Please use this identifier to cite or link to this item: https://sci.ldubgd.edu.ua/jspui/handle/123456789/6016
Title: The total first boundary value problem for equation of hyperbolic type with piecewise constant coefficients and \delta- singularities
Authors: Тацій, Роман Мар'янович
Карабин, Оксана Олександрівна
Чмир, Оксана Юріївна
Keywords: quasi-differential equation
the boundary value problem
the Cauchy matrix
the Dirac function
the eigenvalues problem
the method of Fourier and the method of eigenfunctions
Issue Date: 2019
Publisher: Researches in Mathematics and Mechanics
Abstract: For the first time a new formal solving scheme of the general first boundary value problem for a hyperbolic type equation with piecewise constant coefficients and \delta - singularities was proposed and justified. In the basis of the solving scheme is a concept of quasi-derivatives, a modern theory of systems of linear differential equations, the classical Fourier method and a reduction method. The advantage of this method is a possibility to examine a problem on each breakdown segment and then to combine obtained solutions on the basis of matrix calculation. Such an approach allows the use of software tools for solving the problem.
Description: The expansion by the eigenfunctions theorem is adapted for the case of differential equations with piecewise constant (by the spatial variable) coefficients. Explicit formulas for finding the solution and its quasi-derivatives for any partial interval of the main interval that are valid for arbitrary finite numbers of the first type break points of the earlier referred coefficients are received. This scheme of problem examination was considered in a case of rectangular Cartesian coordinate system. However, it remains valid in a case of any curvilinear orthogonal coordinates. The advantage of this method is a possibility to examine the problem on each breakdown segment and then using the matrix calculation to write down an analytical expression of the solution. Such an approach allows the use of software tools for solving the problem. The received results have a direct application to applied problems.
URI: http://hdl.handle.net/123456789/6016
Appears in Collections:2019



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