# ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE

**Authors:**Kukushkin M.V.^{1}-
**Affiliations:**- Institute of Applied Mathematics and Automatization

**Issue:**Vol 23, No 2 (2017)**Pages:**32-43**Section:**Статьи**URL:**https://journals.ssau.ru/est/article/view/5150**DOI:**https://doi.org/10.18287/2541-7525-2017-23-2-32-43

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## Full Text

## Abstract

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

## About the authors

### M. V. Kukushkin

Institute of Applied Mathematics and Automatization
**Author for correspondence.**

Email: morenov@ssau.ru

Russian Federation

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