Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications

Abstract

This research introduces a numerical approach that employs the midpoint technique successively to solve differential equations that contain Atangana-Baleanu and Caputo fractional derivatives. The stability and consistency of the proposed approach has been demonstrated through the application of the technique suggested for solving the perturbed problem. Furthermore, the error analysis has been carried out for a nonlinear equation with Caputo derivatives. The effectiveness of this method is illustrated by examining various examples incorporating nonlocal kernels. Finally, by the use of an example, the well-known Fitzhugh-Nagumo model with Atangana-Baleanu fractional derivatives is simulated by using the proposed scheme. The method exhibits improved accuracy, consistency, and stability compared to the classical midpoint method, with particularly good performance for the Atangana-Baleanu and Caputo cases due to a specially designed predictor component. The novelty of the work lies in extending the method to fractional operators with non-local and non-singular kernels, offering a more robust approach than existing techniques in the literature.