A Spectral Galerkin Framework for the Fractional Reaction–Subdiffusion Equation Using Legendre Cardinal Functions

Abstract

In this work, we introduce an efficient spectral Galerkin method for solving one-dimensional time-fractional reaction–subdiffusion equations. Traditional fractional operators often pose severe computational challenges and exhibit weak singularities near the initial time. To address these issues, we analytically transform the governing differential equation into a weakly singular Volterra integral equation. The numerical solution is then constructed in a two-dimensional tensor-product space using orthogonal Legendre cardinal functions on both Gauss and Gauss–Lobatto grids. A major computational advantage of this cardinal basis is that it eliminates the need for expensive numerical quadratures when assembling the operational matrices. From a theoretical view, we establish a rigorous a priori error bound and a fully computable a posteriori error estimator based on the residual. The analysis confirms that the convergence rate is governed algebraically by the Sobolev regularity of the exact solution, naturally accelerating to exponential (spectral) convergence for sufficiently smooth profiles. Extensive numerical experiments validate these theoretical claims, demonstrating that the proposed framework offers significant improvements in accuracy and efficiency.