Derivatives with Non-Singular Kernels from the Caputo-Fabrizio Definition and Beyond: Appraising Analysis with Emphasis on Diffusion Models

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.

Keywords: Caputo-Fabrizio derivative, non-singular kernels, diffusion equation.