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The research of Can Huang, is supported by National Natural Science Foundation of China under grant 11401500, 91630204 and the Fundamental Research Funds for the Central Universities under grant 20720150007.
The research of Zhimin Zhang, was supported in part by the National Natural Science Foundation of China under grants 11471031, 91430216, and the US National Science Foundation through grant DMS-1419040.
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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- Spectral Methods for Substantial Fractional Differential Equations
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