High-order numerical method for Caputo–Hadamard fractional reaction-diffusion equations using nonuniform temporal mesh

This paper presents a high-order numerical scheme for solving the Caputo–Hadamard time-fractional reaction–diffusion equation. To address the inherent initial singularity of the solution, the method employs the L1 approximation formula on a specially designed nonuniform temporal mesh. This mesh is defined by the points tk=a+k(4k2−1)3β, where β=3(T−a)N(2N+1)(2N−1), whose graded structure effectively captures the singular behavior near the initial time t=a. For spatial discretization, a fourth-order compact difference formula is applied on a uniform grid to ensure high accuracy. Rigorous theoretical analysis demonstrates that the proposed scheme is unconditionally stable and achieves the optimal convergence rate of O(Nα−2+h4), where N and h represent the temporal and spatial discretization parameters, respectively. Systematic numerical experiments comprehensively validate the theoretical findings, clearly confirming the predicted convergence rates. Furthermore, experiments conducted across various fractional derivative orders and parameter configurations consistently demonstrate the effectiveness and robustness of the proposed method.