Derivatives with Non-Singular Kernels from the Caputo-Fabrizio Definition and Beyond: Appraising Analysis with Emphasis on Diffusion Models
Pp. 269-341 (73)
This chapter presents an attempt to collate existing data about fractional
derivatives with non-singular kernels conceived by Caputo and Fabrizio in 2015.
The idea attracted immediately the interest of the researcher and the text
encompasses the consequent developments of the idea with new derivatives of
Riemann-Liouville type and the generalization with kernels expressed by the
Mittag-Leffler function. The chapter especially stresses the attention on diffusion
equations where the Caputo-Fabrizio time-fractional derivative naturally appears as
a relaxation term when the constitutive equation relating the flux and the gradient
contains either Cattaneo exponential kernel or Jeffrey kernel. Four models are
considered demonstrating the technology of diffusion model derivation. A special
section is devoted to a spatial derivative of Caputo-type with exponential nonsingular
kernel for materials exhibiting spatial memory. Critical comments and
suggestions are devoted to the formalistic fractionalization approach and the
outcomes of this reasonless operation.
Caputo-Fabrizio derivative, non-singular kernels, diffusion equation.
Dept. Chem. Eng, UCTM, Sofia 1756, 8 Kl. Ohridsky Blvd, Bulgaria.