A Spectral Galerkin Method with Hybrid Müntz–Jacobi Basis for Space–Time Fractional Advection–Diffusion Equations

Abstract

For the numerical treatment of space-time fractional advection-diffusion equations, we designed a spectral Galerkin method. We construct the approximation space using a hybrid system of Müntz-Jacobi basis functions, combining the stability and orthogonality properties of Jacobi polynomials and the singularity-resolving capacity of Müntz-type monomials. The basis is reconstructed to satisfy homogeneous boundary conditions, thereby ensuring both consistency with the physical constraints and numerical stability. By including the dominant singular exponents of the exact solution into the approximation space, the suggested framework achieves exponential-type convergence with an accurate representation of endpoint singularities with high precision. We introduce the computational formulation, including the fractional integral and derivative of the new basis and the formulation of the Galerkin discretization. A series of numerical examples is provided, demonstrating the robustness and accuracy of the method for different fractional parameters, and agree with the theoretical error estimates. In general, the hybrid Müntz-Jacobi Galerkin framework achieves a flexible and powerful paradigm for fractional partial differential equations, with natural extensions to multi-term fractional operators, multi-dimensional models, and nonlinear systems.