Numerical Solution Of Nonlinear Stochastic Integral Equations By Block Pulse Functions

The stochastic It?-Volterra integral equation has been widely used in many Fields,such as automatic control,biology,economics,medicine and sociology,but there is only a small number of equations give real solutions.Therefore,it is of great theoretical and practical significance to study how to give its numerical solution.This paper introduces a kind of effective numerical method based on block pulse functions and gives a numerical solution of nonlinear stochastic It?-Volterra integral equation.Based on the disjointness,orthogonality and completeness of block pulse functions,we obtained the integral operator matrix and stochastic integral operator matrix.Combined with another related properties and theorems,nonlinear stochastic integral equation can be converted to algebraic equations,and the error analysis can prove the convergence of the method is great.The main contents of this paper includes:Firstly,we introduce the research background,research status at home and abroad and the innovation of this paper.Secondly,we review the basic knowledge of one-dimensional block pulse functions,discusses how to use its related properties to establish integral operator matrix and stochastic integral operator matrix,and convert the nonlinear stochastic It?-Volterra integral equation into algebraic equation.By MATLAB,we give the numerical results.Analyze the numerical results of examples and compared it with other methods.Thirdly,This chapter introduces the definition of two-dimensional block pulse functions,gives the integral operator matrix and stochastic integral operator matrix of two-dimensional block pulse function and explores how to solve the two-dimensional nonlinear stochastic It?-Volterra integral equation by MATLAB programming.Finally,the results are summarized and the future work is prospected.