A high order multi step method for solving system of fractional differential equations

Abstract

The primary objective of this research paper is to present an advanced method utilizing quadratic interpolation to address the challenges encountered in solving a system of fractional initial value problems. By employing this high-order technique, we aim to improve the accuracy and efficiency of numerical computations associated with fractional differential equations. In our proposed methodology, we strive to enhance the conventional block-by-block approach by mitigating the interdependence of unknown solutions at each block iteration, except for the initial two steps. This modification aims to maintain the favorable stability characteristic inherent in block-by-block schemes, while effectively reducing coupling effects. We meticulously examine the convergence properties of the proposed method and conduct a rigorous analysis of the associated errors that prove that the numerical solution convergence to the true solution with an order of 3 + β when the parameter 0 < β ≤ 1. To substantiate the theoretical claims put forth in our research, we present a series of numerical examples. These examples serve as empirical evidence to demonstrate the effectiveness and practical applicability of the proposed method. The presented method superiority over alternative approaches is highlighted through a comprehensive comparison of the numerical results.