Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Christopher Nicholas Angstmann,Byron Alexander Jacobs,Bruce Ian Henry,Zhuang Xu

doi:10.3390/math8112023

Christopher Nicholas Angstmann, Byron Alexander Jacobs + Show 2 more

Open Access

https://doi.org/10.3390/math8112023

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Journal: Mathematics	Publication Date: Nov 13, 2020
Citations: 14	License type: CC BY 4.0

Affiliation: UNSW Sydney, University of Johannesburg

Abstract

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

Highlights

In mathematical models based on evolution equations it is standard to restrict consideration to models whose solutions exist in a space of continuous functions
This is true in cases of evolution equations with fractional order differential operators, such as the Riemann–Liouville operator [1]
From the formulation presented in this work, we suggest that any numerical method appropriate for ODEs may be used to accurately solve initial values problems (IVPs) with CF operators, and as such many efficient, highly accurate methods are available to these equations

Summary

Introduction

In mathematical models based on evolution equations it is standard to restrict consideration to models whose solutions exist in a space of continuous functions This is true in cases of evolution equations with fractional order differential operators, such as the Riemann–Liouville operator [1]. There has been a great deal of attention focussed on fractional differential operators based on integrals with non-singular kernels Included in this are the Caputo–Fabrizio (CF) operator [3] and the Atangana–Baleanu in the sense of a Caputo (ABC). We briefly discuss the impacts of these results on numerical methods for the solution of IVPs involving CF operators and point out a seemingly overlooked simple approach to the numerical evaluation of such equations We repeat this treatment for the ABC operator and again show that solutions, in general, will have a discontinuity at the origin. We consider a more traditional fractional derivative, the Caputo derivative, and show that solutions to Caputo IVPs can not feature such discontinuities

Caputo–Fabrizio Operator

Example Solutions for CF Initial Value Problems

Numerical Considerations for Equations Involving the CF Operator

The Atangana–Baleanu Operator

Example ABC Initial Value Problems

Singular Kernel Operator Example

Conclusions