Numerical Solution Of Nonlinear Stochastic It(?)-Volterra Integral Equations Based On Wavelet Functions

Stochastic integral equations are widely applied in biology,oceanography,engineer,physical,etc.As we all know,many stochastic Volterra integral equations do not excat solutions,so it makes sense to find more precise approximate solutions to stochastic Volterra integral equations.In this paper,the numerical solution of the nonlinear stochastic integral equations are studied.the basic idea is to get Haar wavelets representations of functions and integrals and stochastic integration operational matrixes by using the definitions and properties of block pulse functions and Haar wavelets.Then,by applying the stochastic integration operational matrixes of Haar wavelets,the of nonlinear stochastic It(?)-Volterra integral equations can be simplify algebraic equations.At the same time,the approximate solution of nonlinear stochastic It(?)-Volterra integral equations is represented by the linear combination of the solution vectors of algebraic equations and Haar wavelets.Moreover the rationality and effectiveness of this approximate solution can be further supported by the error analysis and some examples.The structure of this paper is as follows:In chapter 1,the background of nonlinear stochastic It(?)-Volterra integral equations,the present situation of foreign research,and the innovation of this paper are shown.In chapter 2,some preliminaries of block pulse functions and Haar wavelets are given.In chapter 3,the of nonlinear stochastic It(?)-Volterra integral equations can be simplify algebraic equations by applying the stochastic integration operational matrixes of Haar wavelets.Then,the error analysis is proved some important inequality such as Gronwall inequality.Lastly,the high accuracy and effectiveness of this solution method are supported by some examples.In chapter 4,the solution method of the of the nonlinear stochastic It(?)-Volterra integral equations driven by multiple independent Brownian motions are shown.At the same time,the nonlinear functions of the nonlinear stochastic It(?)-Volterra integral equations driven by multiple independent Brownian motions satisfy the analytic functions.Moreover,the high accuracy and effectiveness of this solution method are supported by some examples.In chapter 5,the research results and prospects of this paper are summarized.