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Efficient spectral methods for PDEs with spectral fractional Laplacian. (English) Zbl 1476.65320

Summary: We develop efficient spectral methods for the spectral fractional Laplacian equation and parabolic PDEs with spectral fractional Laplacian on rectangular domains. The key idea is to construct eigenfunctions of discrete Laplacian (also referred to Fourier-like basis) by using the Fourierization method. Under this basis, the non-local fractional Laplacian operator can be trivially evaluated, leading to very efficient algorithms for PDEs involving spectral fractional Laplacian. We provide a rigorous error analysis of the proposed methods for the case with homogeneous boundary conditions, as well as ample numerical results to show their effectiveness.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
35S15 Boundary value problems for PDEs with pseudodifferential operators
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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