On an operator method for constructing solutions to integral and differential equations with a singularity

Abstract

In this paper, we consider new classes of fractional integral and derivative. These operators are generalizations of well-known integrals and the Riemann-Liouville derivative and the Caputo derivative. The article considers an operator method for solving integral and differential equations of fractional order. This method is based on the construction of normalized systems with respect to integral and differential operators. The algorithm for constructing normalized systems is given in the form of four steps. This method is first used to construct solutions to linear integral equations with constant coefficients. A theorem on the existence and uniqueness of a solution of the considered integral equation is proved. Solutions are defined explicitly and their representability in terms of multivariant functions of the Mittag-Leffler type is shown. These solutions are defined explicitly and they are represented in terms of multivariant functions of the Mittag-Leffler type.