Non-instantaneous Impulses on Random Time in Differential Equations with Ordinary/Fractional Derivatives

Abstract

In some real world phenomena a process may change instantaneously at uncertain moments and act non instantaneously on finite intervals. In modeling such processes it is necessarily to combine deterministic differential equations with random variables at the moments of impulses. The presence of randomness in the jump condition changes the solutions of differential equations significantly. The study combines methods of deterministic differential equations and probability theory. Note differential equations with random impulsive moments differs from the study of stochastic differential equations with jumps (see, for example, [105, 127–131, 134]). We will define and study nonlinear differential equations subject to impulses starting abruptly at some random points and their action continue on intervals with a given finite length. Inspired by queuing theory and the distribution for the waiting time, we study the cases of exponentially distributed random variables, Erlang distributed random variables and Gamma distributed random variables between two consecutive moments of impulses and the intervals where the impulses act are with a constant length.