Spectral Methods for Substantial Fractional Differential Equations

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

18 Scopus Citations
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Detail(s)

Original languageEnglish
Pages (from-to)1554-1574
Journal / PublicationJournal of Scientific Computing
Volume74
Issue number3
Online published19 Jul 2017
Publication statusPublished - Mar 2018

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.

Research Area(s)

  • Collocation method, Generalized Laguerre polynomials, Petrov–Galerkin, Spectral method, Substantial fractional differential equation, Superconvergence

Citation Format(s)

Spectral Methods for Substantial Fractional Differential Equations. / Huang, Can; Zhang, Zhimin; Song, Qingshuo.
In: Journal of Scientific Computing, Vol. 74, No. 3, 03.2018, p. 1554-1574.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review