Inverse Source Problems Of Stochastic Parabolic Partial Differential Equations

In real life,many problems are accompanied by more or less random phenomena.In order to make the mathematical model reflect the actual problem better,when we can describe the actual problem as the inverse problem of partial differential equation,most of the random terms are added to form the inverse problem of stochastic partial differential equation,so the inverse problem of stochastic partial differential equation is closely linked with more practical problems.This makes more and more scholars begin to devote themselves to the research of the inverse problem of stochastic partial differential equations.In this paper,we discuss the inverse source problem for a kind of stochastic parabolic partial differential equations with multiplicative noise,in which the initial and boundary conditions are both zero.The aim is to use the observed values at the end time,i.e.the final values,to invert the source term f(x)in the equation.Different from the existing literature on solving inverse source problem,the paper firstly gives the series expression of the solution of the positive problem by using the method of separation of variables,and then gives another expression of the solution of the positive problem by using the Laplace eigenvalue method.According to the two expressions equal at the end of the time,an equation containing the source term f(x)is obtained,using the equation to find the exact solution of the source term f(x).Expectations are taken on both sides of the equation to ensure that the exact solution exists.This paper also proves the uniqueness of the solution and gives the conditional stability of the exact solution under certain conditions.Since the inverse source problem is ill-posed,this paper adopts the Tikhonov regularization method to solve the problem,and obtains the regularization solution according to the inversion essence of the Tikhonov regularization method.In order to prove the effectiveness of this regularization solution,the error estimates of the exact solution f(x) and the regularization solution f_?^?(x)are obtained by selecting appropriate prior regularization parameters in theoretical from.The comparative images of the regular solution f_?^?(x) and the exact solution f(x) as well as the relative error and absolute error values are obtained by numerical experiments.The results show that the regularization solution is effective,so it is feasible and effective to use the Tikhonov regularization method to solve this kind of inverse source problem.