Couple of the Variational Iteration Method and Fractional-Order Legendre Functions Method for Fractional Differential Equations

Fukang Yin,Fengshun Lu,Hongze Leng,Junqiang Song

doi:10.1155/2014/928765

Abstract

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique.

Highlights

Fractional calculus plays a significant role in modeling physical and engineering processes which are found to be best described by fractional differential equations (FDEs)
Motivated and inspired by the ongoing research in these areas, we extend the above method to FDEs by employing fractional-order Legendre functions, instead of Legendre wavelets
In order to improve the performance of fractional variational iteration method (FVIM), we introduce fractional-order Legendre functions (FLFs) to approximate uk(x) and the inhomogeneous term gint(x) as uk+1 = CkT+1Ψ, gint (x) = GTΨ, (33)

Summary

Introduction

Fractional calculus plays a significant role in modeling physical and engineering processes which are found to be best described by fractional differential equations (FDEs). Considerable attention has been paid to developing an efficient and fast convergent method for FDEs. Recently, some analytical or numerical methods are introduced to find the solutions of nonlinear PDEs, such as Adomian’s decomposition method (ADM) [1, 2], homotopy perturbation method (HPM) [3,4,5], variational iteration method (VIM) [6,7,8], orthogonal polynomials method [9,10,11], and wavelets method [12,13,14,15,16,17].

Preliminaries and Notations

Fractional-Order Legendre Functions

Modified Variation Iteration Method Using FLFs

Applications and Results

Conclusion