Generalized Fractional Nonlinear Birth Processes

摘要

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 upsilon is an element of (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 upsilon is an element of (1, +infinity), as well as upsilon is an element of (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 upsilon is an element of (0, 1] and upsilon is an element of (1, +infinity), 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 upsilon is an element of (1, +infinity)) or its inverse (for upsilon is an element of (0, 1]).