Highly efficient numerical algorithm for nonlinear space variable order fractional reaction-diffusion models

Abstract

In this paper, we present a new novel highly efficient numerical method for non-linear variable order space fractional reaction-diffusion equations. The method is developed using Gaussian quadrature pole rational approximation instead of Padé rational approximation to avoid complex arithmetic. Issues related to computational efficiency and stability of the method are addressed using partial fraction decomposition technique and an easy to implement algorithm is developed. Two linear systems are required to be solved with the same real-valued discretization matrix. The stability and convergence of the method is discussed analytically and demonstrated through numerical experiments by solving some test problems from the literature. The effects of variable order diffusion on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability. MSC Classification: 26A33 , 35K57 , 65M12