AVERAGING PRINCIPLE FOR IMPULSIVE IMPLICIT STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

The paper formulates an averaging principle of a family of impulsive implicit stochastic fractional differential equations, in the Caputo sense. Three basic properties of the system are: 1) fractional order 2a 2b the derivative is not fixed but varies in response to a Poisson process. These models happen in the viscoelastic systems of random shocks and hereditary properties. The first consequence of our work is that there exist and are unique mild solutions to the non-Lipschitz equations when local integrability is assumed on the coefficients. The main result is that the original impulsive system of fractionation can be approximated by an averaged system that has no impulses and has no explicit structure, and has explicit error limits in a mean-square sense. This is based on the application of fractional calculus, halting time techniques and BurkholderDavisGundy inequality to control the terms of stochastic integral. We find that the solution of the original equation tends to the solution of the averaged equation on finite time intervals as the small parameter ε goes to 0 with a rate proportional to ε(alpha-1/2). The theoretical results are confirmed using a numerical example that simulates a fractional RLC circuit with random switchings with the averaged equation requiring less time to compute (73% less) but the averaged equation has a mean-square error less than 4.2. Our results are a generalization of classical averaging theory to systems of fractional-order dynamics with impulses and implicit dependence, not covered by the existing literature. Applications This technology can be used to solve stochastic control problems, in jump process finance, and in memory-effects mechanics of materials. It can be applied to minimise models of more complex stochastic fractional systems that can occur in engineering and physics.