Abstract

A linear boundary value problem for essentially loaded differential equations is investigated. Using the properties of essentially loaded differential equation and assuming the invertibility of the matrix compiled through the coefficients at the values of the derivative of the desired function at load points, we reduce the considering problem to a two-point boundary value problem for loaded differential equations. The method of parameterization is used for solving the problem. The linear boundary value problem for loaded differential equations by introducing additional parameters at the loading points is reduced to equivalent boundary value problem with parameters. The equivalent boundary value problem with parameters consists of the Cauchy problem for the system of ordinary differential equations with parameters, boundary condition and continuity conditions. The solution of the Cauchy problem for the system of ordinary differential equations with parameters is constructed using the fundamental matrix of differential equation. The system of linear algebraic equations with respect to the parameters are composed by substituting the values of the corresponding points in the built solutions to the boundary condition and the continuity condition. Numerical method for finding solution of the problem is suggested, which based on the solving the constructed system and the Bulirsch–Stoer method for solving Cauchy problem on the subintervals.

中文翻译：

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一种求解本质加载微分方程边值问题的数值方法。

摘要

研究了基本加载的微分方程的线性边界值问题。利用基本加载的微分方程的性质，并假设在负载点处通过所需函数的导数值处的系数编译的矩阵的可逆性，我们将考虑的问题简化为负载微分方程的两点边值问题。参数化方法用于解决问题。通过在加载点引入附加参数，将加载的微分方程的线性边值问题简化为带参数的等效边值问题。带参数的等价边值问题由带参数的常微分方程组的柯西问题组成，边界条件和连续性条件。利用微分方程的基本矩阵，构造了带参数的常微分方程系统的柯西问题解。通过代入边界条件和连续性条件的已构建解中的对应点的值，可以构成关于参数的线性代数方程组。提出了寻找问题解决方案的数值方法，该方法是在求解已构造系统的基础上，采用Bulirsch-Stoer方法求解子区间上的柯西问题的。通过代入边界条件和连续性条件的已构建解中的对应点的值，可以构成关于参数的线性代数方程组。提出了一种寻找问题解决方案的数值方法，该方法基于对构造系统的求解以及在子区间上解决柯西问题的Bulirsch–Stoer方法。通过代入边界条件和连续性条件的已构建解中的对应点的值，可以构成关于参数的线性代数方程组。提出了一种寻找问题解决方案的数值方法，该方法基于对构造系统的求解以及在子区间上解决柯西问题的Bulirsch–Stoer方法。