On a differential equation with Caputo-Fabrizio fractional derivative of order $1<\beta\leq 2$ and application to mass-spring-damper system

TL;DR

This paper studies a linear differential equation with Caputo-Fabrizio fractional derivative of order between 1 and 2, reducing it to an integer order equation, proving solution uniqueness, and applying it to a mass-spring-damper system.

Contribution

It introduces methods to solve and analyze fractional differential equations with Caputo-Fabrizio derivatives and applies these results to a physical mass-spring-damper model.

Findings

Explicit solutions for the fractional differential equation are derived.

Uniqueness of solutions for the initial value problem is established.

Application to the mass-spring-damper system demonstrates practical relevance.

Abstract

In this work, we investigate a linear differential equation involving Caputo-Fabrizio fractional derivative of order $1<\beta\leq 2$. Under some assumptions the considered equation is reduced to an integer order differential equation and solutions for different cases are obtained in explicit forms. We also prove a uniqueness of a solution of an initial value problem with a nonlinear differential equation containing the Caputo-Fabrizio derivative. Application of our result to the mass-spring-damper motion is also presented.