Numerical Solutions of Second-Order Fractional Differential Equations: Atangana-Baleanu Caputo Fractional Derivative Approach

Abstract

In this work, we develop a numerical framework for solving second-order fractional differential equations. Second-order fractional differential equations involving the Atangana–Baleanu Caputo fractional derivative, which models nonlocal and memory-dependent dynamical behavior. The numerical scheme is constructed using Lagrange polynomial interpolation adapted to the Atangana–Baleanu Caputo operator. A rigorous convergence and stability analysis is carried out using a fixed-point (contraction) argument under a natural Lipschitz condition. The theoretical results are supported by several numerical experiments that cover linear and nonlinear test problems. The contribution of this study lies in the establishment of a stable and accurate ABC-based numerical scheme and the verification that it standardizes its performance across a wide range of source functions. In general, the proposed method appears to be a powerful and efficient tool for modeling and analyzing complex systems that obey fractional dynamics.