Spectral methods utilizing generalized Bernstein‐like basis functions for time‐fractional advection–diffusion equations

Abstract

This paper presents two methods for solving two‐dimensional linear and nonlinear time‐fractional advection–diffusion equations with Caputo fractional derivatives. To effectively manage endpoint singularities, we propose an advanced space‐time Galerkin technique and a collocation spectral method, both employing generalized Bernstein‐like basis functions (GBFs). The properties and behaviors of these functions are examined, highlighting their practical applications. The space‐time spectral methods incorporate GBFs in the temporal domain and classical Bernstein polynomials in the spatial domain. Fractional equations frequently produce irregular solutions despite smooth input data due to their singular kernel. To address this, GBFs are applied to the time derivative and classical Bernstein polynomials to the spatial derivative. A thorough error analysis confirms the technique's accuracy and convergence, offering a robust theoretical basis. Numerical experiments validate the method, demonstrating its effectiveness in solving both linear and nonlinear time‐fractional advection–diffusion equations.