OSCILLATORY CONDITIONS OF SOME CLASSES EQUATIONS SOLUTIONS WITH LAPLASE-BELTRAMY OPERATOR AT DESIGN PROBLEMS IN UNLIMITED AREAS
Abstract
The problems of oscillation equations solutions that describe some kinds of technical model, as a rule deal with boundary problem solutions study. The problem of determining the oscillation and nonoscilllatin of solutions of differential equations has been a very active area in the last three decades. The problem of existence of oscillatory solutions of the first order differential equations with deviating arguments has been considered by many authors. A large group of engineering tasks in physics and mathematics is described by variables that belong to permanent curvature spaces: the Euclidean, hyperbolic (Lobachevsky) and Ryman. A solution of this equation is called oscillatory if it has an infinite sequence of zeros tending to infinity. Otherwise it is called nonoscillatory.
Sufficient conditions of oscillation properties for the solutions of differential equations with partial derivatives in the constant-curvature spaces have been obtained. Uniqueness of boundary value problems solutions for differential equations with partial derivatives of the second and fourth order, determined by the transformation of their solutions in the zero and suggests the possibility of appropriate solutions to the problem. The main research tool was based on the use of average values on the n-dimension sphere in the appropriate space. Using a notion «mean value» concept in spaces with permanent curvature makes possible to investigate on oscillation some types of equations in the partial derivatives. Separate studies were conducted for solutions, which depend on the radial parameter only. This allowed to reduce the study of partial differential equations properties to ordinary differential equations ones. Several theorems and consequences are well-proved in terms of oscillation. Sufficient conditions of some classes differential equations oscillation conditions solutions with the Laplace – Beltramy operator of the even order in the spaces of permanent curvature have been found. Properties of selfadjoint form equations solutions are investigational in the thoroidal coordinates system in unlimited areas have been investigated. Properties, that provide the terms of oscillation conditions of equation, that depend only on a radial constituent, have been studied in the article. The obtained results can be used for the evaluation of physical-mechanical characteristics of composites and coatings based on them.
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