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Journal of Scientific Computing

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19-07-2017

Spectral Methods for Substantial Fractional Differential Equations

Authors: Can Huang, Zhimin Zhang, Qingshuo Song

Published in: Journal of Scientific Computing | Issue 3/2018

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Abstract

In this paper, a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial fractional differential equations are proposed, analyzed, and tested. We modify a class of generalized Laguerre polynomials to form our trial basis and test basis. After a proper scaling of these bases, our Petrov–Galerkin method results in diagonal and well-conditioned linear systems for certain types of fractional differential equations. In the meantime, we provide superconvergence points of the Petrov–Galerkin approximation for associated fractional derivative and function value of true solution. Additionally, we present explicit fractional differential collocation matrices based upon Laguerre–Gauss–Radau points. It is noteworthy that the proposed methods allow us to adjust a parameter in the basis according to different given data to maximize the convergence rate. All these findings have been proved rigorously in our convergence analysis and confirmed in our numerical experiments.

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Title: Spectral Methods for Substantial Fractional Differential Equations
Authors: Can Huang
Zhimin Zhang
Qingshuo Song
Publication date: 19-07-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0506-8

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