Three-dimensional Hyperbolic Equations Research Articles

The uniqueness of the solution of a mixed problem for three-dimensional hyperbolic equations with type and order degeneracy property

Abstract The relevance of the stated subject is conditioned upon the presence of a real possibility to simulate vibrations of elastic membranes in space according to the Hamilton principle using degenerate three-dimensional hyperbolic equations, which is of particular practical importance from the standpoint of the prospects for mathematical modelling of the heat propagation process in oscillating elastic membranes. The purpose of this paper is to study the sequence of the procedure for mathematical modelling of heat propagation in oscillating elastic membranes which is leading to degenerate three-dimensional hyperbolic equations. The methodological approach of this study is based on a combination of theoretical study of the possibilities of constructing mathematical models of heat propagation in oscillating elastic membranes with the practical application of methods for constructing three-dimensional hyperbolic equations with type and order degeneracy to find a single solution to a mixed problem. In the course of this study, the results were presented in the form of a mathematical proof of the possibility of obtaining a single solution to a mixed problem for three-dimensional hyperbolic equations with type and order degeneracy. The results obtained in this study and the conclusions formulated on their basis are of significant practical importance for developers of methods of mathematical modelling of heat propagation processes in oscillating artificial membranes, which is of key importance from the standpoint of prospects for improving methods of mathematical modelling of processes occurring in technical devices used in various fields of modern industries.

КОМПОЗИЦИЯЛЫҚ МАТЕРИАЛДАР ҚҰРЫЛЫМЫ ЖӘНЕ КОМПОЗИТТЕР МЕХАНИКАСЫНЫҢ ЕСЕПТЕРІ

It is known that in space during mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, then we get degenerating multidimensional hyperbolic equations. Therefore, the analysis of electromagnetic fields in complex environments (for example, if the conductivity of the medium changes) is reduced to degenerating multidimensional hyperbolic equations. It is also known that oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Therefore, by studying mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerating multidimensional hyperbolic equations. When studying these applications, it becomes necessary to obtain a clear representation of the solutions to the investigated problems. The mixed problem for degenerating multidimensional hyperbolic equations in generalized spaces is well researched. This task is also studied in the works of S. A. Aldashev, where it is shown that its correctness significantly depends on the height of the cylindrical region under consideration. A.V. Bitsadze drew attention to the importance of studies of multidimensional hyperbolic equations with degeneration of type and order. Mixed problems for these equations have not previously been studied. In this work, the solvability of a mixed problem is shown and a clear form of a classical solution for three-dimensional hyperbolic equations with degeneration of type and order is obtained.

CORRECTNESS OF A MIXED PROBLEM FOR DEGENERATE THREE-DIMENSIONAL HYPERBOLIC-PARABOLIC EQUATIONS

In mathematical modeling of electromagnetic fields in space, the nature of electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate three-dimensional hyperbolic equations. If the medium has a high conductivity, then we come to degeneratethree-dimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the mediums conductivity changes) is reduced to degenerate three-dimensional hyperbolic-parabolic equations. The mixed problem for multidimensional hyperbolic equations is well studied and has been previously considered in the works of various authors. In the articles of Professor S.A. Aldashev, the unique solvability of the mixed problem for degenerate multidimensional hyperbolic equations is proved. It is known that mixed problems for multidimensional hyperbolic-parabolic equations have not been studied much. The paper finds a new class of degenerate three-dimensional hyperbolic-parabolic equations for which the mixed problem has a unique solution and gives an explicit representation of its classical solution.

On the convergence of the method of iterative approximate factorization of difference operators of high-order accurate bicompact scheme for nonstationary three-dimensional hyperbolic equations

On the convergence of the method of iterative approximate factorization of difference operators of high-order accurate bicompact scheme for nonstationary three-dimensional hyperbolic equations

О сходимости метода итерируемой приближенной факторизации операторов высокоточной бикомпактной схемы для нестационарных трехмерных уравнений гиперболического типа

О сходимости метода итерируемой приближенной факторизации операторов высокоточной бикомпактной схемы для нестационарных трехмерных уравнений гиперболического типа

Meshless Method for a Kind of Three-Dimensional Hyperbolic Equation

The paper deals with a numerical modeling of a three-dimensional linear hyperbolic problem. The modeling is carried out using the meshless element free Galerkin (EFG) method. In the mathematical formulation, the variational method is employed to derive the discrete equations and the essential boundary conditions are enforced by the penalty method. The effectiveness of EFG for three-dimensional hyperbolic problems is investigated by numerical example in this paper.

Invariants of two- and three-dimensional hyperbolic equations

We consider linear hyperbolic equations of the form u t t = ∑ i = 1 n u x i x i + ∑ i = 1 n X i ( x 1 , … , x n , t ) u x i + T ( x 1 , … , x n , t ) u t + U ( x 1 , … , x n , t ) u . We derive equivalence transformations which are used to obtain differential invariants for the cases n = 2 and n = 3 . Motivated by these results, we present the general results for the n-dimensional case. It appears (at least for n = 2 ) that this class of hyperbolic equations admits differential invariants of order one, but not of order two. We employ the derived invariants to construct interesting mappings between equivalent equations.

A Classical Solution Weakened on the Axis for a Centrally Symmetric Mixed Problem for a Three-Dimensional Hyperbolic Equation of Even Order in Hölder Spaces

A Classical Solution Weakened on the Axis for a Centrally Symmetric Mixed Problem for a Three-Dimensional Hyperbolic Equation of Even Order in Hölder Spaces

A Classical Solution, Weakened on the Axis, of a Centrally Symmetric Mixed Problem for a Three-Dimensional Hyperbolic Equation in Hölder Spaces

A Classical Solution, Weakened on the Axis, of a Centrally Symmetric Mixed Problem for a Three-Dimensional Hyperbolic Equation in Hölder Spaces