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

Part of: Real functions General theory in ordinary differential equations Miscellaneous topics of analysis in the complex domain Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  09 May 2017

Changtao Sheng  and
Jie Shen
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:shen7@purdue.edu (J. Shen), ctsheng@stu.xmu.edu.cn (C. T. Sheng)
*Corresponding author. Email addresses:shen7@purdue.edu (J. Shen), ctsheng@stu.xmu.edu.cn (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.

Keywords

Boundary value problemsgeneralized Jacobi functionsspectral element methodfractional differential equationserror bounds

MSC classification

Secondary: 26A33: Fractional derivatives and integrals 30E25: Boundary value problems 34A08: Fractional differential equations 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods 65L70: Error bounds
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 437 - 464
Copyright
Copyright © Global-Science Press 2017 

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