Analytical and Numerical Study of Nonlinear Variable-Order Time Fractional Reaction-Diffusion Coupled Equations Arising in Biological and Chemical Processes

In this article, we analyze a category of time-fractional variable-order reaction-diffusion equations coupled in the Caputo sense that are created with the modeling of complicated biological and chemical processes. Furthermore, it is shown that the solutions exist and are unique, and then the system is subjected to the Ulam-Hyers stability, which confirms the model’s reliability and robustness. An advanced solution method based on shifted second-kind Airfoil polynomials is proposed for the numerical solution, where the polynomials are used to derive an operational matrix for variable-order fractional derivatives that is then applied to the original system using the collocation method to convert it into an equivalent set of algebraic equations. The system created is solved in order to obtain very precise approximations of the unknown functions. The proposed method is illustrated through several numerical experiments that not only show its accuracy but also its efficiency. The results obtained prove that the method is superior to the currently existing numerical techniques for fractional reaction-diffusion systems.