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

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.

[1]  D. Baleanu,et al.  A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative , 2020, Advances in difference equations.

[2]  D. Baleanu,et al.  A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative , 2020 .

[3]  Francisco J. Mendoza,et al.  Riemann-Liouville derivative over the space of integrable distributions , 2020, Electronic Research Archive.

[4]  E. C. Oliveira,et al.  On the mistake in defining fractional derivative using a non-singular kernel , 2019, 1912.04422.

[5]  Haci Mehmet Baskonus,et al.  New numerical simulations for some real world problems with Atangana–Baleanu fractional derivative , 2019, Chaos, Solitons & Fractals.

[6]  Asifa Tassaddiq,et al.  Analysis of differential equations involving Caputo–Fabrizio fractional operator and its applications to reaction–diffusion equations , 2019, Advances in Difference Equations.

[7]  Sania Qureshi,et al.  Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu , 2019, Chaos, Solitons & Fractals.

[8]  Dumitru Baleanu,et al.  New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator , 2019, Mathematics.

[9]  Yuri Luchko,et al.  Desiderata for Fractional Derivatives and Integrals , 2019, Mathematics.

[10]  Changpin Li,et al.  Several Results of Fractional Differential and Integral Equations in Distribution , 2018, Mathematics.

[11]  J. A. Tenreiro Machado,et al.  A critical analysis of the Caputo-Fabrizio operator , 2018, Commun. Nonlinear Sci. Numer. Simul..

[12]  Hasan Bulut,et al.  Cancer treatment model with the Caputo-Fabrizio fractional derivative , 2018 .

[13]  V. E. Tarasov.  No nonlocality. No fractional derivative , 2018, Commun. Nonlinear Sci. Numer. Simul..

[14]  D. Zorica,et al.  Properties of the Caputo-Fabrizio fractional derivative and its distributional settings , 2018 .

[15]  A. Atangana,et al.  Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative , 2017 .

[16]  Andrea Giusti,et al.  A comment on some new definitions of fractional derivative , 2017, Nonlinear Dynamics.

[17]  Dumitru Baleanu,et al.  Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer , 2017 .

[18]  Ilyas Khan,et al.  Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model , 2016 .

[19]  Obaid J. Algahtani,et al.  Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model , 2016 .

[20]  E. D. Goufo.  Application of the Caputo-Fabrizio Fractional Derivative without Singular Kernel to Korteweg-de Vries-Burgers Equation∗ , 2016 .

[21]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[22]  Michele Caputo,et al.  Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels , 2016 .

[23]  Vasily E. Tarasov,et al.  On chain rule for fractional derivatives , 2016, Commun. Nonlinear Sci. Numer. Simul..

[24]  Chenkuan Li.  Several Results of Fractional Derivatives in Dʹ(R+) , 2015 .

[25]  Badr Saad T. Alkahtani,et al.  New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative , 2015, Arabian Journal of Geosciences.

[26]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[27]  S. Arabia,et al.  Properties of a New Fractional Derivative without Singular Kernel , 2015 .

[28]  G. Shilov,et al.  Properties and Operations , 2014 .

[29]  Vasily E. Tarasov,et al.  No violation of the Leibniz rule. No fractional derivative , 2013, Commun. Nonlinear Sci. Numer. Simul..

[30]  Changpin Li,et al.  On Riemann-Liouville and Caputo Derivatives , 2011 .

[31]  Shuqin Zhang,et al.  Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives , 2009 .

[32]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[33]  Gaston H. Gonnet,et al.  On the LambertW function , 1996, Adv. Comput. Math..

[34]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .