Approximate solution to a mixed boundary problem for the elastic semi-infinite layer

The elasticity problem for a semi-infinite layer is under consideration, when a static compressive load distributed across a rectangular area is acting on the upper layer's face, lower layer's face is in the smooth contact with a rigid foundation and side edge is rigidly fixed. The method of Popov G.Ya. is used here, which is based on the introduction the new functions represented through the linear combinations of the displacements' derivatives. The system of Lamé equations is reduced to two jointly solved equations and one separately solved equation, the boundary conditions are also separated. The integral transforms reduce the problem to a one-dimensional inhomogeneous vector boundary value problem, which is solved by the matrix differential calculus. The integral equation obtained in the process is solved approximately using the method of orthogonal polynomials with analyzing singularity of the unknown function.