Abstract

A modification of the parameterization method is proposed to solve a linear two-point boundary value problem for a Fredholm integro-differential equation. The domain of the problem is partitioned and additional parameters are set as the values of the solution at interior points of the partition subintervals. Definition of a regular pair consisting of a partition and chosen interior points is given. The original problem is transformed into a multipoint boundary value problem with parameters. For fixed values of parameters, we get a special Cauchy problem for a system of integro-differential equations on the subintervals. Using the solution to this problem, the boundary condition and continuity conditions of solutions at the interior mesh points of the partition, we construct a system of linear algebraic equations in parameters. It is established that the solvability of the problem under consideration is equivalent to that of the constructed system.

中文翻译：

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Fredholm积分微分方程线性边值问题的参数化方法的修改

摘要

提出了一种参数化方法的改进方案，以解决Fredholm积分微分方程的线性两点边值问题。对问题的范围进行了分区，并在分区子间隔的内部点处将其他参数设置为解决方案的值。给出了由分区和选定内部点组成的规则对的定义。原始问题转化为带参数的多点边值问题。对于固定参数值，对于子区间上的积分-微分方程组，我们得到一个特殊的柯西问题。利用该问题的解，该分区的内部网格点处的解的边界条件和连续性条件，我们构造了一个参数线性代数方程组。