Error Analysis of the L1 Scheme on a Modified Graded Mesh for a Caputo–Hadamard Fractional Diffusion Equation

The L1 scheme on a modified graded mesh is proposed to solve a Caputo–Hadamard fractional diffusion equation with order α∈(0,1). Firstly, an improved graded mesh frame is innovatively constructed, and its mathematical properties are verified. Subsequently, a new truncation error bound for the L1 discretisation format of Caputo–Hadamard fractional-order derivatives is established by means of a Taylor cosine expansion of the integral form, and a second-order central difference method is used to achieve high-precision discretisation of spatial derivatives. Furthermore, a rigorous analysis of stability and convergence under the maximum norm is conducted, with special attention devoted to validating that the L1 approximation scheme manifests an optimal convergence order of 2−α when deployed on the modified graded mesh. Finally, the theoretical results are substantiated through a series of numerical experiments, which validate their accuracy and applicability.