Finite Difference/Fractional Pertrov–Galerkin Spectral Method for Linear Time-Space Fractional Reaction–Diffusion Equation

Mahmoud A. Zaky ()
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Mahmoud A. Zaky: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia

Mathematics, 2025, vol. 13, issue 11, 1-13

Abstract: Achieving high-order accuracy in finite difference/spectral methods for space-time fractional differential equations often relies on very restrictive and usually unrealistic smoothness assumptions in the spatial and/or temporal domains. For spatial discretization, spectral methods using smooth basis functions are commonly employed. However, spatial–fractional derivatives pose challenges, as they often lack guaranteed spatial smoothness, requiring non-smooth basis functions. In the temporal domain, finite difference schemes on uniformly graded meshes are commonly employed; however, achieving accuracy remains challenging for non-smooth solutions. In this paper, an efficient algorithm is adopted to improve the accuracy of finite difference/Pertrov–Galerkin spectral schemes for a time-space fractional reaction–diffusion equation, with a hyper-singular integral fractional Laplacian and non-smooth solutions in both time and space domains. The Pertrov–Galerkin spectral method is adapted using non-smooth generalized basis functions to discretize the spatial variable, and the L1 scheme on a non-uniform graded mesh is used to approximate the Caputo fractional derivative. The unconditional stability and convergence are established. The rate of convergence is O N μ − γ + K − min { ρ β , 2 − β } , achieved without requiring additional regularity assumptions on the solution. Finally, numerical results are provided to validate our theoretical findings.

Keywords: fractional Pertrov–Galerkin method; integral fractional Laplacian; non-smooth solutions; L1 scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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