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

• Published in: Advances in difference equations Vol. 2012; no. 1; pp. 1 - 19
• Main Authors: Bhrawy, Ali H, Al-Shomrani, Mohammed M
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 09.02.2012; Springer Nature B.V; BioMed Central Ltd
• Subjects: Analysis; Approximation; Boundary conditions; Boundary value problems; Derivatives; Difference and Functional Equations; Functional Analysis; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Quadratures; Spectra; collocation method; direct method; shifted Legendre polynomials; multi-term FDEs; tau method; Gauss-Lobatto quadrature; multi-point boundary conditions
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/1687-1847-2012-8

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.