A Hybrid Spectral Element Method for Fractional Two-Point Boundary Value Problems. Issue 2 (9th May 2017)

• Record Type: Journal Article
• Authors: Sheng, Changtao; Shen, Jie
• Is Part Of: Numerical mathematics. Volume 10:Issue 2(2017)
• Page Start: 437
• Page End: 464
• Publication Date: 2017-05-09
• Subjects: 26A33, -- 30E25, -- 34A08, -- 65L60, -- 65L70; Boundary value problems, -- generalized Jacobi functions, -- spectral element method, -- fractional differential equations, -- error bounds; Numerical analysis -- Periodicals; Numerical analysis; Periodicals; 518.05
• DOI: 10.4208/nmtma.2017.s11
• Languages: English
• ISSNs: 1004-8979

Abstract

Abstract: We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions at t = 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates under L 2 ( I )-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.