Some Aspects of the Numerical Analysis of a Fractional Duffing Oscillator with a Fractional Variable Order Derivative of the Riemann-Liouville Type

TL;DR

This paper investigates numerical methods for solving a fractional Duffing oscillator model with variable fractional order derivatives, comparing explicit finite-difference and predictor-corrector schemes in terms of convergence and accuracy.

Contribution

It introduces and verifies numerical schemes for fractional Duffing oscillators with variable order derivatives, highlighting the efficiency of the predictor-corrector method.

Findings

Predictor-corrector method converges faster than explicit finite-difference scheme.

Both schemes' accuracy improves with increased grid nodes.

Numerical estimates of accuracy approach unity as grid density increases.

Abstract

In this paper, we consider some aspects of the numerical analysis of the mathematical model of fractional Duffing with a derivative of variable fractional order of the Riemann-Liouville type. Using numerical methods: an explicit finite-difference scheme based on the Grunwald-Letnikov and Adams-Bashford-Moulton approximations (predictor-corrector), the proposed numerical model is found. These methods have been verified with a test case. It is shown that the predictor-corrector method has a faster convergence than the method according to the explicit finite-difference scheme. For these schemes, using Runge's rule, estimates of the computational accuracy were made, which tended to unity with an increase in the number of calculated grid nodes.