The Approximation Problem Of Stochastic Vibration Equations With Variable Damping

In recent years,much research has been made on the approximation of stochas-tic vibration equations,but the study of stochastic vibration equations with vari-able damping is less.In the actual model,the random vibration system with vari-able damping exists widely.For example,the stability of orbiting satellite model,cli-mate model,filter model and so on.We mainly studies the approximation problem of the stochastic vibration equation with variable damping in this paper.We obtain the boundedness and existence and uniqueness of the solution by using It(?) formu-la and Gronwall lemma.By using the average principle,we get the average approxi-mation equation of X_t~? when ??0,and get the more efficient approximation equation of X_t~? when ? is very small.Further more,by using the method of partial estimation,it is proved that the solution of the original equation is approximated by the solution of the corresponding deterministic equation when the singular perturbation tends to 0.This article is divided into four parts.The first chapter is the introduction which mainly introduces the research background,research status,research significance and the main work of this paper.The second chapter is the preliminaries which aim to let readers understand the contents better.In this chapter,we briefly review the proba-bility basic knowledge and some concepts of It(?) type stochastic differential equation-s and we mainly introduce the mean principle and Smoluchowski-Kramers approx-imation.In the third chapter,we get the boundedness and existence and uniqueness of the solution by using the Gronwall lemma and Fubini theorem and other knowl-edge.In the fourth chapter,we get the effective approximation of the original equation with the average principle and prove the Smoluchowski-Kramers approximation again by the method of partial estimation.