Математичне моделювання коливних процесів у необмеженому кусково-однорідному клиновидному суцільному циліндрі

Abstract

The theory of boundary value problems for differential equations with partial derivatives develops intensively and its results are important for the development of many sections of mathematics. Its achievements are applied in the mathematical modeling of various processes and phenomenon of physics, mechanics, biology, medicine, economics, engineering. It is well known that the complexity of a boundary-value problem significantly depends on the coefficients of equations and the geometry of domain in which the problem is considered. Properties of solutions of boundary value problems for linear, quasilinear, and some classes of nonlinear equations in single-connected domains have been studied in sufficient detail. However, many important applied problems of thermal physics, thermomechanics, theory of elasticity, theory of electrical circuits, theory of vibrations lead to boundary value problems for differential equations with partial derivatives not only in homogeneous domains when the coefficients of the equations are continuous, but also in piecewise homogeneous and inhomogeneous domains when the coefficients of the equations are piecewise continuous. In this article the exact analytical solutions of mathematical models of oscillating processes (hyperbolic initial-boundary problem of conjugation) for unlimited piecewise-homogeneous wedge-shaped solid cylinder are obtained by means of the method of integral and hybrid integral transforms, in combination with the method of main solutions (influence matrices and Green's matrices). The obtained solutions are of algorithmic character, continuously depend on the parameters and data of problem and can be used in further theoretical research and in practical engineering calculations of real processes which are modeled by hyperbolic boundary-value problems that are described by a cylindrical coordinate system (problems of acoustics, hydrodynamics, the theory of vibrations of mechanical systems).