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 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.

scholarly journals Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 110
Author(s):  
Abdukomil Risbekovich Khashimov ◽  
Dana Smetanová
Keyword(s):  
Quadratic Forms ◽  
Nonlocal Problem ◽  
General Boundary ◽  
Fredholm Integral Equations ◽  
Third Order ◽  
Existence Of A Solution ◽  
Energy Integrals ◽  
Multiple Characteristics ◽  
Uniqueness Of The Solution ◽  
Non Local

The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used.

scholarly journals The Keldysh problem for a mixed-type three-dimensional equation with three singular coefficients

2021 ◽  
pp. 29-46
Author(s):  
К.Т. Каримов
Keyword(s):  
Uniqueness Theorem ◽  
Mixed Type ◽  
Three Dimensional ◽  
Type Equation ◽  
Rectangular Parallelepiped ◽  
Mixed Type Equation ◽  
One Dimensional ◽  
Singular Coefficients ◽  
Dimensional Equation ◽  
Keldysh Problem

В данной статье изучена задача Келдыша для трехмерного уравнения смешанного типа с тремя сингулярными коэффициентами в прямоугольном параллелепипеде. На основании свойства полноты систем собственных функций двух одномерных спектральных задач, доказана теорема единственности. Решение поставленной задачи построено в виде суммы двойного ряда Фурье-Бесселя. In this article, we study the Keldysh problem for a three-dimensional mixed-type equation with three singular coefficients in a rectangular parallelepiped. Based on the completeness property of systems of eigenfunctions of two one-dimensional spectral problems, a uniqueness theorem is proved. The solution to the problem posed is constructed as the sum of a double Fourier-Bessel series.

scholarly journals On a nonlocal problem for mixed parabolic–hyperbolic type equation with nonsmooth line of type changing

2014 ◽  
Vol 07 (02) ◽  
pp. 1450030 ◽  
Author(s):  
E. T. Karimov ◽  
N. A. Rakhmatullaeva
Keyword(s):  
Boundary Problem ◽  
Integral Equations ◽  
Original Problem ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Smooth Curves ◽  
Energy Integrals ◽  
Uniqueness Of The Solution

In this paper, we investigate a boundary problem with nonlocal conditions for mixed parabolic–hyperbolic type equation with three lines of type changing. Considered domain contains a rectangle as a parabolic part and three domains bounded by smooth curves and type-changing lines as a hyperbolic part of the mixed domain. Applying method of energy integrals we prove the uniqueness of the solution for the considered problem. The proof of the existence will be done by reducing the original problem into the system of the second kind Volterra integral equations.

scholarly journals The Dirichlet Problem for the Perturbed Elliptic Equation

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2108 ◽  
Author(s):  
Ulyana Yarka ◽  
Solomiia Fedushko ◽  
Peter Veselý
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Fourier Method ◽  
Eigenvalues And Eigenfunctions ◽  
Uniqueness Of The Solution ◽  
Basis System

In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier method. This article focuses on the Dirichlet problem for the elliptic equation perturbed by the selected variable. We established the spectral properties of the perturbed operator. In this work, we found the eigenvalues and eigenfunctions of the perturbed task operator. Further, we proved the completeness, minimal spanning system, and Riesz basis system of eigenfunctions of the perturbed operator. Finally, we proved the theorem on the existence and uniqueness of the solution to the boundary value problem for a perturbed elliptic equation.

ON AN ANALOGUE OF THE HOLMGREN'S PROBLEM FOR 3D SINGULAR ELLIPTIC EQUATION

2012 ◽  
Vol 05 (02) ◽  
pp. 1250021 ◽  
Author(s):  
J. J. Nieto ◽  
E. T. Karimov
Keyword(s):  
Elliptic Equation ◽  
Explicit Form ◽  
Green's Function ◽  
Unique Solvability ◽  
Explicit Solution ◽  
Hypergeometric Functions ◽  
Three Dimensional ◽  
Function Solution ◽  
Singular Elliptic Equation ◽  
Singular Coefficients

In the present paper an analogue of the Holmgren's problem for a three-dimensional elliptic equation with singular coefficients has been studied for the unique solvability. The uniqueness for the solution of considered problem is proved by an energy integral's method. Applying a method of Green's function, solution of the problem is found in an explicit form. Moreover, decomposition formulas, formulas of differentiation and some adjacent relations for Lauricella's hypergeometric functions were used to find explicit solution for aforementioned problem, which contains Appell's hypergeometric functions.

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