Approximate Solution for Time-Delayed Convective Fisher Equation by Adm-Padé Technique

Abstract

We present an approximate solution to the time-delayed convective Fisher equation using ADM-Padé technique which is a combination of Adomian decomposition method and Padé approximation. This technique gives the approximate solution with faster convergence and higher accuracy than using ADM alone. 221 Asian-European J. Math. 2010.03:221-233. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/22/15. For personal use only. 222 A. Alharbi & E. S. Fahmy is clear that when τ = 0 equation (1.1) reduces to the convective Fisher equation.Here we will find an approximate solution for the modified model for time delayed convective Fisher equation using Adomian's Decomposition method and then apply the Padé approximants to the series solution derived from the ADM to improve the accuracy and speed up the convergence.Since the beginning of the 1980s Adomian [6] has presented and developed a so-called decomposition method for solving linear or nonlinear problems such as ordinary differential equations. Adomians decomposition method consists of splitting the given equation into linear and nonlinear parts. Then, ADM inverts a derivative operator contained in the linear operator on both sides, and identifies the initial and/or boundary conditions and the terms involving the independent variable alone as initial approximation. Next, ADM decomposes the unknown function into a series whose components are to be determined, and decomposes the nonlinear function in terms of special polynomials called Adomians polynomials. Finally, ADM finds the successive terms of the series solution by recurrent relation using these Adomians polynomials. ADM is quantitative rather than qualitative, analytic, requiring neither linearization nor perturbation, and continuous with no resort to discretization and consequent computer-intensive calculations. Some applications [3] of the method show its advantages. However, ADM has some drawbacks. By using ADM, we get a series solution, in practice a truncated series solution. Although the series can be rapidly convergent in a very small region, it has very slow convergence rate in the wider region we examine, and the truncated series solution is an inaccurate solution in that region, which will greatly restrict the application area of the method. Many examples given in [13] can be used to support this assertion.We have used an extension of ADM [10], which can improve the convergence rate of the series solution. Padé approximant [5] approximates a function by the ratio of two polynomials. The coefficients of the powers occurring in the polynomials are determined by the coefficients in the power series expansion of the function. Generally, the Padé approximant can enlarge the convergence domain of the truncated power series, and can improve greatly the convergence rate of the truncated power series. In order to improve the accuracy of ADM, we propose to modify Adomians series solution for the time delayed convective Fisher equation by using the Padé approximant.Since the ADM serie...