# Adyghe Int. Sci. J. Vol. 23, No 2. P. 11–17.

Contents of this issue

DOI: https://doi.org/10.47928/1726-9946-2023-23-2-11-17
EDN: NGAJVN

MATHEMATICS

 MSC 34А08, 34В05 Original Article

# Initial value problem for differential equation of fractional order with variable coefficients and with variable delay

Junior Researcher of Department of Fractional Calculus, Institute of Applied Mathematics and Automation of KBSC RAS, (360017, 89 A Shortanova St., Nalchik, Russia), https://orcid.org/0000-0001-7612-8850, madina.mazhgihova@yandex.ru

Abstract. In this paper, for a linear ordinary differential equation with variable coefficients, with a Dzhrbashyan – Nersesyan fractional differentiation operator of the first order and with variable delay, the method of steps for solving the initial problem is implemented.
Fractional operators take into account the history of the process under consideration. There is also a time delay during the processes. The delay occurs because there is always a time duration for some processes. Therefore, differential equations containing both a fractional derivative and an delay argument are more realistic when describing mathematical models of various processes.
The equation under study is equivalently reduced to a Volterra integral equation of the second kind. The general representation of the solution is explicitly written out.

Keywords: fractional order differential equation, fractional derivative, Dzhrbashyan – Nersesyan derivative, delay differential equation, Volterra integral equation, variable delay, initial value problem, method of steps

Acknowledgments: the author are thankful to the anonymous reviewer for his valuable remakes.

For citation. M. G. Mazhgikhova Initial value problem for differential equation of fractional order with variable coefficients and with variable delay. Adyghe Int. Sci. J. 2023. Vol. 23, No. 2. P. 11-17.
DOI: https://doi.org/10.47928/1726-9946-2023-23-2-11-17, EDN: NGAJVN.

The author has read and approved the final version of the manuscript.
Submitted 14.06.2023; approved after reviewing 20.06.2023; aeeepted for publication 23.06.2023