General Fractional Calculus Operators with the Sonin kernels and Some of Their Applications

Abstract

In this survey paper, the general fractional integrals and derivatives with the Sonin kernels, the regularized general fractional derivatives, the sequential generalized fractional derivatives as well as some of their applications and the fractional differential equations with these derivatives are discussed. We start with a short summary of different kinds of the general Fractional Calculus operators with the Sonin kernels and then proceed with a presentation of some recent results including the fundamental theorems of Fractional Calculus for the general fractional derivatives of arbitrary order, for the regularized general fractional derivatives of arbitrary order, and for the sequential general fractional derivatives. As an application of these results, the generalized convolution Taylor formulas and the generalized convolution Taylor series with the coefficients and remainders in terms of the general fractional derivatives and the regularized general fractional derivatives are presented. Finally, we discuss a method for construction of solutions to the fractional differential equations with the general fractional derivatives in terms of the so called convolution series.