Optimal Error Estimates of Fast L1 Scheme for Variable‐Order Fractional Diffusion Equations

ABSTRACT This paper develops a fast finite difference scheme on graded meshes for variable‐order time‐fractional diffusion equations with weak initial singularities. The proposed method combines 1 approximation with the sum‐of‐exponentials technique for kernel approximation, achieving both high accuracy and computational efficiency. We establish sharp error estimates for the scheme and rigorously prove its numerical stability and convergence using maximum principle analysis. Numerical experiments confirm the theoretical results, demonstrating that the accelerated scheme maintains optimal convergence rates while significantly reducing computational costs compared to existing approaches.