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.
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