# Approximate Traveling Wave Solutions of  Coupled Whitham-Broer-Kaup Shallow Water  Equations by Homotopy Analysis Method

Authors: M. M. Rashidi; D. D. Ganji; S. Dinarvand
Journal: Differential Equations and Nonlinear Mechanics
 (2008)

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## Abstract

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave
solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water.
Comparisons are made between the results of the proposed method and exact solutions.
The results show that the homotopy analysis method is an attractive method in solving the systems
of nonlinear partial differential equations.


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## Body

## 1. Introduction

In 1992, Liao [1] employed the basic ideas of the homotopy in topology to propose method for nonlinear problems, namely, homotopy analysis method (HAM), [2–6]. This method
has many advantages over the classical methods; mainly, it is independent of
any small or large quantities. So, the HAM can be applied no matter if
governing equations and boundary/initial conditions contain small or large
quantities or not. The HAM also avoids discretization and provides an efficient
numerical solution with high accuracy, minimal calculation, and avoidance of
physically unrealistic assumptions. Furthermore, the HAM always provides us
with a family of solution expressions in the auxiliary parameter h; the
convergence region and rate of each solution might be determined conveniently
by the auxiliary parameter h. This method has been successfully applied
to solving many types of nonlinear problems [7–11].A substantial
amount of research work has been invested in the study of linear and nonlinear
systems of partial differential equations (PDEs). Systems of nonlinear partial
differential equations  arise in many scientific models such as the propagation of shallow
water waves and the Brusselator model of the chemical reaction-diffusion model.Here, we consider the coupled Whitham-Broer-Kaup (WBK)
equations which have been studied by Whitham [12], Broer [13], and Kaup [14].
The equations describe the propagation of shallow water waves, with different
dispersion relations. The WBK equations are as follows:
(1.1)∂u∂t+u∂u∂x+∂v∂x+β∂2u∂x2=0,∂v∂t+∂∂x(uv)+α∂3u∂x3−β∂2v∂x2=0,
where u=u(x,t) is the horizontal velocity, v=v(x,t) is the height
that deviates from equilibrium position
of the liquid, and α,β are constants which are represented in
different diffusion powers [15].
The exact solutions of u(x,t) and v(x,t) are given by [16]
(1.2)u(x,t)=λ−2k(α+β2)0.5coth[k(x+x0)−λt],v(x,t)=−2k2(α+β2+β(α+β2)0.5)csch2[k(x+x0)−λt],
where λ,k, and x0 are
arbitrary constants. Above system is a very good model to describe dispersive
waves. If α=1 and β=0, then
the system represents the modified Boussinesq (MB) equations [16]. If α=0 and β≠0, then
the system represents the classical long-wave equations that describe shallow
water wave with dispersion [15].This Letter has
been organized as follows. In Section2, the basic concept of the HAM is introduced. In Section
3, we extend the application of the HAM to construct approximate solutions for
the coupled WBK equations. Numerical experiments are presented in Section 4.


## 2. Basic Concepts of HAM

Let us consider the following differential
equation:(2.1)𝒩[w(τ)]=0,
where 𝒩 is a nonlinear operator; τ denotes independent variable; w(τ) is an unknown function, respectively. For
simplicity, we ignore all boundary or initial conditions, which can be treated
in the similar way. By means of generalizing the traditional homotopy method,
Liao [5]
constructs the so-called zero-order deformation equation:
(2.2)(1−p)ℒ[φ(τ;p)−w0(τ)]=ph𝒩[φ(τ;p)],
where p∈[0,1] is the embedding parameter; h≠0 is a nonzero auxiliary parameter; ℒ is an auxiliary linear operator; w0(τ) is an initial guess of w(τ);φ(τ;p)  is an unknown function, respectively. It is
important that one has great freedom to choose auxiliary things in HAM.
Obviously, when p=0 and p=1, it holds
(2.3)φ(τ;0)=w0(τ),φ(τ;1)=w(τ),
respectively. Thus, as p increases from 0 to 1, the solution φ(τ;p) varies from the initial guess w0(τ) to the solution w(τ). Expanding φ(τ;p) in Taylor series with respect to p, we have
(2.4)φ(τ;p)=w0(τ)+∑m=1+∞wm(τ)pm,
where
(2.5)wm(τ)=1m!∂mφ(τ;p)∂pm|p=0.
If the auxiliary linear operator, the
initial guess, and the auxiliary parameter h are so properly chosen, the series (2.4)
converges at p=1, then we have
(2.6)w(τ)=w0(τ)+∑m=1+∞wm(τ),
which must be one of
solutions of original nonlinear equation, as proved by Liao [5].
As h=−1, (2.2)
becomes
(2.7)(1−p)ℒ[φ(τ;p)−w0(τ)]+p𝒩[φ(τ;p)]=0,
which is used mostly in the homotopy
perturbation method, whereas the solution is obtained directly, without using Taylor series [17, 18].According to definition (2.5),
the governing equation can be deduced from the zero-order deformation (2.2).
Define the vector
(2.8)w→n={w0(τ),w1(τ),…,wn(τ)}.
Differentiating (2.2) m times with respect to the embedding parameter p and then
setting p=0 and finally dividing them by m!, we
have the so-called mth-order deformation equation:
(2.9)ℒ[wm(τ)−χmwm−1(τ)]=hRm(w→m−1),
where
(2.10)Rm(w→m−1)=1(m−1)!∂m−1𝒩[φ(τ;p)]∂pm−1|p=0,χm={0,m≤1,1,m>1.It should be emphasized
thatwm(τ) for m≥1 is governed by the linear (2.9)
with the linear boundary conditions that come from original problem, which can
be easily solved by symbolic computation software such as Maple and
Mathematica.


## 3. Application

First, we consider the coupled Whitham-Broer-Kaup (WBK)
equations (1.1), with the initial conditions
(3.1)u(x,0)=λ−2k(α+β2)0.5coth[k(x+x0)],v(x,0)=−2k2(α+β2+β(α+β2)0.5)csch2[k(x+x0)].
For application of the
homotopy analysis method, we choose the initial approximations
(3.2)u0(x,t)=u(x,0)=λ−2k(α+β2)0.5coth[k(x+x0)],v0(x,t)=v(x,0)=−2k2(α+β2+β(α+β2)0.5)csch2[k(x+x0)],
and the linear operator
(3.3)ℒ[φ(x,t;p)]=∂φ(x,t;p)∂t
with the property
(3.4)ℒ(c)=0,
where c is constant. From (1.1), we define a system
of nonlinear operators as
(3.5)𝒩1[φ1(x,t;p),φ2(x,t;p)]=∂φ1(x,t;p)∂t+φ1(x,t;p)∂φ1(x,t;p)∂x+∂φ2(x,t;p)∂x+β∂2φ1(x,t;p)∂x2,𝒩2[φ1(x,t;p),φ2(x,t;p)]=∂φ2(x,t;p)∂t+∂∂x(φ1(x,t;p)φ2(x,t;p))+α∂3φ1(x,t;p)∂x3−β∂2φ2(x,t;p)∂x2.
Using the above definition,
we construct the zero-order deformation equations:
(3.6)(1−p)ℒ[φ1(x,t;p)−u0(x,t)]=ph1𝒩1[φ1(x,t;p),φ2(x,t;p)],(1−p)ℒ[φ2(x,t;p)−v0(x,t)]=ph2𝒩2[φ1(x,t;p),φ2(x,t;p)].
Obviously, when p=0 and p=1,
(3.7)φ1(x,t;0)=u0(x,t),φ1(x,t;1)=u(x,t),φ2(x,t;0)=v0(x,t),φ2(x,t;1)=v(x,t).
Thus, as the embedding parameter p increases from 0 to
1, φ1(x,t;p) and φ2(x,t;p) vary from the
initial approximations u0(x,t) and v0(x,t) to the solutions u(x,t) and v(x,t), respectively. Expanding φ1(x,t;p) and φ2(x,t;p) in Taylor
series with respect to p, we
have
(3.8)φ1(x,t;p)=u0(x,t)+∑m=1+∞um(x,t)pm,φ2(x,t;p)=v0(x,t)+∑m=1+∞vm(x,t)pm,
where
(3.9)um(x,t)=1m!∂mφ1(x,t;p)∂pm|p=0,vm(x,t)=1m!∂mφ2(x,t;p)∂pm|p=0.
If the auxiliary linear operator, the initial approximations, and the auxiliary parameters h1 and h2 are so properly
chosen, the above series converge at p=1, then we have
(3.10)u(x,t)=u0(x,t)+∑m=1+∞um(x,t),v(x,t)=v0(x,t)+∑m=1+∞vm(x,t),
which must be one of
solutions of original system. Define the vectors
(3.11)u→n={u0(x,t),u1(x,t),…,un(x,t)},v→n={v0(x,t),v1(x,t),…,vn(x,t)}.
We gain the mth-order deformation equations:
(3.12)ℒ[um(x,t)−χmum−1(x,t)]=h1R1,m(u→m−1,v→m−1),ℒ[vm(x,t)−χmvm−1(x,t)]=h2R2,m(u→m−1,v→m−1),
subject to initial conditions
(3.13)um(x,0)=0,vm(x,0)=0,
where
(3.14)R1,m(u→m−1,v→m−1)=∂um−1(x,t)∂t+∑n=0m−1un(x,t)∂um−1−n(x,t)∂x+∂vm−1(x,t)∂x+β∂2um−1(x,t)∂x2,R2,m(u→m−1,v→m−1)=∂vm−1(x,t)∂t+∂∂x(∑n=0m−1un(x,t)vm−1−n(x,t))+α∂3um−1(x,t)∂x3−β∂2vm−1(x,t)∂x2,χm={0,m≤1,1,m>1.
Obviously, the solution of
the mth-order deformation equations (3.12) for m≥1 becomes
(3.15)um(x,t)=χmum−1(x,t)+h1ℒ−1[R1,m(u→m−1,v→m−1)],vm(x,t)=χmvm−1(x,t)+h2ℒ−1[R2,m(u→m−1,v→m−1)].
For simplicity, we suppose h1=h2=h. From (3.2) and (3.15), we now successively obtain
(3.16)u1(x,t)=2(α+β2)0.5λhk2tcsch2[k(x+x0)],v1(x,t)=4α+4β(β+(α+β2)0.5)λhk3tcoth[k(x+x0)]csch2[k(x+x0)],u2(x,t)=−2(α+β2)0.5λhk2tcsch3[k(x+x0)](λhktcosh[k(x+x0)]−(1+h)sinh[k(x+x0)]),v2(x,t)=−2α−2β(β+(α+β2)0.5)λhk3tcsch4[k(x+x0)]×(λhkt(2+cosh[2k(x+x0)])−(1+h)sinh[2k(x+x0)]),u3(x,t)=13(α+β2)0.5λhk2tcsch4[k(x+x0)]×(−3(1+h)2+4λ2h2k2t2+(3+h(6+h(3+2λ2k2t2)))×cosh[2k(x+x0)]−6λh(1+h)ktsinh[2k(x+x0)]),v3(x,t)=13α+13β(β+(α+β2)0.5)λhk3tcsch5[k(x+x0)]×((−3(1+h)2+22λ2h2k2t2)cosh[k(x+x0)]+(3+h(6+3h+2λ2hk2t2))×cosh[3k(x+x0)]−6λh(1+h)kt(3sinh[k(x+x0)]+sinh[3k(x+x0)]))⋮
We used 10 terms in evaluating the approximate solutions uapp=∑i=09ui and vapp=∑i=09vi.The series
solutions contain the auxiliary parameterh. The validity of the method
is based on such an assumption that the series (2.4) converges at p=1. It is
the auxiliary parameter h which ensures that this assumption can be
satisfied. As pointed out by Liao [5], in general, by means of the so-called h-curve,
it is straightforward to choose a proper value of h which ensures that
the solution series is convergent. In this way, we choose h=−0.8 in following computational works.


## 4. Numerical Experiments

We now obtain
numerical solutions of the coupled Whitham-Broer-Kaup (WBK)
equations. In order to verify the efficiency of the proposed method in
comparison with exact solutions, we report
the absolute errors fork=0.2,λ=0.005,x0=10, and different values of α and β, in the following examples.Example 4.1.
Consider the WBK equations (1.1), with the initial conditions
(3.1), and the exact solutions (1.2). In Table 1, we show the absolute error for α=0.5 and β=1.Table 1
Absolute errors foru(x,t) and v(x,t) given by the HAM for h=−0.8, when k=0.2,λ=0.005,x0=10,α=0.5, and β=1.
xt=1t=2t=3uvuvuv19.9250E-059.0532E-051.9973E-051.8221E-043.0144E-042.7505E-0434.3988E-053.9582E-058.8512E-057.9651E-051.3358E-041.2021E-0451.9644E-051.7569E-053.9526E-053.5351E-055.9649E-055.3350E-0578.8025E-067.8509E-061.7711E-051.5797E-052.6727E-052.3839E-05Example 4.2.
Whenα=1 and β=0, the WBK equations are reduced to the modified Boussinesq (MB) equations
[16]. We show the absolute error for MB equations in Table 2.Table 2
Absolute errors foru(x,t) and v(x,t) given by the HAM for h=−0.8, when k=0.2,λ=0.005,x0=10,α=1, and β=0.
xt=1t=2t=3uvuvuv18.1037E-053.3226E-051.6308E-046.6872E-052.4613E-041.0094E-0433.5916E-051.4527E-057.2270E-052.9233E-051.0906E-044.4119E-0551.6039E-056.4478E-063.2273E-051.2974E-054.8703E-051.9579E-0577.1872E-062.8813E-061.4461E-055.7975E-062.1823E-058.7490E-06Example 4.3.
Whenα=0 and β=0.5, the WBK equations are reduced to the
approximate long-wave (ALW) equations in shallow water [15]. Table 3 shows the
absolute error in this case.Table 3
Absolute errors foru(x,t) and v(x,t) given by the HAM for h=−0.8, when k=0.2,λ=0.005,x0=10,α=0, and β=0.5.
xt=1t=2t=3uvuvuv14.0519E-051.6613E-058.1538E-053.3436E-051.2306E-045.0472E-0531.7958E-057.2634E-063.6135E-051.4616E-055.4533E-052.2059E-0558.0197E-063.2239E-061.6137E-056.4871E-062.4352E-059.7899E-0673.5936E-061.4407E-067.2306E-062.8988E-061.0911E-054.3745E-06


## 5. Conclusions

In this study, the
homotopy analysis method (HAM) was used for finding the approximate traveling
wave solutions of the Whitham-Broer-Kaup (WBK)
equations in shallow water. A very good agreement between the results of the
HAM and exact solutions was
observed, which confirms the validity of the HAM. It should be emphasized that the HAM provides us
with a convenient way to control the convergence of approximation series, which
is a fundamental qualitative difference in analysis between the HAM and other
methods. Furthermore, as the HAM does not require discretization, it is not
affected by computation round off errors, and large computer memory as well as consumed time which are issues in the
calculation procedure. The results show that the HAM is powerful mathematical
tool for solving systems of nonlinear partial differential equations having
wide applications in engineering.



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