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A Hybrid Spectral Element Method for Fractional Two-Point Boundary Value Problems

Published online by Cambridge University Press:  09 May 2017

Changtao Sheng*
Affiliation:
Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing and School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China
Jie Shen*
Affiliation:
Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing and School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China Department of Mathematics, Purdue University, West Lafayette, IN 47907-1957, USA
*
*Corresponding author. Email addresses:[email protected] (J. Shen), [email protected] (C. T. Sheng)
*Corresponding author. Email addresses:[email protected] (J. Shen), [email protected] (C. T. Sheng)
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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 L2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

Type
Research Article
Copyright
Copyright © Global-Science Press 2017 

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