Comprehensive Numerical Analysis of Time-Fractional Reaction–Diffusion Models with Applications to Chemical and Biological Phenomena

Abstract

This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional derivative is treated in the Caputo sense, addressing both linear and nonlinear scenarios. The proposed schemes were rigorously evaluated for stability and convergence. Additionally, the effectiveness of the developed schemes was validated through various linear and nonlinear models, including the Allen–Cahn equation, the KPP–Fisher equation, and the Complex Ginzburg–Landau oscillatory problem. These models were tested in one-, two-, and three-dimensional spaces to investigate the diverse patterns and dynamics that emerge. Comprehensive numerical results were provided, showcasing different cases of the fractional order parameter, highlighting the schemes’ versatility and reliability in capturing complex behaviors in fractional reaction–diffusion dynamics.