This paper explores non-axisymmetric boundary value problems for the Laplace equation. Neumann's, Dirichlet's and mixed boundary conditions are involved, supposing their periodic behaviour. Boundary value problems arise as auxiliary issues in many practical applications. Among them there are problems related to numerical simulation of vibrations of fluid-filled elastic shells of revolution, coupled vibrations of elastic circular plates resting on a sloshing liquid, crack propagation in elastic mediums, and more. The common feature in these problems is the necessity to obtain the numerical solution of the Laplace equation under different boundary conditions. As these problems are auxiliary, it is necessary to obtain their numerical solutions with high accuracy. The most effective method to solve these problems is the boundary elements method (BEM). Here a new variant of BEM is proposed for the axisymmetric calculation domain with given periodic functions for boundary conditions. The shape of the calculation domain allows us to reduce surface integral equations to one-dimensional ones. In doing so, we must evaluate elliptic-like inner integrals with high accuracy, to elaborate the method of calculation of the outer integrals with logarithmic, Cauchy or Hadamard finite part singularities. An efficient method for evaluating elliptic-like integrals was developed using a special series for integrands, and the quadrature equations were obtained for highprecision calculation of outer integrals. The method developed can be used to determine free vibration modes and frequencies for elastic fluid-filled shells of revolution.

The paper presents a problem of gravitational-capillarity wave propagation in the frame of boundary integral equations. The wave propagation is considered in rigid compound shells of revolution. The liquid is supposed to be an ideal and incompressible one, and its flow is irrotational. The boundary value problem is formulated for Laplace's equation to obtain the velocity potential. Non-penetration boundary conditions are used at the shell's wetted surface, as well as kinematic and dynamic boundary conditions are given on the free liquid surface. Effects of surface tension are included in the Bernoulli's equation as additional pressure that is proportional to the free surface mean curvature. It allows us to consider coupled effects of both gravitational and capillarity waves. The problem is reduced to a system of singular integral equations. For their numerical simulation, the boundary element method is in use. The singular integral equations in implementation of a discrete model are transformed to linear algebraic ones, and eigenvalue problems are solved for different capillarity length numbers. Benchmark numerical investigations are presented including different kinds of compound rigid shells.