格林函数以及在常微分方程中的应用

Green functions and their applications to ordinary differential equations

本文综述了各种常用的常微分方程初、边值问题解的格林函数表示以及格林函数的计算法.对于一阶线性常微分方程初值问题,给出格林函数计算公式以及解的格林函数表示;对于二阶线性常微分方程的边值问题和初值问题,给出格林函数计算格式,以及解的格林函数表示;对于满足一般初值条件的Sturm-Liouville问题给出解的格林函数卷积表示和格林函数计算法;对于二阶线性常微分方程的非混合和混合边值问题考虑了解的格林函数表示和格林函数计算法;最后,给出了高阶线性常微分方程边值问题解的表示和其格林函数算法.

In this paper,we are devoted to a review of the Green functions and the Green function method for solving the initial and boundary value problems of various ordinary differential equations.For the initial value problem of linear ordinary differential equation of first order,the formula of Green function and the solution represented by Green function are given.For the boundary value problem and initial value problem of linear ordinary differential equation of second order,the calculation of Green function and the solution represented by Green function are shown,respectively.The calculation of Green function and the solution representation by Green function for Sturm-Liouville problem satisfying general initial value conditions are established.For the un-mixed and mixed boundary value problems of linear ordinary differential equations of second order,we also consider how to find Green function and the solution represented by Green function,respectively.Finally,For the boundary value problem of higher order linear ordinary differential equations,the Green function and the solution expressed by Green function are obtained.