Müntz Spectral Methods for the Time-Fractional Diffusion Equation

In this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov–Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than {H^{1}}). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov–Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods.