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

TL;DR

This paper introduces a stability-enhanced nonstandard finite difference framework for solving 1D and 2D Caputo-type time-fractional diffusion equations, improving stability and accuracy over traditional schemes through novel approximations and rigorous analysis.

Contribution

It proposes a new nonstandard L1 approximation for Caputo derivatives and develops explicit NSFD schemes with enhanced stability for fractional diffusion equations.

Findings

Enhanced stability regions compared to standard schemes

Numerical experiments confirm improved accuracy and stability

Convergence of the proposed NSFD schemes is established

Abstract

Standard finite difference (SFD) schemes often suffer from limited stability regions, especially when applied in explicit setup to partial differential equations. To address this challenge, 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). A nonstandard L1approximation is proposed for the Caputo fractional derivative, and its local truncation error is derived analytically. This nonstandard L1 formulation is used to construct the NSFD scheme for a Caputo-type time-fractional initial value problem. The absolute stability of the resulting scheme is rigorously examined using the boundary locus method, and its performance is validated through numerical simulations on test examples for various choices of denominator functions. Based on this framework, two explicit NSFD schemes are developed for 1D and 2D cases of the Caputo-type TFDE. Their stability is further assessed through the discrete energy method, with particular focus on the expansion of stability region relative to SFD schemes. The convergence of the proposed NSFD schemes is also established. Finally, a comprehensive set of numerical experiments is conducted to demonstrate the accuracy and stability advantages of proposed methods, with results presented through tables and graphical illustrations.