Approximate solutions of one dimensional systems with fractional derivative

TL;DR

This paper introduces a straightforward numerical method for approximating the fractional Caputo derivative using quadratic interpolation, enabling the solution of fractional differential equations and revealing an equivalence between fractional oscillators and dissipative ordinary oscillators.

Contribution

It presents a simple explicit quadratic interpolation method for evaluating the fractional Caputo derivative, facilitating numerical solutions of fractional differential equations.

Findings

The method effectively approximates the fractional Caputo derivative.

It establishes an equivalence between fractional oscillators and dissipative ordinary oscillators.

The approach simplifies numerical analysis of non-local fractional systems.

Abstract

The fractional calculus is useful to model non-local phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution of ordinary fractional differential equations. Due to the non-locality of the fractional derivative, we may establish an equivalence between fractional oscillators and ordinary oscillators with a dissipative term.