Müntz Spectral Methods for the Time-Fractional Diffusion Equation

In this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct...

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Bibliographic Details
Published in:Journal of computational methods in applied mathematics Vol. 18; no. 1; pp. 43 - 62
Main Authors: Hou, Dianming, Hasan, Mohammad Tanzil, Xu, Chuanju
Format: Journal Article
Language:English
Published: Minsk De Gruyter 01.01.2018
Walter de Gruyter GmbH
Subjects:
41A05
41A10
41A25
45K05
65M70
65N35
Approximation
Exact solutions
Fractional Jacobi Polynomials
Galerkin method
Mathematical analysis
Müntz Polynomials
Polynomials
Singularity
Spectra
Spectral Accuracy
Spectral methods
Time-Fractional Diffusion Equation
Weighted Sobolev Spaces
ISSN:1609-4840, 1609-9389
Online Access:Get full text
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Summary:In this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov–Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than ). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov–Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods.
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ISSN:1609-4840
1609-9389
DOI:10.1515/cmam-2017-0027