Comparison study between restrictive Taylor, restrictive Padé approximations and Adomian decomposition method for the solitary wave solution of the General KdV equation

Abstract

Some accurate finite difference sophisticated methods for solving initial boundary value problem for partial differential equations gives its exact solution with certain values of the mesh sizes of space and time as done [H.N.A. Ismail, On the convergence of the restrictive Padé approximation to the exact solutions of IBVP of parabolic and hyperbolic types, Appl. Math. Comput., accepted; H.N.A. Ismail, G.S.E. Salem, On the convergence of the restrictive Taylor approximation to the exact solutions of IBVP for parabolic, hyperbolic, convection diffusion, and KdV equations, Appl. Math. Comput., in press]. The restrictive Padé and restrictive Taylor approximations are very promising methods. In recent publications [H.N.A. Ismail, K.R. Raslan, G.S.E. Salem, Solitary wave solutions for the General KdV equation by Adomian decomposition method, 28th International Conference for Statistics and Computer Science and its Applications, Cairo, Appl. Math. Comput., June, accepted; A.M. Wazwaz, Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method, Chaos, Soliton. Fract. 12 (2001) 2283–2293; A.M. Wazwaz, Solitary wave solutions for the modified KdV equation by Adomian decomposition method, Int. J. Appl. Math. 3(4) (2000) 361–368], we have dealt with the numerical solutions of the Korteweg–de-Vries (KdV) and modified Korteweg–de-Vries (MKdV) equations. We extend this study to a more general nonlinear equation, which is the General Korteweg–de-Vries (GKdV) equation. The method applied here is Adomian decomposition method, which has been developed by George Adomian [Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994], restrictive Taylor and restrictive Padé approximations. Numerical examples are tested to illustrate the pertinent feature of the proposed algorithms.