The Prandtl integrodifferential equation and the contact problem for a piecewise homogeneous plate

An inhomogeneous Prandtl singular integrodifferential equation on a number axis with a coefficient that takes different complex values on the positive and negative semiaxes is considered. The problem of reinforcing a piecewise homogeneous elastic plate by two semi-infinite stringers of different stiffness, rigidly attached to the plate along the dividing line of the materials, is reduced to this equation. The Prandtl equation is reduced, using the Mellin integral transformation to a system of difference equations with periodic coefficients whose periods are twice as great as the difference in the equations. This system is reduced by means of “special” diagonalization of its matrix and conformal mapping, first to a matrix Riemann boundary-value problem with a coefficient matrix of the substitution type, and then to a scalar Riemann boundary-value problem on a two-sheet Riemann surface, whose solution is constructed explicitly in quadratures. The solution of the Prandtl equation is found by solving the Riemann problem using an inverse Mellin transform.