caa a22 4500 605519668 CHVBK 20210128100731.0 cr unu---uuuuu 210128e20150901xx s 000 0 eng 10.1007/s11009-013-9369-0 doi (NATIONALLICENCE)springer-10.1007/s11009-013-9369-0 Generalized Fractional Nonlinear Birth Processes [Elektronische Daten] [Mohsen Alipour, Luisa Beghin, Davood Rostamy] 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]). Springer Science+Business Media New York, 2013 Generalized fractional birth process nationallicence Fractional Caputo derivative nationallicence Fractional Riemann-Liouville derivative nationallicence Mittag-Leffler functions nationallicence Stable subordinator nationallicence Alipour Mohsen Faculty of Basic Science, Babol University of Technology, P.O. Box 47148-71167, Babol, Iran aut Beghin Luisa Department of Statistical Sciences, Sapienza University of Rome, Rome, Italy aut Rostamy Davood Department of Mathematics, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin, Iran aut Methodology and Computing in Applied Probability Springer US; http://www.springer-ny.com 17/3(2015-09-01), 525-540 1387-5841 17:3<525 2015 17 11009 https://doi.org/10.1007/s11009-013-9369-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s11009-013-9369-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Alipour Mohsen Faculty of Basic Science, Babol University of Technology, P.O. Box 47148-71167, Babol, Iran aut NATIONALLICENCE 700 1- Beghin Luisa Department of Statistical Sciences, Sapienza University of Rome, Rome, Italy aut NATIONALLICENCE 700 1- Rostamy Davood Department of Mathematics, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin, Iran aut NATIONALLICENCE 773 0- Methodology and Computing in Applied Probability Springer US; http://www.springer-ny.com 17/3(2015-09-01), 525-540 1387-5841 17:3<525 2015 17 11009