RIEMANN-LIOUVILLE OPERATORI QATNASHGAN KASR TARTIBLI ISSIQLIK TARQALISH TENGLAMASIGA QO'YILGAN KOSHI MASALASI

The field of mathematical analysis, called fractional calculus and devoted to the study and application of derivatives and integrals of arbitrary order, has a long history and rich content, due to penetration into it and relationships with a wide variety of issues in the theory of functions, integral and differential equations, etc. It is located in constant development, which feeds on the ideas and results of various directions in mathematical analysis. The fractional calculus of functions of one and many variables continues to develop intensively at the present time, as evidenced by both a large stream of publications and international conferences specially devoted to questions of fractional calculus. It is possible that it is precisely the constant development of the theory of fractional integro-differentiation, and in recent decades its great branching, especially in the case of functions of many variables, that is the reason for the lack of monographs on this theory. Meanwhile, this circumstance could not but serve as a brake on the development of fractional calculus. A number of very significant and fundamental results have been published in original papers, many of which are difficult to access and little known.

This article is devoted to the solution of the  Koshi problem for a differential equation with fractional Riemann-Liouville derivatives. The work uses the spectral method and the integral Laplace transform to find the solution of the direct problem in explicit form through an infinite series with a three-parameter Mittag-Leffler function. Further, when sufficient conditions are met with respect to the given functions, it is proved that the constructed solution is regular.

Keywords. Riemann-Liouville fractional derivative, Mittag-Leffler function, Laplace transform, initial conditions, Koshi problem