Approximate Solution of Fractional Nonlinear Partial Differential Equations by the Legendre Multiwavelet Galerkin Method

M A Mohamed,M Sh Torky

doi:10.1155/2014/192519

Abstract

The Legendre multiwavelet Galerkin method is adopted to give the approximate solution for the nonlinear fractional partial differential equations (NFPDEs). The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. We presented the numerical results and a comparison with the exact solution in the cases when we have an exact solution to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

Highlights

Nowadays, fractional differential equations have garnered a great deal of attention and appreciation recently due to its ability to provide an accurate description of different nonlinear phenomena
The main characteristic of this approach is using these properties together with the Galerkin method to reduce the nonlinear fractional partial differential equations (NFPDEs) to the solution of nonlinear system of algebraic equations
Fractional differential equations have garnered a great deal of attention and appreciation recently due to its ability to provide an accurate description of different nonlinear phenomena

Summary

Introduction

Fractional differential equations have garnered a great deal of attention and appreciation recently due to its ability to provide an accurate description of different nonlinear phenomena. The numerical methods used to deal with these equations [9] and they have largely been using some semianalytical techniques to solve these equations such as, differential transform method [10,11,12,13,14,15,16,17], Adomian decomposition method [18,19,20,21], Laplace decomposition method [22,23,24], homotopy perturbation method [25,26,27,28,29], and variational iteration method [30,31,32]. The aim of this paper is to expand the application of Legendre multiwavelet Galerkin method to provide approximate solutions for initial value problems of fractional nonlinear partial differential equations and to make comparison with that obtained by other numerical methods

Preliminaries and Notations

Properties of Legendre Multiwavelets

Solution of Nonlinear Fractional Partial Differential Equations

Illustrative Example

Conclusion