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
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
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