Finite Difference/Fractional Pertrov–Galerkin Spectral Method for Linear Time-Space Fractional Reaction–Diffusion Equation
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| Publicado en: | Mathematics vol. 13, no. 11 (2025), p. 1864 |
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| Autor principal: | |
| Publicado: |
MDPI AG
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| Materias: | |
| Acceso en línea: | Citation/Abstract Full Text Full Text - PDF |
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| Resumen: | 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 <inline-formula>ONμ−γ+K−min{ρβ,2−β},</inline-formula> achieved without requiring additional regularity assumptions on the solution. Finally, numerical results are provided to validate our theoretical findings. |
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| ISSN: | 2227-7390 |
| DOI: | 10.3390/math13111864 |
| Fuente: | Engineering Database |