Integral and integro-differential equations with an exponential kernel and applications

Abstract

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.