Barnyak M. Ya.
Construction of the solutions of boundary-value problems for the laplace equation in domains of revolution with edged boundary
Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 579-595
A method for the construction of high-precision approximate solutions of boundary-value problems for the Laplace equation in domains with corner points is proposed. We consider boundary-value problems for the three-dimensional Laplace equation in domains in the form of bodies of revolution whose meridional section has corner points. The solutions of the problems are constructed by using variational methods. For the numerical realization of these methods, we construct special solutions of the Laplace equation with singularities (or with singularities of their partial derivatives) on a certain ray originating at a corner point and directed outside the domain. To illustrate the proposed method, we construct the solutions of the Neumann problem and the problem of natural oscillations of ideal liquid in a spherical cavity.
Construction of solutions for the problem of free oscillations of an ideal liquid in cavities of complex geometric form
Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1587–1600
We consider the problem of free oscillations of an ideal incompressible liquid in cavities of complex geometric form. The domain filled with liquid is divided into subdomains of simpler geometric form. The original problem is reduced to the spectral problem for a part of the domain filled with liquid. To this end, we use solutions of auxiliary boundary-value problems in subdomains. We construct approximate solutions of the problem obtained using the variational method. We also consider the problem of the rational choice of a system of coordinate functions. Results of the numerical realization of the proposed method are presented.