A Stochastic Galerkin Spectral Method For Solving Differential Equations

The spectral method is a common numerical method for solving ordinary differential equations and partial differential equations.It has the characteristics of high precision and simple implementation process.Initial value problems for stochastic ordinary differential equations with random variables and initial bound ary value problems for stochastic partial differential equations are widely used to describe uncertainties.In this paper,we design a random Galerkin spectral method for the initial value problem of stochastic ordinary differential equations and the initial bound-ary value problem of stochastic partial differential equations,and try to solve the uncertainty problem numerically by this method.By solving specific examples:the initial value problem of first-order stochastic ordinary differential equations,the initial boundary value problem of second-order stochastic nonlinear Burgers equation,the boundary value problem of one-dimensional elliptic equations,and the boundary value problem of two-dimensional elliptic equations,the continu-ous introduction is given in detail.The stochastic random variables obey the Gaussian distribution,the Gamma distribution,the Beta distribution,and the uniform distribution based on the Hermite polynomial,the Laguerre polynomial,the Jacobi polynomial,the Legendre polynomial and the discrete random vari-ables,respectively,obey the Poisson distribution,the binomial distribution,the negative binomial distribution,and the hypergeometric distribution.The imple-mentation of the Galerkin spectral method based on the Charlier polynomial,the Krawtchouk polynomial,the Meixner polynomial,and the Hahn polynomi-al,respectively:first assume that the random variables in the equation can be approximated by the expansion based on the basis function ?i(x)Then the ap-proximation expansion of the random variable is substit uted into the equation,and the initial value problem,the boundary value problem and the initial boundary value problem of the ordinary differential equation or partial differential equation with unknown function are obtained by the random Galerkin spectrum method.The original stochastic differential equations are obtained for equations without random variables.Approximate information.The random Galerkin spectral method based on the basis function ?i(x)has the advantages of high precision and simple implementation process.The algo-rithm implementation process and numerical examples show the advantages of the spectrum method.The numerical methods proposed in this paper are gen-eral and can be used to deal with more general stochastic ordinary differential equations or stochastic partial differential equations.