On Using the Homotopy Perturbation Method for Finding the Travelling Wave Solutions of Generalized Nonlinear Hirota- Satsuma Coupled KdV Equations

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.7(2009) No.2,pp On Using the Homotopy Perturbation Method for Finding the Travelling Wave Solutions of Generalized Nonlinear Hirota- Satsuma Coupled KdV Equations E.M. E. Zayed,T. A. Nofal, Khaled A. Gepreel Mathematics Department, Faculty of Sciences, Taif University,Saudi Arabia (Received 0 March 2008, accepted 0 December 2008) Abstract: In this paper, we use the homotopy perturbation method to find the travelling wave solutions for the generalized nonlinear Hirota- Satsuma coupled KdV equations. The results reveal that the homotopy perturbation method is very effective, convenient and quite accurate to systems of nonlinear equations.it is predicted that the homotopy perturbation method can be found widely applicable in engineering and physics. Keywords: the homotopy perturbation method; travelling wave solutions; Hirota- Satsuma coupled KdV system; nonlinear partial differential equations 1 Introduction Nonlinear partial differential equations are known to describe a wide variety of phenomena not only in physics, where applications extend over magneto fluid dynamics, water surface gravity waves, electromagnetic radiation reactions, and ion acoustic waves in plasma, just to name a few, but also in biology and chemistry, and several other fields. It is one of the important tasks in the study of the nonlinear partial differential equations to seek exact and explicit solutions. In the past several decades both mathematicians and physicists have made many attempts in this direction. Various methods for obtaining exact solutions to nonlinear partial differential equations have been proposed. Among these are the Bäcklund transformation method [1,2], Hirota s bilinear method [], the inverse scattering transform method [4], extended tanh method [5-7], Adomian pade approximation [8-10],Variational method [11-14], The variational iteration method [15,16], Various Lindstedt-Poincare methods [17-20] and others [21-27]. In the present paper, we shall use the homotopy perturbation method to construct travelling wave solutions for the following generalized nonlinear Hirota -Satsuma coupled KdV system: u t = 1 4 u xxx + uu x + ( v 2 + w) x, v t = 1 2 v xxx uv x, w t = 1 2 w xxx uw x. (1) The homotopy perturbation method was first proposed by He [28-]. The homotopy perturbation method does not depend on a small parameter in the equation. Using the homotopy technique in topology, a homotopy is constructed with an embedding parameter p [0, 1] which is considered as a small parameter. We close this introduction with the remark that Fan [4] has discussed the system (1.1) using a different approach. This system describes interactions of two long waves with different dispersion relations. Corresponding author. address: emezayed@hotmail.com Copyright c World Academic Press, World Academic Union IJNS /21

2 160 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp Basic idea of He s homotopy perturbation method We illustrate the following nonlinear differential equation []: with the boundary conditions A(u) f(r) = 0, r Ω, (2) B(u, u ) = 0, r Γ, () n where A is a general differential operator, B is a boundary operator, f(r) is an analytic function and Γ is the boundary of the domain Ω. Generally speaking, the operator A can be divided into two parts which are L and N, where L is linear but N is nonlinear. Equation (2.1) can therefore be rewritten as follows : L(u) + N(u) f(r) = 0. (4) By the homotopy technique, we construct a homotopy v(r, p) : Ω [0, 1] R which satisfies or H(v, p) = (1 p)[l(v) L(u 0 )] + p[a(v) f(r)] = 0, (5) H(v, p) = L(v) L(u 0 ) + pl(u 0 ) + p (N(v) f(r)) = 0, (6) where p [0, 1] is an embedding parameter, u 0 is an initial approximation of the equation (2.1) which satisfies the boundary conditions (2.2). Obviously, from equations (2.4) and (2.5) we have H(v, 0) = L(v) L(u 0 ) = 0, (7) H(v, 1) = A(v) f(r) = 0. (8) The changing process of p from zero to unity is just that of v(r, p) from u 0 (r) to u(r). In topology, this is called deformation, and L(v) L(u 0 ) and A(v) f(r) are called homotopy. According to the homotopy perturbation method, we can first use the embedding parameter p as a small parameter, and assume that the solution of equations (2.4) and (2.5) can be written as a power series in p as follows: v = v 0 + p v 1 + p 2 v (9) On setting p = 1 results in the approximate solution of equation (2.), we have: u = lim p 1 v = v 0 + v 1 + v (10) The combination of the perturbation method and the homotopy method is called the homotopy perturbation method, which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantage of the traditional perturbation techniques. The series (2.9) is convergent to most cases. However, the convergent rate depends on the nonlinear operator A(v) ( the following opinions are suggested by He []): (1) The second derivative of N(v) with respect to v must be small because the parameter may be relatively large, i.e., p 1. (2) The norm of L 1 N must be smaller than one so that the series converges. v Analysis of the method To investigate the travelling wave solution of equation (1.1), we first construct a homotopy as follows: (1 p)(v 1 u 0) + p(v v 1 v 1v 1 + 6v 2v 2 v ) = 0, (1 p)(v 2 v 0) + p(v v 2 + v 1v2 ) = 0, (11) (1 p)(v w 0) + p(v v + v 1v ) = 0, IJNS for contribution: editor@nonlinearscience.org.uk

3 E.M. E. Zayed, et al.: On Using the Homotopy Perturbation Method for Finding 161 where primes denotes differentiation with respect to x, and dot denotes differentiation with respect to t, and the initial approximations are follows: and v 1,0 (x, t) = u 0 (x, t) = u(x, 0), v 2,0 (x, t) = v 0 (x, t) = v(x, 0), (12) v,0 (x, t) = w 0 (x, t) = w(x, 0), v 1 = v 1,0 + pv 1,1 + p 2 v 1,2 + p v 1, +... v 2 = v 2,0 + pv 2,1 + p 2 v 2,2 + p v 2, +... (1) v = v,0 + pv,1 + p 2 v,2 + p v, +... where v i,j, i, j = 1, 2,,...are functions to be determined. Substituting equations (.2) and (.) into (.1) and arranging the coefficients of p powers, we have [v 1,1 1 4 v 1,0 + v 1,0 v 1,0 v 1,0 + 6v 2,0v 2,0 v,0 ]p + [v 1,2 1 4 v 1,1 (v 1,0v 1,1 + v 1,1v 1,0 ) + 6(v 2,1v 2,0 + v 2,0v 2,1 ) v,1 ]p2 + [v 1, 1 4 v 1,2 (v 1,0v 1,2 + v 1,1v 1,1 + v 1,2v 1,0 ) + 6(v 2,0v 2,2 + v 2,1v 2,1 + v 2,2v 2,0 ) + v,2 ]p +... = 0, (14) [v 2, v 2,0 + v 2,0 + v 1,0 v 2,0 ]p + [v 2, v 2,1 + (v 1,0v 2,1 + v 1,1v 2,0 )]p2 + [v 2, v 2,2 + (v 1,0v 2,2 + v 1,1v 2,1 + v 1,2v 2,0 )]p +... = 0, (15) [v, v,0 + v,0 + v 1,0 v,0 ]p + [v, v,1 + (v 1,0v,1 + v 1,1v,0 )]p2 + [v, v,2 + (v 1,0v,2 + v 1,2v,0 + v 1,1v,1 )]p +... = 0. (16) In order to obtain the unknowns of v i,j (x, t), i, j = 1, 2,,...,we construct and solve the following system which includes nine equations with nine unknowns variables, considering the initial conditions of v i,j (x, t), i, j = 1, 2, and having the initial approximations (.2): v 1,1 1 4 v 1,0 + v 1,0 v 1,0 v 1,0 + 6v 2,0v 2,0 v,0 = 0, v 1,2 1 4 v 1,1 (v 1,0v 1,1 + v 1,1 v 1,0) + 6(v 2,1 v 2,0 + v 2,0 v 2,1) v,1 = 0, v 1, 1 4 v 1,2 (v 1,0v 1,2 + v 1,1 v 1,1 + v 1,2 v 1,0) + 6(v 2,0 v 2,2 + v 2,1 v 2,1 + v 2,2 v 2,0) + v,2 = 0, v 2, v 2,0 + v 2,0 + v 1,0 v 2,0 = 0, v 2, v 2,1 + (v 1,0v 2,1 + v 1,1 v 2,0) = 0, v 2, v 2,2 + (v 1,0v 2,2 + v 1,1 v 2,1 + v 1,2 v 2,0) = 0, v, v,0 + v,0 + v 1,0 v,0 = 0, v, v,1 + (v 1,0v,1 + v 1,1 v,0) = 0, v, v,2 + (v 1,0v,2 + v 1,2 v,0 + v 1,1 v,1) = 0 (17) IJNS homepage:

4 162 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp From equation (2.9), if the first three approximations are sufficient, we will obtain. 4 Application u(x, t) = lim p 1 v 1 (x, t) = v(x, t) = lim p 1 v 2 (x, t) = w(x, t) = lim p 1 v (x, t) = v 1,k (x, t), (18) k=0 v 2,k (x, t), (19) k=0 v,k (x, t), (20) Firstly, we consider the solutions of the equation (1.1) with the following initial conditions [4]: k=0 u(x, 0) = 1 (λ 2k2 ) k 2 csch 2 (kx), v(x, 0) = b 0 1 k k 2 6λcoth(kx), w(x, 0) = c 0 2 kb 0 k 2 6λcoth(kx), (21) where k, b 0, c 0 and λ are arbitrary constants. To calculate the terms of the homotopy series (.8)-(.10) for u(x, t), v(x, t) and w(x, t),we substitute the initial conditions(4.1) into the system (.7) and finally using the Maple, the solutions of the equation can be obtained as follows: v 1,0 (x, t) = u(x, 0) = 1 (λ 2k2 ) k 2 csch 2 (kx), (22) v 1,1 (x, t) = 2tλk csch 2 (kx) coth(kx), (2) v 1,2 (x, t) = t 2 λ 2 k 4 csch 2 (kx)[csch 2 (kx) + 2 ], (24) v 1, (x, t) = 4t λ k 5 csch 2 (kx) coth(kx)[csch 2 (kx) + 2 ], (25) v 2,0 (x, t) = u(x, 0) = b 0 1 k k 2 6λcoth(kx), (26) v 2,1 (x, t) = 1 tλk2 k 2 6λcsch 2 (kx), (27) v 2,2 (x, t) = 1 t2 λ 2 k k 2 6λcsch 2 (kx) coth(kx), (28) v 2, (x, t) = 1 t λ k 4 k 2 6λcsch 2 (kx)[csch 2 (kx) + 2 ], (29) v,0 (x, t) = c 0 2 kb 0 k 2 6λcoth(kx), (0) v,1 (x, t) = 2 k2 tλb 0 k 2 6λcsch 2 (kx), (1) v,2 (x, t) = 2 k t 2 λ 2 b 0 k 2 6λcsch 2 (kx) coth(kx), (2) v, (x, t) = 2 t λ k 4 b 0 k 2 6λcsch 2 (kx)[csch 2 (kx) + 2 ]. () IJNS for contribution: editor@nonlinearscience.org.uk

5 E.M. E. Zayed, et al.: On Using the Homotopy Perturbation Method for Finding 16 In this manner the other components can be obtained. Substituting (4.2)- (4.1) into (.8)-(.10), we obtain: u(x, t) = 1 (λ 2k2 ) k 2 csch 2 (kx) 2tλk csch 2 (kx) coth(kx) t 2 λ 2 k 4 csch 2 (kx)[csch 2 (kx) + 2 ] 4t λ k 5 csch 2 (kx) coth(kx)[csch 2 (kx) + 2 ] +..., (4) v(x, t) = b 0 1 k k 2 6λcoth(kx) 1 tλk2 k 2 6λcsch 2 (kx) 1 t2 λ 2 k k 2 6λcsch 2 (kx) coth(kx) 1 t λ k 4 k 2 6λcsch 2 (kx)[csch 2 (kx) + 2 ] +..., (5) w(x, t) = c 0 2 k2 b 0 k 2 6λcoth(kx) 2 k2 tλb 0 k 2 6λcsch 2 (kx) 2 k t 2 λ 2 b 0 k 2 6λcsch 2 (kx) coth(kx) 2 t λ k 4 b 0 k 2 6λcsch 2 (kx)[csch 2 (kx) + 2 ] +... (6) Using the Taylor series, we obtain the closed form solutions as follows: u(x, t) = 1 (λ 2k2 ) k 2 csch 2 [k(x tλ)], (7) v(x, t) = b 0 1 k k 2 6λcoth[k(x tλ)], (8) w(x, t) = c 0 2 kb 0 k 2 6λcoth[k(x tλ)]. (9) With the initial conditions (4.1), the solitrary wave solutions of (1.1) of bell-type for u(x, t) and kink-type for v(x, t) and w(x, t) are in full agreement with the ones constructed by Fan [4]. 5 Comparing the results with the exact solutions To demonstrate the convergence of the homotopy perturbation method, the results of the numerical example are presented and only few terms are required to obtain accurate solutions. The accuracy of the homotopy perturbation method for the generalized Hirota-Satsuma coupled KdV equation is controllable, and absolute errors are very small with the present choice of x, t. These results are listed in the Table1, it is seen that the implemented method achieve a minimum accuracy of four and maximum accuracy of eleven significant figures for equation (1.1), for the first three approximations. Both the exact results and approximate solutions obtained for the first approximations are plotted Fig1-Fig. There are no visible differences in the two solutions of each diagrams. It is also evident that when more terms for homotopy perturbation method, are computed the numerical results get much more closer to the corresponding exact solutions with the initial conditions (4.1). 6 Conclusions In this paper, the homotopy perturbation method was used for finding soliton wave solutions of a generalized Hirota-Satsuma coupled KdV equation with initial conditions. It can be concluded that the homotopy perturbation method is very powerful and efficient technique in finding exact solutions for wide classes of problems. It is worth pointing out that the homotopy perturbation method presents a rapid convergence solutions. IJNS homepage:

6 164 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp Table 1: The homotopy perturbation results for u(x, t), v(x, t) and w(x, t) for the first three approximation in comparison with the analytic solutions when c 0 = 1.5, b 0 = 1.5, λ =.01 and k = 0.5, for the solitary wave solutions with the initial conditions (4.1) of equation (1.1) respectively. (x,t) u exact u homotopy v exact v homotopy w exact w homotopy (0.1, 0.1) (0.1, 0.2) (0.1, 0.) (0.2, 0.1) (0.2, 0.2) (0.2, 0.) (0., 0.1) (0., 0.2) (0., 0.) (a) u homotopy (b) u exact Figure 1: The homotopy perturbation results for u(x, t) for the first three approximation shown in (a) in comparison with the analytic solutions (b) when c 0 = 1.5, b 0 = 1.5, =.01andk = 0.5, for the solitary wave solutions with the initial conditions (4.1) of equation (1.1). (a) v homotopy (b) v exact Figure 2: The homotopy perturbation results for v(x, t) for the first three approximation shown in (a) in comparison with the analytic solutions (b) c 0 = 1.5, b 0 = 1.5, =.01andk = 0.5, for the solitary wave solutions with the initial conditions (4.1) of equation (1.1). IJNS for contribution: editor@nonlinearscience.org.uk

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