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Numerical solution of a singular integral equation related with a dynamic contact interaction problem

A singular integral equation with a fixed singularity to which the problem of contact interaction of two quarters of spaces in the conditions of harmonic oscillations of longitudinal shear is reduced is considered. A quarters of the space is situated so that the half-space composed of them has a stepped boundary. In the contact area, the conditions for ideal coupled are satisfied. The unknown function in this equation is the contact stresses. For the numerical solution of this equation, a method that takes into account the asymptotic behavior of contact stresses at the edge point is proposed. The basis of this method is the use of special quadrature formulas for singular integrals obtained in the article. When obtaining these formulas, the unknown function was approximated by an interpolation polynomial, in which the roots of the Laguerre polynomials are the points of interpolation. The values of the unknown function at the interpolation points are found by the collocation method, herewith the collocation points of collocationare the roots of the special function. An approximate formula for calculating contact stresses can have practical application. The effectiveness of the proposed method is demonstrated by the numerical example.