Narrow Results

A class of mixed boundary value problems (BVPs) arising in the study of scattering of surface water waves by the edges of floating structures comprising of elastic plates, with or without cracks, is examined for their solutions. It is observed that the simplest possible method of solution of such BVPs is the one that involves solution of an over-determined system of Linear Algebraic Equations. Such over-determined systems of equations are best solved by the method of least squares. Numerical results for useful practical quantities such as the "reflection" and "transmission" coefficients are obtained for one of the problems considered here.

The harmonic Dirichlet problem in a planar domain with smooth cracks of an arbitrary shape is considered in case, when the solution is not continuous at the ends the cracks. The well–posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, the integral representation for a solution is obtained. With the help of the integral representation, the properties of the solution are studied. It is proved that a weak solution of the Dirichlet problem in question does not exist typically, though the classical solution exists.