Multistage Parker–Sochacki Method for Fractional ODE and PDE Models: Application to the Brusselator System

Multistage PSM for Fractional Brusselator Models

Résumé

This study presents a novel extension of the Parker–Sochacki method to the numerical solution of the fractional-order Brusselator model, formulated in both ordinary differential equation (ODE) and partial differential equation (PDE) settings. The Brusselator serves as a canonical model for oscillatory behavior in chemical and biological systems, and its fractional-order formulation captures memory effects inherent in such dynamics. For the PDE form, we combine the Parker–Sochacki method with the method of lines, enabling efficient treatment of spatially extended systems. A multistage implementation is developed using the Caputo fractional derivative, with an a priori step-size selection based on convergence analysis to ensure stability and accuracy. Numerical simulations demonstrate the method’s effectiveness in resolving the dynamical behavior of the system across different fractional orders. Comparisons with benchmark results from the literature confirm the accuracy and computational efficiency of the proposed scheme. The results highlight the strong influence of the fractional order on system dynamics, and underscore the method’s potential for solving a wide range of nonlinear fractional differential equations arising in epidemiology, chemical kinetics, and related fields.