TY - JOUR A1 - Diethelm, Kai A1 - Garrappa, Roberto A1 - Stynes, Martin T1 - Why fractional derivatives with nonsingular kernels should not be used JF - Fractional Calculus and Applied Analysis N2 - 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. Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20666 N1 - Accepted version des Artikels. Published source: Diethelm, K., Garrappa, R., Giusti, A. et al. Why Fractional Derivatives with Nonsingular Kernels Should Not Be Used. Fract Calc Appl Anal 23, 610–634 (2020). https://doi.org/10.1515/fca-2020-0032 VL - 23 SP - 610 EP - 634 ER - TY - JOUR A1 - Diethelm, Kai A1 - Garrappa, Roberto A1 - Stynes, Martin T1 - Good (and Not So Good) Practices in Computational Methods for Fractional Calculus JF - Mathematics N2 - The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results. Y1 - 2020 U6 - https://doi.org/10.3390/math8030324 VL - 8 ER -