Jacobi‐Galerkin Spectral Method for the Fractional Diffusion Equation

ABSTRACT An efficient numerical method for approximating the fractional diffusion equation, combining the Jacobi spectral method for spatial discretization with the leapfrog scheme for temporal discretization is proposed in this paper. Instead of the Petrov‐Galerkin one as in literature, the spatial discretization for the source problem is conducted using a standard Galerkin weak formulation. This choice ensures that the resulting discretized linear system retains symmetry and positive definiteness, offering a significant advantage. Furthermore, we provide a rigorous analysis of the stability and convergence of the proposed method. Theoretically, we prove that the spatial discretization via the Jacobi spectral method achieves asymptotically exponential convergence, while the temporal discretization using the leapfrog scheme attains a second‐order convergence rate. To validate the accuracy and efficiency of our approach, we compare it with a classical finite difference method. Finally, several numerical examples are presented to demonstrate the feasibility, stability, and high accuracy of the proposed algorithm.