Summary A convolution integral equation of the first kind and integro-differential equation of the second kind with the kernel $e^{-\gamma |y-\eta|}$ on a finite and semi-infinite interval are analyzed. For the former equation necessary and sufficient conditions for the existence and uniqueness of the solution are obtained, and when the solution exists, a closed-form representation for the solution is derived. On the basis of these results new integral relations for the spheroidal functions and Laguerre polynomials are obtained. The integro-differential equations on a finite and semi-infinite interval are transformed into a vector and scalar Riemann–Hilbert problem, respectively. Both problems are solved in closed-form. An application of these solutions to bending problems of a strip-shaped and a half-plane-shaped plate contacting with an elastic linearly deformable three-dimensional foundation characterized by the Korenev kernel $AK_0(\delta r)$ ($A$ and $\delta$ are parameters, $K_0(\cdot)$ is the modified Bessel function, and $r=\sqrt{(x-\xi)^2+(y-\eta)^2}$) is considered.

中文翻译：

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具有指数核的积分和积分微分方程及应用

小结 分析了有限和半无限区间上核$e^{-\gamma |y-\eta|}$的第一类卷积积分方程和第二类积分微分方程。对于前一种方程，得到解存在唯一性的充要条件，当解存在时，导出解的闭式表示。在这些结果的基础上，得到了球体函数和拉盖尔多项式的新积分关系。有限和半无限区间上的积分-微分方程分别转换为向量和标量 Riemann-Hilbert 问题。这两个问题都以封闭形式解决。