Exponential/fractional Jacobi spectral Galerkin and collocation schemes for the multi-dimensional time-fractional diffusion equations on the semi-infinite domain

Abstract

Conventional spectral Galerkin and collocation techniques based on classical orthogonal polynomials are well known for their spectral accuracy when applied to partial differential equations with smooth solutions. Nevertheless, their effectiveness may diminish when solutions exhibit weak singularities. This work proposes robust spectral approaches tailored for time-fractional diffusion equations defined on one- and two-dimensional semi-infinite domains, with the temporal fractional derivative interpreted in the Caputo sense. Due to the typical presence of singularities near the initial time, standard spectral methods may not yield accurate approximations. To overcome this challenge, we introduce spectral Galerkin and collocation strategies utilizing fractional Jacobi and exponential Jacobi basis functions. The fractional Jacobi functions (JFs) effectively capture temporal singularities at $t = 0$, while the exponential JFs are suited to handle the unbounded spatial domains. These formulations are naturally extended to higher dimensions. A suite of numerical experiments in one and two spatial dimensions demonstrates the accuracy and reliability of the proposed schemes, particularly in the treatment of the nonlocal fractional operators.

Keywords

• fractional diffusion equations
• Jacobi polynomials
• spectral methods
• semi-infinite domain
• Galerkin method
• collocation method

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