A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative (Journal of Computational Mathematics, 2024)
In this paper, we consider numerical solutions of the fractional diffusion equation withthe $α$ order time fractional derivative defined in the Caputo-Hadamard sense. A highorder time-stepping scheme is constructed, analyzed, and numerically validated. The contribution of the paper is twofold: 1) regularity of the solution to the underlying equationis investigated, 2) a rigorous stability and convergence analysis for the proposed schemeis performed, which shows that the proposed scheme is $3 + α$ order accurate. Severalnumerical examples are provided to verify the theoretical statement.