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ON A FRACTIONAL INTEGRAL EQUATION DESCRIBING THE SEMI-MARKOV RANDOM WALK PROCESS WITHIN A BAND
  • Konul Omarova,
  • Elshan A. Ibayev,
  • Rovshan Bandaliyev
Konul Omarova
Baku Business University

Corresponding Author:omarovakonulk@gmail.com

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Elshan A. Ibayev
Ministry of Science and Education of the Republic of Azerbaijan
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Rovshan Bandaliyev
Azerbaijan University of Architecture and Construction
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Abstract

One of the important problems in probability theory is finding the distribution of the time of the sojourn of a system (a process) within a specified band. With this purpose, in this paper we consider the semi-Markov random processes with negative drift and positive jumps. An integral equation for the Laplace transform of conditional distribution of the time of the system sojourn within a given band is obtained. In this paper residence time of the system is given by the gamma distribution with parameters and resulting in a fractional order integral equation. In the class of gamma distributions, the resulting general integral equation of convolution type is reduced to a fractional order differential equation with constant coefficients. And also, in the presented paper we obtain exact solutions of the fractional differential equation in the form of infinite series.