Functional difference equations in the problem on the forced oscillations of a fluid in an infinite pool with conical bottom

Abstract:

The model problem under study concerns the stationary forced oscillations of a fluid of small amplitude under the action of the field of gravity in an infinite pool with sources located on its conical bottom with infiltration. A classical solution of that problem is studied in the linear approximation. By the use of the Mellin transform and expansion in spherical functions, the problem is reduced to a set of systems of functional difference equations with meromorphic coefficients that are combinations of associated Legendre functions and their derivatives. Then, the problem on systems of difference equations reduces to singular integral equations. For this, in particular, solutions of some auxiliary first order functional equations with meromorphic coefficients are computed. It is shown that the system of integral equations in question is Fredholm with index zero. Within some assumptions, the classical solution of the problem exists and is unique. Some estimates of the classical solution in the vicinity of the conic point and at infinity are obtained.