In this framework, the necessary and sufficient conditions for the existence and uniqueness of the second-order linear Fredholm–Stieltjes-integral equations, $u(x)=\lambda \underset{a}{\overset{b}{\int }}K(x,y)u(y)dg(y)+f(x),x\in \left[a,b\right]$, are thoroughly derived. Moreover, instead of approximating the integral equation by different numbers of partition n, the optimal number n for the given error tolerance is established. The system of equations is implemented in MAPLE for the Runge method. An efficient scheme is proposed for second-order integral equations. The solution has been compared with an exact and closed-form solution for limited cases.

中文翻译：

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关于二阶线性 Fredholm-Stieltjes 积分方程的数值解

在此框架下，二阶线性 Fredholm-Stieltjes 积分方程存在唯一性的充要条件， $你(X)=\lambda \underset{\mathrm{一种}}{\overset{乙}{\int }}钾(X,是)你(是)dG(是)+F(X),X\in \left[\mathrm{一种},乙\right]$，是彻底推导出来的。此外，不是通过不同数量的分区n来逼近积分方程，而是建立给定容错的最佳数量n。方程组是在 MAPLE 中为 Runge 方法实现的。针对二阶积分方程提出了一种有效的方案。该解决方案已与有限情况下的精确和封闭形式的解决方案进行了比较。