,A3=(1−αy0)−1/2y3′+α2(1−αy0)−3/2y2′y1+α2(1−αy0)−3/2y1′y2+α2(1−αy0)−3/2y0′y3+38α2(1−αy0)−5/2y1′y12+34α2(1−αy0)−5/2y0′y1y2+516α3(1−αy0)−7/2y0′y13,⋮Consequently,
the following approximants are obtained:
(6.15)y0(x)=1,y1(x)=1+Ax,y2(x)=1+Ax−A31−αx3,y3(x)=1+Ax−A31−αx3−αA212(1−α)3/2x4+A10(1−α)x5,y4(x)=1+Ax−A31−αx3−αA212(1−α)3/2x4+A10(1−α)x5−3α2A380(1−α)5/2x5+αA215(1−α)2x6+O(x7),y5(x)=1+Ax−A31−αx3−αA212(1−α)3/2x4+A10(1−α)x5−3α2A380(1−α)5/2x5+αA215(1−α)2x6−α3A448(1−α)7/2x6+O(x7),⋮ The series solution is given as
(6.16)y(x)=1+Ax−A31−αx3−αA212(1−α)3/2x4+(A10(1−α)−3α2A380(1−α)5/2)x5+(αA215(1−α)2−α3A448(1−α)7/2)x6+O(x7).Now, we
investigate the mathematical behavior of the solution y(x) in order to determine the initial slope y′(0). This goal can be achieved by forming diagonal Pade
approximants [34, 44, 47] which have the advantage of manipulating the
polynomial approximation into a rational function to gain more information
about y(x).
It is well-known that Pade approximants will converge on the entire real axis [20, 22, 28, 34, 43, 44, 47], if y(x) is free of singularities on the real axis. It
is of interest to note that Pade approximants give results with no greater
error bounds than approximation by polynomials. More importantly, the diagonal
approximant is the most accurate approximant; therefore we will construct only
the diagonal approximants in the following discussions. Using the boundary
condition y(∞)=0,
the diagonals approximant [M/M]
vanishes if the coefficient of x with the highest power in the numerator vanishes. The
computational work can be performed by using the mathematical software MAPLE.
The physical behavior indicates that y(x) is a decreasing function, hence y′(0)<0.
Using this fact, and following [20, 34, 44], complex roots and nonphysical
positive roots should be excluded. Based on this, the [2/2] Pade approximant
produced the slope A to be
(6.17)A=−2(1−α)1/43α,and using [3/3]
Pade approximants we find
(6.18)A=−(−4674α+8664)1−α−144γ57α,where
(6.19)γ=5(1−α)(1309α2−2280α+1216).Using (6.17)–(6.19) gives the
values of the initial slope A=y′(0) listed in Table 1. The formulas (6.17) and (6.19)
suggest that the initial slope A=y′(0) depends mainly on the parameter α,
where 0<α<1.
Table 3 shows that the initial slope A=y′(0) increases with the increase of α.
The mathematical structure of y(x) was successfully enhanced by using the Pade
approximants. Table 4 indicates the values of y(x) [34, 44] and by using the [2/2] and [3/3]
approximants for specific value of α=0.5.Table 1
Numerical values forα=f′′(0) for 0<k<1 by using diagonal Pade approximants [43].
n[2/2][3/3][4/4][5/5][6/6]0.2−0.3872983347−0.3821533832−0.3819153845−0.3819148088−0.38191218541/3−0.5773502692−0.5615999244−0.5614066588−0.5614481405−0.5614419340.4−0.6451506398−0.6397000575−0.6389732578−0.6389892681−0.63897347940.6−0.8407967591−0.8393603021−0.8396060478−0.8395875381−0.83960567690.8−1.007983207−1.007796981−1.007646828−1.007646828−1.007792100Table 2
Numerical values forα=f′′(0) for k>1 by using diagonal Pade approximants [43].
nα4−2.48395403210−4.026385103100−12.843343151000−40.655382185000−104.8420672Table 3
The initial slopesA=y′(0) for various values of α.
αB[2/2]=y′(0)B[3/3]=y′(0)0.1−3.556558821−1.9572089530.2−2.441894334−1.7864755160.3−1.928338405−1.4782708430.4−1.606856838−1.2318018090.5−1.373178096−1.0255297040.6−1.185519607−0.84003460850.7−1.021411309−0.66120478930.8−0.8633400217−0.47766972860.9−0.6844600642−0.2772628386Table 4
The values ofy(x) for α=0.5 for x=0.1 to 1.0.
Xy kiddery[2/2]y[3/3]0.10.88165882830.86330606410.89791670280.20.76630767810.73012622610.79852281990.30.65653799950.60330541400.70411297030.40.55440240320.48488987170.61650379010.50.46136502950.37616038690.53705337960.60.37831093150.27773116280.46656256690.70.30559765460.18968433710.40624260330.80.24313254730.11171051650.35608016990.90.19046236810.043236732360.31799666141.00.15876898260.016467508470.2900255005Table 5
(Error estimates). The absolute error between the exact and the series solutions. Higher accuracy can be obtained by introducing some more components of the series solution.
xitj0.010.020.040.10.20.5−12.80886E−141.79667E−121.15235E−102.83355E−81.83899E−64.74681E−4−0.86.27276E−144.01362E−122.57471E−106.33178E−84.10454E−61.04489E−3−0.66.08402E−143.90188E−122.25663E−106.18024E−84.02299E−61.03093E−3−0.41.16573E−147.41129E−134.82756E−111.23843E−88.53800E−62.46302E−4−0.25.53446E−143.53395E−122.25663E−105.47485E−83.47264E−68.35783E−408.63198E−145.53357E−122.54174E−108.65197E−85.54893E−61.37353E−30.25.56222E−143.55044E−122.27779E−105.60362E−83.63600E−69.29612E−40.41.14353E−147.14928E−134.49107E−111.03370E−85.93842E−79.61260E−50.66.06182E−143.87551E−122.47218E−105.97562E−83.76275E−68.79002E−40.86.23945E−143.99519E−122.55127E−106.18881E−83.92220E−69.36404E−412.79776E−141.78946E−121.14307E−102.77684E−81.76607E−64.28986E−4Table 6
The absolute error between the exact and the series solutions. Higher accuracy can be obtained by introducing some more components of the series solution.
xitj0.010.020.040.10.20.5−17.77156E−161.36557E−148.57869E−132.09264E−101.33823E−83.25944E−6−0.81.11022E−161.99840E−151.12688E−132.73880E−111.74288E−94.14094E−7−0.62.22045E−161.09912E−147.28861E−131.78030E−101.14025E−82.79028E−6−0.41.11022E−162.32037E−141.50302E−123.67002E−102.34944E−85.74091E−6−0.26.66134E−163.23075E−142.04747E−124.99918E−103.19983E−97.81509E−604.44089E−163.49720E−142.24365E−125.47741E−103.50559E−88.55935E−60.25.55112E−163.19744E−142.04714E−124.99820E−103.19858E−87.80749E−60.43.33067E−162.32037E−141.50324E−123.66815E−102.34706E−85.72641E−60.63.33067E−161.12133E−147.28528E−121.77772E−101.13695E−82.77022E−60.83.33067E−161.99840E−151.13132E−132.76944E−111.78208E−94.41936E−717.77156E−161.38778E−148.58313E−132.09593E−101.34244E−83.28504E−6Example 6.3  (see [32, 51]).
Consider the following singularly perturbed sixth-order Boussinesq equation:(6.20)utt=uxx+(p(u))xx+αuxxxx+βuxxxxxx,taking α=1,β=0,andp(u)=3u2, the model equation is given as
(6.21)utt=uxx+3(u2)xx+uxxxx,with initial conditions
(6.22)u(x,0)=2ak2ekx(1+aekx)2,ut(x,0)=2ak31+k2(1−aekx)ekx(1+aekx)3,where a and k are arbitrary
constants. The exact solution u(x,t) of the problem is given as [32, 51]
(6.23)u(x,t)=2ak2exp(kx+k1+k2t)(1+aexp(kx+k1+k2t))2.The correct functional is given as
(6.24)un+1(x,t)=2ak2ekx(1+aekx)2+2ak31+k2(1−aekx)ekx(1+aekx)3t+∫0tλ(∂2un∂t2−(u˜n)xx−(u˜n)xxxx−3∑n=0∞Bn)dt,where Bn are Adomian’s polynomials for nonlinear
operator F(u)=u2(x) and can be generated for all types of nonlinearities
according to the algorithm developed in [42–44, 51] which yields the following:
(6.25)B0=(u02)xx,B1=2u0u1xx+4u0xu1x+2u0xxu1,B2=2u0u2xx+4u0xu2x+2u0xxu2+2u1xu1xx+2(u1x)2B3=(2u0u3+2u1u2)xx,⋮ Making the correct functional
stationary, the Lagrange multiplier can be identified as λ=x−t,
consequently,
(6.26)un+1(x,t)=2ak2ekx(1+aekx)2+2ak31+k2(1−aekx)ekx(1+aekx)3t+∫0t(x−t)(∂2un∂t2−(u˜n)xx−(u˜n)xxxx−3∑n=0∞Bn)dt.The following approximants are
obtained:
(6.27)u0(x,t)=2ex(1+ex)2,u1(x,t)=2ex(1+ex)2+2ak31+k2(1−aekx)ekx(1+aekx)3t+2ex(1−4ex+e2x)(1+ex)4t2,u2(x,t)=2ex(1+ex)2+2ak31+k2(1−aekx)ekx(1+aekx)3t+2ex(1−4ex+e2x)(1+ex)4t2−22ex(−1+ex)(1−10ex+e2x)3(1+ex)5t3+ex(1−4ex+e2x)(1−44ex+78e2x−44e3x+e4x)3(1+ex)8t4,u3(x,t)=2ex(1+ex)2+2ak31+k2(1−aekx)ekx(1+aekx)3t+2ex(1−4ex+e2x)(1+ex)4t2−22ex(−1+ex)(1−10ex+e2x)3(1+ex)5t3+ex(1−4ex+e2x)(1−44ex+78e2x−44e3x+e4x)3(1+ex)8t4−2ex(−1+ex)(1−56ex+246e2x−56e3x+e4x)15(1+ex)7t5+ex(1−452ex+19149e2x−207936e3x+807378e4x−1256568e5x)45(1+ex)12t6+ex(807378e6x−207936e7x+19149e8x−452e9x+e10x)45(1+ex)12t6,⋮The series solution is given as
(6.28)u(x,t)=2ex(1+ex)2+2ak31+k2(1−aekx)ekx(1+aekx)3t+2ex(1−4ex+e2x)(1+ex)4t2−22ex(−1+ex)(1−10ex+e2x)3(1+ex)5t3+ex(1−4ex+e2x)(1−44ex+78e2x−44e3x+e4x)3(1+ex)8t4+8e2x(1−10ex+20e2x−10e3x+e4x)(1+ex)8t4−2ex(−1+ex)(1−56ex+246e2x−56e3x+e4x)15(1+ex)7t5+ex(1−452ex+19149e2x−207936e3x+807378e4x−1256568e5x)45(1+ex)12t6+ex(807378e6x−207936e7x+19149e8x−452e9x+e10x)45(1+ex)12t6+⋯.Example 6.4  (see [32, 51]).
Consider the following singularly perturbed sixth-order Boussinesq equation:(6.29)utt=uxx+(u2)xx−uxxxx+12uxxxxxx,with initial conditions
(6.30)u(x,0)=−105169sech4(x26),ut(x,0)=−210194/13sech4(x/26)tanh(x/26)2197.The exact solution of the problem
is given as
(6.31)u(x,t)=−105169sech4[126(x−97169t)].Applying the modified variational
iteration method, we obtain
(6.32)un+1(x,t)=−105169sech4(x26)+−210194/13sech4(x/26)tanh(x/26)2197t+∫0tλ(∂2un∂t2−(u˜n)xx+(u˜n)xxxx−12(u˜n)xxxxxx+∑n=0∞bn)dt.Making the correct functional
stationary, the Lagrange multiplier can be identified as λ=x−t, consequently
(6.33)un+1(x,t)=−105169sech4(x26)+−210194/13sech4(x/26)tanh(x/26)2197t+∫0t(x−t)(∂2un∂t2−(u˜n)xx+(u˜n)xxxx−12(u˜n)xxxxxx+∑n=0∞Bn)dt,where Bn are Adomian’s polynomials for nonlinear
operator F(u)=u2(x) and can be generated for all types of nonlinearities
according to the algorithm developed in [42–44, 51]. Consequently, the
following approximants are obtained:
(6.34)u0(x,t)=−105169sech4(x26),u1(x,t)=−105169sech4(x26)−105194/13sech6(x/26)sinh(2x/13)2197t−105371293(−291+194cosh(2x13))sech6x26t2,u2(x,t)=−105169sech4(x26)−105194/13sech6(x/26)sinh(2x/13)2197t−105371293(−291+194cosh(2x13))sech6x26t2+395sech7(x/26)52206766144(108162522sinhx26−16642522sinh3x26)t3+(−334200sech5(x26)+354247cosh(213x)sech5(x26)−47164cosh(2213x)sech5(x26))t4+(3201cosh3(3213x)sech5(x26)−388cosh(4213x)sech5(x26))t4+⋯,⋮The series solution is obtained as
(6.35)u(x,t)=−105169sech4(x26)−105194/13sech6(x/26)sinh(2x/13)2197t−105371293(−291+194cosh(2x13))sech6x26t2+395sech7(x/26)52206766144(108162522sinhx26−16642522sinh3x26)t3+(−334200sech5(x26)+354247cosh(213x)sech5(x26)−47164cosh(2213x)sech5(x26))t4+(3201cosh3(3213x)sech5(x26)−388cosh(4213x)sech5(x26))t4+⋯.Example 6.5.
Consider the following linear third-order dispersive
KdV equation:(6.36)ut+2ux+uxxx=0,t>0,with initial
condition
(6.37)u(x,0)=sinx.The correct
functional is given as
(6.38)un+1(x,t)=un(x,t)+∫0xλ(s)(∂un∂t+2∂u˜n∂x+∂3u˜n∂x3)ds.Making the
correct functional stationary, the Lagrange multiplier can be identified as λ=−1, consequently,
(6.39)un+1(x,t)=un(x,t)−∫0x(∂un∂t+2∂u˜n∂x+∂3u˜n∂x3)ds.Applying the
modified variational iteration method,
(6.40)un+1(x,t)=un(x,t)−∫0x(∂un∂t+2∑n=0∞∂u˜n∂x+∑n=0∞∂3u˜n∂x3)ds.Consequently, the following
approximants are obtained:
(6.41)u0(x,t)=sinx,u1(x,t)=sinx−tcosx,u2(x,t)=sinx(1−t22!)−tcosx,u3(x,t)=sinx(1−t22!)−cosx(t−t33!),⋮The series
solution is given by
(6.42)u(x,t)=sinx(1−t22!+t44!−⋯)−cosx(t−t33!+t55!+⋯),and in a closed
form by
(6.43)u(x,t)=sin(x−t).If we change the
initial condition as u(x,0)=cosx, than the following closed-form solution will
be obtained:
(6.44)u(x,t)=cos(x−t).Example 6.6.
Consider the following linear third-order dispersive
KdV equation in a two-dimensional space:(6.45)ut+uxxx+uyyy=0,t>0,with initial
condition
(6.46)u(x,y,0)=cos(x+y).The correct
functional is given as
(6.47)un+1(x,y,t)=un(x,y,t)+∫0xλ(s)(∂un∂t+2∂3u˜n∂x3+∂3u˜n∂y3)ds.Making the correct
functional stationary, the Lagrange multiplier can be identified as λ=−1, consequently,
(6.48)un+1(x,y,t)=un(x,y,t)−∫0x(∂un∂t+2∂3u˜n∂x3+∂3u˜n∂y3)ds.Applying the
modified variational iteration method,
(6.49)un+1(x,y,t)=un(x,y,t)−∫0x(∂un∂t+2∑n=0∞∂3u˜n∂x3+∑n=0∞∂3u˜n∂y3)ds.Consequently,
the following approximants are obtained:
(6.50)u0(x,y,t)=cos(x+y),u1(x,y,t)=cos(x+y)−2tsin(x+y),u2(x,y,t)=cos(x+y)(1−(2t)22!)−(2t)sin(x+y),u3(x,y,t)=cos(x+y)(1−(2t)22!)−sin(x+y)((2t)−(2t)33!),⋮The series
solution is given by
(6.51)u(x,y,t)=cos(x+y)(1−(2t)22!+⋯)−sin(x+y)((2t)−(2t)33!+⋯),and in a closed
form by
(6.52)u(x,y,t)=cos(x+y+2t).If we change the
initial condition as u(x,y,0)=sin(x+y), than the following closed-form solution will
be obtained:
(6.53)u(x,y,t)=sin(x+y+2t).Example 6.7  ([42]).
Consider the following
singular fourth-order parabolic partial differential equation in two space
variables:(6.54)∂2u∂t2+2(1x2+x46!)∂4u∂x4+2(1y2+y46!)∂4u∂y4=0,with initial
conditions
(6.55)u(x,y,0)=0,∂u∂t(x,y,0)=2+x66!+y66!,and the boundary
conditions
(6.56)u(12,y,t)=(2+(0.5)66!+y66!)sint,u(1,y,t)=(2+16!+y66!)sint,∂2u∂x2(12,y,t)=(0.5)424sint,∂2u∂x2(1,y,t)=124sint,∂2u∂y2(x,12,t)=(0.5)424sint,∂2u∂y2(x,1,t)=124sint.The correct functional is given as
(6.57)un+1(x,t)=u0(x,t)+∫0tλ(ξ)(∂2un∂t2+2(1x2+x46!)∂4u˜n∂x4+(2(1y2+y46!)∂4u˜n∂y4))dξ,where u˜n is considered as a restricted variation.
Making the above functional stationary, the Lagrange multiplier can be
identified as λ=ξ−t, consequently,
(6.58)un+1(x,t)=u0(x,t)+∫0t(ξ−t)(∂2un∂t2+2(1x2+x46!)∂4u˜n∂x4+(2(1y2+y46!)∂4u˜n∂y4))dξ.Applying the
modified variational iteration method, we have
(6.59)un+1(x,t)=u0(x,t)+∫0t(ξ−t)(∂2un∂t2+2(1x2+x46!)∑n=0∞(∂4u˜n∂x4)+(2(1y2+y46!)∑n=0∞(∂4u˜n∂y4)))dξ.Consequently, the following
approximants are obtained as:
(6.60)u0(x,t)=(2+x66!+y66!)tu1(x,t)=(2+x66!+y66!)(t−t33!)u2(x,t)=(2+x66!+y66!)(t−t33!+t55!),u3(x,t)=(2+x66!+y66!)(t−t33!+t55!−t77!),u4(x,t)=(2+x66!+y66!)(t−t33!+t55!−t77!+t99!),⋮The solution is
given by
(6.61)u(x,y,t)=(2+x66!+y66!)(t−t33!+t55!−t77!+t99!+⋯)=(2+x66!+y66!)sint.Remark 6.8.
It is worth mentioning that Ghorbani and Saberi-Nadjafi [49] and Ghorbani [50]
introduced He polynomials which are compatible to Adomian’s polynomials, are
easier to calculate, and hence make the solution procedure simpler.


## 7. Conclusion

In this paper,
we applied the modified variational iteration method (MVIM) for solving singular
and nonsingular initial and boundary value problems. The proposed technique is
applied on boundary layer problem, unsteady flow of gas, Boussinesq equations, third-order
dispersive and fourth-order parabolic partial differential equations. The Pade
approximants were employed in order to make the work more concise and for
better understanding of the solution behavior. It may be concluded that the
proposed frame work is very powerful and efficient in finding the analytical
solutions for singular and nonsingular boundary value problems. The method
gives more realistic series solutions that converge very rapidly in physical
problems.Figure 1
(α=−2.483954032).Figure 2Figure 3Figure 4Figure 5
(t=1).Figure 6
(t=1).



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