A stability enhanced nonstandard finite difference framework for solving one and two dimensional nonlocal differential equations

This study investigates the efficacy of nonstandard finite difference (NSFD) schemes in enhancing stability of explicit SFD schemes for 1D and 2D Caputo-type time-fractional diffusion equations (TFDEs). The Caputo fractional derivative introduces a nonlocal temporal memory effect, allowing the system dynamics to depend on the entire history of the solution rather than only its current state. Unlike existing NSFD approaches for fractional problems which apply nonstandard discretizations only to integer-order terms, present work introduces a nonstandard L1 approximation of the Caputo fractional derivative on a graded mesh. The local truncation error of this approximation is derived, and its performance is validated through numerical simulations on test examples for various choices of denominator functions. Its absolute stability on a uniform mesh is rigorously examined using the boundary locus method. Based on this framework, explicit NSFD schemes for 1D and 2D Caputo-type TFDEs are developed on a uniform mesh. Their stability is further assessed using the discrete energy method, with particular focus on expanding the stability region. The convergence of the proposed NSFD schemes is also established. Finally, numerical experiments are conducted to demonstrate the accuracy and stability advantages of the proposed methods. The results are presented through tabular and graphical illustrations.