A special relationship in spheroidal wave functions and its application to contact problems

Abstract

A spectral and kindred relationship are set up by methods of the theory of the generalized potential /1/ for an integral operator generated by a symmetric difference kernel in the form of a Macdonald function in two identical semi-infinite intervals {(− ∞, − α), (α, ∞)} that contain spheroidal wave functions. The formula for the expansion of an arbitrary function in these functions is also set up by a well-known method /2/. On the basis of the results obtained, a solution is then constructed for the integral equation of the contact problem of the impression of two identical stamps with half-plane bases into a half-space being deformed in a power-law form in the formulation of /3/. This contact problem can be described by the same integral equation when the elastic modulus of a linearly elastic half-space changes with depth according to a power law /1/. The spectral relationships in classical orthogonal polynomials for extensive classes of integral operators in mathematical physics are presented in /4,5/, where the method of orthogonal polynomials based on them is also elucidated, and numerous applications of this method are given to contact and mixed problems of elasticity theory. We also mention /6–9/ which are related directly to the investigation presented here.