Why Fractional Derivatives with Nonsingular Kernels Should Not Be Used

• Published in: Fractional calculus & applied analysis Vol. 23; no. 3; pp. 610 - 634
• Main Authors: Diethelm, Kai, Garrappa, Roberto, Giusti, Andrea, Stynes, Martin
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 01.06.2020; De Gruyter; Nature Publishing Group
• Subjects: 34A08; Abstract Harmonic Analysis; Analysis; Approximation; Calculus; Convolution; Convolution integrals; Derivatives; Differential equations; Editorial Survey; Existence theorems; Fractional calculus; fractional derivative; fractional integral; Functional Analysis; fundamental theorem of calculus; Integral Transforms; Integrals; Kernels; Mathematical analysis; Mathematics; non-singular kernel; Numerical analysis; Operational Calculus; Primary: 26A33; Secondary: 35R11; non-singular kernel; fundamental theorem of calculus; Primary: 26A33; fractional integral; Secondary: 35R11; 34A08
• DOI: 10.1515/fca-2020-0032
• ISSN: 1311-0454

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.