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

作者：Bhrawy Ali H^*; Al Shomrani Mohammed M

来源：Advances in Difference Equations, 2012, 2012(1): 8.

DOI：10.1186/1687-1847-2012-8

• multi-term FDEs
• multi-point boundary conditions
• tau method
• collocation method
• direct method
• shifted Legendre polynomials
• Gauss-Lobatto quadrature

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.

• 出版日期2012