AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 309420 10.1155/2014/309420 309420 Research Article Application of the Homotopy Analysis Method for Solving the Variable Coefficient KdV-Burgers Equation Lu Dianchen Liu Jie Hong Baojian 1 Faculty of Science Jiangsu University Zhenjiang, Jiangsu 212013 China ujs.edu.cn 2014 2732014 2014 22 01 2014 21 02 2014 27 3 2014 2014 Copyright © 2014 Dianchen Lu and Jie Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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  and Adomian’s decomposition method  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 Section 2, 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,rΩ, subject to boundary condition (2)B(u,un)=0,rΓ, 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 operator A 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 of p 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 of H(u,p)=0 can be written as a power series in p: (6)u~(x,t,p)=i=0ui(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 p1, 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, when α(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 . 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×IR, (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 , 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.

When m1,  v1(x,t) degenerates to the following solitary wave solution: (12)v1.1(x,t)=c0-β6k2tanhξtanhξ+sechξ+sechξ+β3k2tanh2ξ-12k2sech2ξ(tanhξ+sechξ+sechξ)2.

When m0, v1(x,t) degenerates to the trigonometric function solution (13)v1.2(x,t)=c0+12k2βcosξsinξ+cosξ+1-12k2βcos2ξ(sinξ+cosξ+1)2, where ξ=kx+[-c0k+βk3(4m2-5)]t+ξ0,k, c0, andξ0 are any constant, m is the module, and 0m1.

One can easily prove that H(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 p1.

Let (14)u~(x,t,p)=i=0ui(x,t)pi=u0+pu1+p2u2+ be the solution of H(u,p)=0; by  we can know that this series is uniformly convergent in the p[0,1]. Thus, it yields that (15)u=i=0ui(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)=-12π0t-+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)=12π0t-+(-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+12π0t-+(-α(τ)v1ξξ-v1u1ξhhhhhhhhhhhlh-u1v1ξ-α(τ)u1ξξ)hhhhhhhhhhlh×cos[-λ3β(t-τ)hhhhhhlhhhhhhhhhhlll+λ(x-ξ)-λ3β(t-τ)]dxdξdτ,

where ξ=kx+[-c0k+βk3(4m2-5)]t+ξ0, k, c0, and ξ0 are any constant, m is the module, and 0m1. 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)=-12π0t-+α(τ)v1ξξhhhhhhhhhhhll×cos[-λ3β(t-τ)hhhhhhhhhhhhhhhhlll+λ(x-ξ)-λ3β(t-τ)]dxdξdτ. When m1 and m0, u2*(x,t) degenerates to the following approximate solutions: (24)u2.1*(x,t)=c0-6βk2tanhξtanhξ+sechξ+sechξ+β3k2tanh2ξ-12k2sech2ξ(tanhξ+sechξ+sechξ)2-12π0t-+(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-6k2sinξ+12k2cosξsinξ+cosξ+1-12k2βcos2ξ(sinξ+cosξ+1)2-12π0t-+cos[-λ3β(t-τ)+λ(x-ξ)]hhhhhhhhhhhllh×(α(τ)v1ξ+v1u1ξhhhhhhhhhhhhhhlh+u1v1ξ+α(τ)u1ξξ)dxdξdτ.

By comparing the higher power coefficients of p, 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (no. 61070231), the Outstanding Personal Program in Six Fields of Jiangsu Province, China (Grant no. 2009188), and the Graduate Student Innovation Project of Jiangsu Province (Grant no. CXLX13_673).

He J. H. A simple perturbation approach to Blasius equation Applied Mathematics and Computation 2003 140 2-3 217 222 10.1016/S0096-3003(02)00189-3 MR1953895 ZBL1028.65085 He J. H. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons and Fractals 2005 26 3 695 700 2-s2.0-18844426016 10.1016/j.chaos.2005.03.006 He J. H. Homotopy perturbation method for bifurcation of nonlinear problems International Journal of Nonlinear Sciences and Numerical Simulation 2005 6 2 207 208 2-s2.0-17844387391 Fu H. S. Cao L. Han B. A homotopy perturbation method for well log constrained seismic waveform inversion Chinese Journal of Geophysics 2004 55 9 2173 2179 Shi L. F. Zhou X. C. Homotopic mapping solution of soliton for a class of disturbed Burgers equation Acta Physica Sinica 2010 59 5 2915 2918 2-s2.0-77953038414 Ganji D. D. Sadighi A. Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations International Journal of Nonlinear Sciences and Numerical Simulation 2006 7 4 411 418 2-s2.0-33748919061 Gorji M. Ganji D. D. Soleimani S. New application of He's homotopy perturbation method International Journal of Nonlinear Sciences and Numerical Simulation 2007 8 3 319 328 2-s2.0-34548590668 Shi Y. R. Xu X. J. Wu Z. X. Wang Y. H. Yang H. J. Duan W. S. K. P. Application of the homotopy analysis method to solving nonlinear evolution equations Acta Physica Sinica 2006 55 4 1555 1560 MR2247598 ZBL1202.65130 Mo J. Q. Yao J. S. Homotopic mapping method for the solution of soliton to a perturbed KdV equation Acta Physica Sinica 2008 57 12 7419 7422 MR2516984 Shi L. F. Mo J. Q. Soliton-like homotopic approximate analytic solution for a disturbed nonlinear evolution equation Acta Physica Sinica 2009 58 12 8123 8126 MR2666910 ZBL1212.35067 Liao S. Comparison between the homotopy analysis method and homotopy perturbation method Applied Mathematics and Computation 2005 169 2 1186 1194 10.1016/j.amc.2004.10.058 MR2174713 ZBL1082.65534 Shi Y. R. Yang H. J. Application of a homotopy analysis method to solving a dissipative system Acta Physica Sinica 2010 59 1 67 74 MR2662677 He J. H. Homotopy perturbation method for solving boundary value problems Physics Letters A 2006 350 1-2 87 88 10.1016/j.physleta.2005.10.005 MR2199322 ZBL1195.65207 Adomian G. A review of the decomposition method in applied mathematics Journal of Mathematical Analysis and Applications 1988 135 2 501 544 10.1016/0022-247X(88)90170-9 MR967225 ZBL0671.34053 Adomian G. Nonlinear Stochastic Systems and Applications to Physics 1989 Dordrecht, The Netherlands Kluwer Academic Publishers Group 10.1007/978-94-009-2569-4 MR982493 Abbasbandy S. A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method Chaos, Solitons and Fractals 2007 31 1 257 260 2-s2.0-33745859755 10.1016/j.chaos.2005.10.071 Abbasbandy S. Darvishi M. T. A numerical solution of Burgers' equation by modified Adomian method Applied Mathematics and Computation 2005 163 3 1265 1272 10.1016/j.amc.2004.04.061 MR2199478 ZBL1060.65649 Malkov M. A. Spatial chaos in weakly dispersive and viscous media: a nonperturbative theory of the driven KdV-Burgers equation Physica D: Nonlinear Phenomena 1996 95 1 62 80 10.1016/0167-2789(96)00043-7 Guan K. Y. Gao G. Qualitative analysis and traveling wave solutions of the Burgers-KdV equation Science in China A 1987 30 1 64 73 Liu S. D. Liu S. S. KdV-Burgers equation modelling of turbulence Science in China A 1991 34 9 938 945 Hong B. New Jacobi elliptic functions solutions for the variable-coefficient MKdV equation Applied Mathematics and Computation 2009 215 8 2908 2913 10.1016/j.amc.2009.09.035 MR2563407 ZBL1180.35459 Liao S. Beyond Perturbation: Introduction to the Homotopy Analysis Method 2004 New York, NY, USA CRC Press MR2058313