General fractional integrals and derivatives and their applications

Abstract

In this survey paper, some recent results obtained for the general fractional integrals and derivatives and for the regularized general fractional derivatives are discussed. We start with a short historical survey of the general Fractional Calculus operators with the Sonin kernels and continue with a presentation of the recent developments regarding the general fractional derivatives and their applications. In particular, we introduce the first and the second 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 consider some Cauchy problems for the fractional differential equations with the general fractional derivatives.