An Extension of the Picard Theorem to Fractional Differential Equations with a Caputo-Fabrizio Derivative

H R Marasi,H Aydi,A Soltani Joujehi,Ali Aral

doi:10.1155/2021/6624861

Abstract

In this paper, we consider fractional differential equations with the new fractional derivative involving a nonsingular kernel, namely, the Caputo-Fabrizio fractional derivative. Using a successive approximation method, we prove an extension of the Picard-Lindelöf existence and uniqueness theorem for fractional differential equations with this derivative, which gives a set of conditions, under which a fractional initial value problem has a unique solution.

Highlights

Due to the demonstrated applications of fractional operators in various and widespread fields of many sciences, such as mathematics, physics, chemistry, engineering, and statistics [1,2,3,4], various operators of a fractional calculus have been found to be remarkably popular for modelling of numerous varied problems in these sciences
In [16], we found a comparison approach of two latest fractional derivatives models, namely, Atangana-Baleanu and CaputoFabrizio, for a generalized Casson fluid and obtained exact solutions
By Picard’s theorem, we can study the existence and uniqueness of a solution of first-order differential equations. This theorem can be applied to ensure the existence of a unique solution of higher-order ordinary differential equations and for systems of differential equations

Summary

Introduction

Due to the demonstrated applications of fractional operators in various and widespread fields of many sciences, such as mathematics, physics, chemistry, engineering, and statistics [1,2,3,4], various operators of a fractional calculus have been found to be remarkably popular for modelling of numerous varied problems in these sciences. The new definition suggested by Caputo and Fabrizio [5], which has all the characteristics of the old definitions, assumes two different representations for the temporal and spatial variables They claimed that the classical definition given by Caputo appears to be convenient for mechanical phenomena, related with plasticity, fatigue, damage, and with electromagnetic hysteresis. When these effects are not present, it seems more appropriate to use the new Caputo-Fabrizio operator. We obtain an extension of Picard’s theorem for differential equations with the Caputo-Fabrizio fractional derivative This theorem provides conditions for which a fractional initial value problem involving the CaputoFabrizio derivative has a unique solution. The proof of this extension of Picard’s theorem provides a way of constructing successive approximations to the solution

Preliminaries

Extension of Picard Theorem

Conclusion