A Newton interpolation based predictor-corrector numerical method for fractional differential equations with an activator-inhibitor case study

TL;DR

This paper introduces a new predictor-corrector numerical method based on Newton interpolation for fractional differential equations, improving accuracy and applicability, demonstrated through error analysis and a biological activator-inhibitor case study.

Contribution

A novel predictor-corrector scheme utilizing an improved Atangana-Seda formula for fractional derivatives, enhancing numerical solutions for fractional differential equations.

Findings

Accurate approximation of classical and fractional derivatives.

Effective simulation of fractional differential equations.

Successful application to the Gierer-Meinhardt model.

Abstract

This paper presents a new predictor-corrector numerical scheme suitable for fractional differential equations. An improved explicit Atangana-Seda formula is obtained by considering the neglected terms and used as the predictor stage of the proposed method. Numerical formulas are presented that approximate the classical first derivative as well as the Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Simulation results are used to assess the approximation error of the new method for various differential equations. In addition, a case study is considered where the proposed scheme is used to obtained numerical solutions of the Gierer-Meinhardt activator-inhibitor model with the aim of assessing the system's dynamics.