On the formulation of fractional Adams-Bashforth method with Atangana-Baleanu-Caputo derivative to model chaotic problems

TL;DR

This paper develops a fractional Adams-Bashforth method using the Atangana-Baleanu-Caputo derivative to model chaotic systems, providing theoretical analysis and numerical simulations demonstrating its effectiveness.

Contribution

It introduces a new fractional Adams-Bashforth method based on the Atangana-Baleanu derivative, with proven existence and uniqueness of solutions for chaotic differential equations.

Findings

Method successfully models chaotic systems with fractional derivatives.

Simulation results show good agreement for various fractional orders.

The approach extends classical methods to nonlocal, nonsingular fractional derivatives.

Abstract

Mathematical analysis with numerical application of the newly formulated fractional version of the Adams-Bashforth method using the Atangana-Baleanu derivative which has nonlocal and nonsingular properties is considered in this paper. We adopt the fixed point theory and approximation method to prove the existence and uniqueness of the solution via general two-component time-fractional differential equations. The method is tested with three nonlinear chaotic dynamical systems in which the integer-order derivative is modeled with the proposed fractional-order case. Simulation result for different $\alpha$ values in $(0,1]$ is presented.