A shifted Legendre spectral method for fractional-order multi-point boundary value problems

Advances in Difference Equations

Mohammed M Al-Shomrani¹  Ali H Bhrawy² 

[1] Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt;Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

关键词: multi-term FDEs;    multi-point boundary conditions;    tau method;    collocation method;    direct method;    shifted Legendre polynomials;    Gauss-Lobatto quadrature.;   

DOI  :  10.1186/1687-1847-2012-8

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundary value problems of fractional-order with variable coefficients. Here the approximation is based on shifted Legendre polynomials and the quadrature rule is treated on shifted Legendre Gauss-Lobatto points. We also present a Gauss-Lobatto shifted Legendre collocation method for solving nonlinear multi-order FDEs with multi-point boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. Thus we can find directly the spectral solution of the proposed problem. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms.

【 授权许可】

CC BY