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Investigation of fractional order differential equation for boundary functional of a semi-Markov random walk process with negative drift and positive jumps

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  • Bandaliyev, R.A.
  • Ibayev, E.A.
  • Omarova, K.K.

Abstract

Semi-Markov processes established a rich framework for many real-world problems. Semi-Markov processes include Markov processes, Markov chains, renewal processes, Markov renewal processes, Poisson processes, birth and death processes, and etc. Since semi-Markov processes describe more complex mathematical models of various objects, in many cases the obtained mathematical models are investigated numerically. But in some cases it is possible to study such mathematical models using analytical methods. In the presented paper, the authors chose to do the second way. In this paper we study the semi-Markov random walk processes with negative drift and positive jumps. The random variable or the moment, at which the process for the time reaches in zero level is introduced. An integral equation for the Laplace transform of conditional distribution of this random variable is obtained. In this paper length of jump is given by the gamma distribution with parameters α and β 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, the exact solution of the resulting fractional differential equation with constant coefficients has been found. Finally, using form of Laplace transform the expectation and variance of the random variable are found.

Suggested Citation

  • Bandaliyev, R.A. & Ibayev, E.A. & Omarova, K.K., 2021. "Investigation of fractional order differential equation for boundary functional of a semi-Markov random walk process with negative drift and positive jumps," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    • Handle: RePEc:eee:chsofr:v:152:y:2021:i:c:s0960077921007487
    DOI: 10.1016/j.chaos.2021.111394
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    References listed on IDEAS

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    1. Zahra Dehghan Shabani & Rouhollah Shahnazi, 2020. "Spatial distribution dynamics and prediction of COVID‐19 in Asian countries: spatial Markov chain approach," Regional Science Policy & Practice, Wiley Blackwell, vol. 12(6), pages 1005-1025, December.
    2. Abdel-Rehim, E.A. & Hassan, R.M. & El-Sayed, A.M.A., 2021. "On simulating the short and long memory of ergodic Markov and Non-Markov genetic diffusion processes on the long run," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    3. Atangana, Abdon, 2018. "Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 688-706.
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