Uniqueness of the Solution of a Nonlocal Problem for an Elliptic-Hyperbolic Equation with Singular Coefficients

2021 ◽  
Vol 109 (3-4) ◽  
pp. 563-569
Author(s):  
N. V. Zaitseva
Keyword(s):  
Hyperbolic Equation ◽  
Nonlocal Problem ◽  
Singular Coefficients ◽  
Uniqueness Of The Solution

scholarly journals Nonlocal Problem for a Three-dimensional Elliptic Equation with Singular Coefficients in a Rectangular Parallelepiped

2020 ◽  
pp. 533-546
Author(s):  
Kamoliddin T. Karimov
Keyword(s):  
Elliptic Equation ◽  
Separation Of Variables ◽  
Nonlocal Problem ◽  
Three Dimensional ◽  
Fourier Method ◽  
Double Fourier Series ◽  
Rectangular Parallelepiped ◽  
Singular Coefficients ◽  
Energy Integrals ◽  
Uniqueness Of The Solution

The nonlocal problem for an elliptic equation with two singular coefficients in a rectangular parallelepiped is studied. The uniqueness of the solution of the problem is proved with the use of the method of energy integrals. The spectral Fourier method based on the separation of variables is used to prove the existence of solutions. The solution of the problem is constructed as double Fourier series in terms of a sum of trigonometric and Bessel functions. Under some conditions on parameters and given functions the uniform convergence of the constructed series and its derivatives up to the second order inclusive is proved

scholarly journals ON CERTAIN NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION WITH INTEGRAL CONDITIONS OF THE FIRST KIND

2017 ◽  
Vol 17 (5) ◽  
pp. 29-36
Author(s):  
A.V. Duzheva
Keyword(s):  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Nonlocal Problem ◽  
Integral Condition ◽  
Generalized Solution ◽  
Integral Conditions

In this article, we consider a nonlocal problem for hyperbolic equation with integral conditions of the first kind. The main goal of this article is to show the method which allows to reduce posed problem to the problem with integral condition of the second kind. Existence and uniqueness of generalized solution is proved.

scholarly journals ON SMOOTHNESS OF SOLUTION OF ONE NONLOCAL PROBLEM FOR HYPERBOLIC EQUATION

2021 ◽  
Vol 26 (2) ◽  
pp. 15-22
Author(s):  
V. A. Kirichek
Keyword(s):  
Hyperbolic Equation ◽  
Nonlocal Problem ◽  
Generalized Solution ◽  
Integral Boundary ◽  
Apriori Estimates ◽  
The Space L2 ◽  
Elastic Fixation ◽  
The Right ◽  
Derivatives Of ◽  
Smoothness Of Solution

In this paper we consider a nonlocal problem with integral boundary condition for hyperbolic equation. The conditions of the problem contain derivatives of the first order with respect to both x and t,, which can be interpreted as an elastic fixation of the right end rod in the presence of a certain damper, and since the conditions also contain integral of the desired solution, this condition is nonlocal. It is known that problems with nonlocal integral conditions are non-self-adjoint and, therefore, the study of solvability encounters difficulties that are not characteristic of self-adjoint problems. Additional difficulties arise also due to the fact that one of the conditions is dynamic. The attention of the article is focused on studying thesmoothness of the solution of the nonlocal problem. The concept of a generalized solution is introduced, and the existence of second-order derivatives and their belonging to the space L2 are proved. The proof is basedon apriori estimates obtained in this work.

scholarly journals A nonlocal problem for a generalized Trikomi equation with a spectral parameter in the unbounded domain

2021 ◽  
pp. 17-26
Author(s):  
Р.Т. Зуннунов
Keyword(s):  
Half Plane ◽  
Spectral Parameter ◽  
Unbounded Domain ◽  
Nonlocal Problem ◽  
Existence Of A Solution ◽  
Tricomi Equation ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Upper Half Plane ◽  
Generalized Tricomi Equation

В данной статье изучена нелокальная задача для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является верхней полуплоскостью. Единственность решения поставленной задачи доказана методом интегралов энергии. Существование решения поставленной задачи доказана методом функций Грина и интегральных уравнений. In this article, we study a nonlocal problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane. The uniqueness of the solution to the problem posed is proved by the method of energy integrals. The existence of a solution to the problem is proved by the method of Green’s functions and integral equations.

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