3 - Fractional derivatives with nonsingular kernels

Abstract

In this chapter, we investigate the history of the fractional derivatives with nonsingular kernels, which are called the general fractional derivatives with nonsingular kernels. We introduce the family of the general fractional derivatives with Sonine, Mittag-Leffler, Wiman, Prabhakar, Gorenflo–Mainardi, Miller–Ross nonsingular kernel, general fractional derivatives with one-parameter Lorenzo–Hartley nonsingular kernel, and general fractional derivatives with two-parameter Lorenzo–Hartley nonsingular kernel. Moreover, we propose the family of Hilfer-type general fractional derivatives based on them.