A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems

A H Bhrawy,M A Alghamdi

doi:10.1155/2013/306746

Abstract

We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.

Highlights

Many practical problems arising in engineering, physical, biological, and biomedical sciences require solving fractional differential equations (FDEs)
We present an explicit expression for the derivatives of any fractional order for the shifted Legendre basis functions in terms of the shifted Legendre polynomials
We introduce maximum absolute error, using shifted Legendre-Gauss-Lobatto collocation (SL-G-LC) method for ζ = 2.5, η = 1.5, θ = 0.9 with various choices of N. This problem was solved in [12] using shifted JacobiGauss collocation (SJ-GC) method based on Jacobi operational matrix, the results provided by Doha et al [12] have been presented in the third, fourth, and fifth columns of Table 3 for Jacobi parameters α = β = 0, α = β = 1/2, and α = β = 1, respectively

Summary

Introduction

Many practical problems arising in engineering, physical, biological, and biomedical sciences require solving fractional differential equations (FDEs), (see, e.g., [1,2,3,4]). The aim of this paper is to design some spectral techniques based on the shifted Legendre-Galerkin (SLG) method and shifted Legendre-Gauss-Lobatto collocation (SL-G-LC) method in modal basis for the solution of linear and nonlinear multi-term FDEs, respectively. The matrices corresponding to shifted LegendreGalerkin approximation are clearly described, including the modes required to impose nonhomogeneous initial conditions Another goal of this paper is to treat the nonlinear FDEs subject to nonhomogeneous initial conditions by implementing a new pseudo-spectral approximation based on Legendre polynomials. This approach is characterized by the representation of the solution by a truncated series of Legendre-Galerkin basis functions.

Preliminaries and Notations

Legendre-Galerkin Method for Fractional IVPs

Shifted Legendre Pseudospectral Approximation in Modal Basis

Numerical Examples

Conclusion