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Generalized Fractional Nonlinear Birth Processes

Author

Listed:
  • Mohsen Alipour

    (Babol University of Technology)

  • Luisa Beghin

    (Sapienza University of Rome)

  • Davood Rostamy

    (Imam Khomeini International University)

Abstract

We consider here generalized fractional versions of the difference-differential equation governing the classical nonlinear birth process. Orsingher and Polito (Bernoulli 16(3):858–881, 2010) defined a fractional birth process by replacing, in its governing equation, the first order time derivative with the Caputo fractional derivative of order υ ∈ (0, 1]. We study here a further generalization, obtained by adding in the equation some extra terms; as we shall see, this makes the expression of its solution much more complicated. Moreover we consider also the case υ ∈ (1, +∞ ), as well as υ ∈ (0, 1], using correspondingly two different definitions of fractional derivative: we apply the fractional Caputo derivative and the right-sided fractional Riemann–Liouville derivative on ℝ+, for υ ∈ (0, 1] and υ ∈ (1, +∞ ), respectively. For the two cases, we obtain the exact solutions and prove that they coincide with the distribution of some subordinated stochastic processes, whose random time argument is represented by a stable subordinator (for υ ∈ (1, +∞ )) or its inverse (for υ ∈ (0, 1]).

Suggested Citation

  • Mohsen Alipour & Luisa Beghin & Davood Rostamy, 2015. "Generalized Fractional Nonlinear Birth Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 525-540, September.
    • Handle: RePEc:spr:metcap:v:17:y:2015:i:3:d:10.1007_s11009-013-9369-0
    DOI: 10.1007/s11009-013-9369-0
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    References listed on IDEAS

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    1. Luisa Beghin & Claudio Macci, 2012. "Alternative Forms of Compound Fractional Poisson Processes," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-30, October.
    2. Dexter O. Cahoy & Federico Polito, 2012. "Simulation and Estimation for the Fractional Yule Process," Methodology and Computing in Applied Probability, Springer, vol. 14(2), pages 383-403, June.
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    Cited by:

    1. F. G. Badía & C. Sangüesa, 2017. "Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 647-664, June.
    2. Beghin, Luisa, 2018. "Fractional diffusion-type equations with exponential and logarithmic differential operators," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2427-2447.

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