Mathematical modeling of the semi-Markovian random walk processes with jumps and delaying screen by means of a fractional order differential equation

In this paper, we investigate the semi-Markovian random walk processes with jumps and delaying screen in zero. The Laplace transform on time, Laplace- Stieltjes transform on phase of the conditional distribution of semi-Markovian random walk processes with jumps is found. We get a mathematical modeling of the semi-Markov random walk processes with a delaying screen in zero, given in general form by means of integral equation. In this paper, the residence time of the system is given by the gamma distribution with the parameters α and β resulting in a fractional order integral equation. The purpose of this paper is to reduce the fractional order integral equation to a fractional order differential equation. Finally, we find the exact solution of fractional order differential equation. © 2018 John Wiley & Sons, Ltd.