# Application of the Homotopy Analysis Method for Solving the Variable Coefficient KdV-Burgers Equation

Authors: Dianchen Lu; Jie Liu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The homotopy analysis method is applied to solve the variable coefficient KdV-Burgers equation. With the aid of generalized elliptic method and Fourier’s transform method, the approximate solutions of double periodic form are obtained. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. The results indicate that this method is efficient for the nonlinear models with the dissipative terms and variable coefficients.


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## Body

## 1. Introduction

To solve the nonlinear partial differential equation (NPDE) has been an attractive research topic for mathematicians and physicists. Nonlinear evolution equations with variable coefficients can describe the physical phenomenon more accurately, and it is of great significance to study how to find the solutions of nonlinear evolution equations with variable coefficients. The nonlinear partial differential equations are generally difficult to solve and their exact solutions are difficult to obtain. In recent years, some various approximate methods have been developed such as homotopy analysis method [1–13] and Adomian’s decomposition method [14–17] to solve linear and nonlinear differential equations. However, the above works only studied the solutions of equations with constant coefficients. In this work, we apply the homotopy analysis method to the variable coefficients KdV-Burgers equations and obtain the approximate solution of the Jacobi elliptic function form. This method has the merits of simplicity and easy execution.This paper is arranged in the following manner. In Section2, we present the homotopy analysis method. In Section 3, the homotopy analysis method on the variable coefficient KdV-Burgers equation is presented. Finally, some conclusions are given.


## 2. Basic Idea of Homotopy Analysis Method

To explain the basic idea of the homotopy analysis method, we consider the following nonlinear differential equation:(1)A(u)-f(r)=0,r∈Ω,
subject to boundary condition
(2)B(u,∂u∂n)=0,r∈Γ,
where A is the general differential operator, B is the boundary operator, f(r) is the known analytic function, and Γ is the boundary of the region Ω.Generally speaking, the operatorA can be decomposed into linear part L and nonlinear part N. Equation (2) therefore can be written as
(3)L(u)+N(u)-f(r)=0.
Now we set up homotopy mapping H(u,p):Ω×[0,1]→R, which satisfies
(4)H(u,p)=L(u)-L(v)+p(L(v)+N(u)-f(r)),
where p is parameter, v is auxiliary function, and L(v)+N(v)=0.By (4), we obtain
(5)H(u,0)=L(u)-L(v),H(u,1)=A(u)-f(r)=0.As can be seen from 0 to 1 ofp is the process of L(u)-L(v) to A(u)-f(r) of H(u,p); this is the homotopy deformation.Assume that the solution ofH(u,p)=0 can be written as a power series in p:
(6)u~(x,t,p)=∑i=0∞ui(x,t)pi=u0+pu1+p2u2+⋯.
So when p=0, u~(x,t,0)=u0(x,t) is the solution of L(u)-L(v)=0; when p→1, the approximate solution of A(u)-f(r)=0 is u(x,t)=u0+u1+u2+⋯.


## 3. Application

In this section, we focus on the variable coefficient KdV-Burgers equation(7)ut+uux+α(t)uxx+β(t)uxxx=0,
where α(t) and β(t) are any function about t.It is fascinating to observe that, whenα(t) and β(t) are constant, (7) becomes the well-known KdV-Burgers equation; the equation plays an important role in studying liquid with bubbles inside, the flow of liquid in elastic tubes, and the problems of turbulence [18–20]. When α(t) is constant, β(t)=0, (7) becomes the well-known Burgers equation. When β(t) is constant, α(t)=0, (7) becomes the well-known KdV equation. When β(t)=β is constant, (7) becomes
(8)ut+uux+α(t)uxx+βuxxx=0.Next, we applied the homotopy analysis method to study the approximate solution of (8).In order to get the solution of (8), we lead in homotopy mapping.To aim at (8), we set up homotopy mapping H(u,p):R×I→R,
(9)H(u,p)=L(u)-L(v)+p(L(v)+uux+α(t)uxx),
where R=(-∞,+∞), I=[0,1], v is the auxiliary function, and the linear operator L is expressed as L(u)=ut+βuxxx.By using the generalized elliptic method [21], we can get that the typical KdV equation corresponding to (8),
(10)vt+vvx+βvxxx=0
has the following elliptic function solution:
(11)v1(x,t)=c0-β6k2m2sn(ξ,m)+12k2(m2-1)cn(ξ,m)sn(ξ,m)+cn(ξ,m)+dn(ξ,m)+β3k2m4sn2(ξ,m)-12k2cn2(ξ,m)(sn(ξ,m)+cn(ξ,m)+dn(ξ,m))2.Whenm→1,v1(x,t) degenerates to the following solitary wave solution:
(12)v1.1(x,t)=c0-β6k2tanh⁡ξtanh⁡ξ+sech⁡ξ+sech⁡ξ+β3k2tanh2⁡ξ-12k2sech2⁡ξ(tanh⁡ξ+sech⁡ξ+sech⁡ξ)2.Whenm→0, v1(x,t) degenerates to the trigonometric function solution
(13)v1.2(x,t)=c0+12k2βcos⁡⁡ξsin⁡ξ+cos⁡⁡ξ+1-12k2βcos⁡2ξ(sin⁡ξ+cos⁡⁡ξ+1)2,
where ξ=kx+[-c0k+βk3(4m2-5)]t+ξ0,k, c0, andξ0 are any constant, m is the module, and 0≤m≤1.One can easily prove thatH(u,1)=0 and (8) is the same, so the solution u(x,t) of (8) is the solution of H(u,p)=0 when under the condition p→1.Let(14)u~(x,t,p)=∑i=0∞ui(x,t)pi=u0+pu1+p2u2+⋯
be the solution of H(u,p)=0; by [22] we can know that this series is uniformly convergent in the p∈[0,1]. Thus, it yields that
(15)u=∑i=0∞ui(x,t)=u0+u1+u2+⋯.In order to obtain the approximate solution of (8), we substitute (14) into the equation H(u,p)=0. By taking the auxiliary function v=v1(x,t), and comparing the coefficients of the same power of p, one can obtain that
(16)p0:L(u0)=L(v)=L(v1),(17)p1:L(u1)=-L(v1)-u0u0x-α(t)u0xx=-α(t)u0xx,(18)p2:L(u2)=-u0u1x-u1u0x-α(t)u1xx.From (16) we have
(19)u0(x,t)=v1(x,t).By using the Fourier transform, one can obtain the solution of (17) with the initial condition u1|t=0=0 as follows:
(20)u1(x,t)=-12π∫0t∬-∞+∞α(τ)v1ξξhhhhhhhhhh×cos⁡[-λ3β(t-τ)hhhhlhhhhhhhhhhl+λ(x-ξ)-λ3β(t-τ)]dxdξdτ.
Similarly, one also finds the solution of (18) with the initial condition u2|t=0=0 as
(21)u2(x,t)=12π∫0t∫-∞+∞(-v1u1ξ-u1v1ξ-α(t)u1ξξ)hhhhhhhl×∫-∞+∞cos⁡[-λ3β(t-τ)hhhhhhhhhhhhhhhhl+λ(x-ξ)-λ3β(t-τ)]dxdξdτ.
From (11), (12), (13), (20), and (21) the two-degree approximate solution of (8) can be obtained as follows:
(22)u2*(x,t)=c0-β6k2m2sn(ξ,m)+12k2(m2-1)cn(ξ,m)sn(ξ,m)+cn(ξ,m)+dn(ξ,m)+β3k2m4sn2(ξ,m)-12k2cn2(ξ,m)(sn(ξ,m)+cn(ξ,m)+dn(ξ,m))2+12π∫0t∬-∞+∞(-α(τ)v1ξξ-v1u1ξhhhhhhhhhhhlh-u1v1ξ-α(τ)u1ξξ)hhhhhhhhhhlh×cos⁡[-λ3β(t-τ)hhhhhhlhhhhhhhhhhlll+λ(x-ξ)-λ3β(t-τ)]dxdξdτ,whereξ=kx+[-c0k+βk3(4m2-5)]t+ξ0, k, c0, and ξ0 are any constant, m is the module, and 0≤m≤1. Consider
(23)v0=c0-β6k2m2sn(ξ,m)+12k2(m2-1)cn(ξ,m)sn(ξ,m)+cn(ξ,m)+dn(ξ,m)+β3k2m4sn2(ξ,m)-12k2cn2(ξ,m)(sn(ξ,m)+cn(ξ,m)+dn(ξ,m))2,u1(x,t)=-12π∫0t∬-∞+∞α(τ)v1ξξhhhhhhhhhhhll×cos⁡[-λ3β(t-τ)hhhhhhhhhhhhhhhhlll+λ(x-ξ)-λ3β(t-τ)]dxdξdτ.
When m→1 and m→0, u2*(x,t) degenerates to the following approximate solutions:
(24)u2.1*(x,t)=c0-6βk2tanh⁡ξtanh⁡ξ+sech⁡ξ+sech⁡ξ+β3k2tanh2ξ-12k2sech2⁡ξ(tanh⁡ξ+sech⁡ξ+sech⁡ξ)2-12π∫0t∬-∞+∞(α(τ)u1ξξ+v1u1ξhhhhhhhhhhhllhh+u1v1ξ+α(τ)u1ξξ)hhhhhhhhhhhllhl×cos⁡[-λ3β(t-τ)hhhhhhhhhhhhhhllhllh+λ(x-ξ)-λ3β(t-τ)]dxdξdτ,u2.2*(x,t)=c0+β-6k2sin⁡ξ+12k2cos⁡⁡ξsin⁡ξ+cos⁡⁡ξ+1-12k2βcos⁡2⁡ξ(sinξ+cos⁡ξ+1)2-12π∫0t∬-∞+∞cos⁡[-λ3β(t-τ)+λ(x-ξ)]hhhhhhhhhhhllh×(α(τ)v1ξ+v1u1ξhhhhhhhhhhhhhhlh+u1v1ξ+α(τ)u1ξξ)dxdξdτ.By comparing the higher power coefficients ofp, more higher power approximate solutions of (8) can also be obtained.


## 4. Conclusion

This work studies the variable coefficients KdV-Burgers equations by using the homotopy analysis method, and the two-degree approximate solution of the Jacobi elliptic function form is obtained, which can degenerate to solitary wave approximate solution and trigonometric function approximate solution in the limit case. Our results show that the homotopy analysis method is applicable to the variable solution equations; how to apply this method to high-degree and high-dimensional system remains to be further studied.



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