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Hou, Dianming; Hasan, Mohammad Tanzil; Xu, Chuanju

Müntz spectral methods for the time-fractional diffusion equation. (English) Zbl 1382.65343

Comput. Methods Appl. Math. 18, No. 1, 43-62 (2018).
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 \(H^{1}\)). 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.

Cited in 34 Documents

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R11 Fractional partial differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

time-fractional diffusion equation; singularity; Müntz polynomials; fractional Jacobi polynomials; weighted Sobolev spaces; spectral accuracy; convergence; fractional spectral method; Petrov-Galerkin formulation; error estimate; numerical experiment

Software:

FODE
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Full Text: DOI

References:

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