scholarly journals GREEN'S FUNCTIONS OF SOME BOUNDARY VALUE PROBLEMS FOR BYHARMONIC OPERATORS AND THEIR CORRECT CONSTRICTIONS

2021 ◽  
Vol 2 (336) ◽  
pp. 15-23
Author(s):  
B. D. Koshanov ◽  
A. Baiarystanov ◽  
M. Daurenkyzy ◽  
S. O. Turymbet
Keyword(s):  
Dirichlet Problem ◽  
Green Function ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Neumann Problem ◽  
Boundary Value ◽  
Abstract Theory ◽  
The Poisson Equation ◽  
The Green Function ◽  
Defect Indices

In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

scholarly journals Existence of Positive Solutions for Two-Point Boundary Value Problems of Nonlinear Finite Discrete Fractional Differential Equations and Its Application

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Caixia Guo ◽  
Jianmin Guo ◽  
Ying Gao ◽  
Shugui Kang
Keyword(s):  
Green Function ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Fractional Differential ◽  
The Green Function

This paper is concerned with the two-point boundary value problems of nonlinear finite discrete fractional differential equations. On one hand, we discuss some new properties of the Green function. On the other hand, by using the main properties of Green function and the Krasnoselskii fixed point theorem on cones, some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established.

scholarly journals SOLVABILITY OF THE NONLINEAR BOUNDARY VALUE PROBLEMS USING THE GREEN FUNCTION

2020 ◽  
Vol 225 (02) ◽  
pp. 76-81
Author(s):  
Ngô Thị Kim Quy
Keyword(s):  
Green Function ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
The Green Function

Hàm Green có ứng dụng rộng rãi trong nghiên cứu các bài toán giá trị biên. Đặc biệt, hàm Green là công cụ quan trọng để chỉ ra sự tồn tại và duy nhất nghiệm của các bài toán. Trong bài báo này, chúng tôi nghiên cứu tính giải được của bài toán biên phi tuyến đối với phương trình vi phân có sử dụng hàm Green. Khác với cách tiếp cận của các tác giả khác, chúng tôi đưa bài toán ban đầu về phương trình toán tử đối với hàm vế phải. Xét hàm này trong miền bị chặn xác định, với một số điều kiện dễ kiểm tra chứng tỏ rằng toán tử này có tính chất co. Điều này bảo đảm bài toán gốc có nghiệm duy nhất.

scholarly journals ON THE UNIQUENESS VIOLATION OF THE DIRICHLET PROBLEM SOLUTION FOR SECOND-ORDER SYSTEMS

2021 ◽  
Vol 3 (1) ◽  
pp. 83-94
Author(s):  
V. Kyrychenko ◽  
◽  
Ye. Lesina ◽  
◽  
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Unit Disc ◽  
Boundary Value ◽  
Second Order ◽  
Important Place ◽  
Homogeneous Dirichlet Problem ◽  
Uniqueness Of The Solution ◽  
Constant Complex

The study of the issues of the correct posedness of boundary value problems for differential equations and systems occupies an important place in modern research. When considering correctness, the question of unique solvability of this problem is of paramount importance. In particular, the problem of violation of the uniqueness of the solution of boundary value problems for general differential equations in bounded domains with algebraic boundary is of interest. The property of nontrivial solvability of the homogeneous Dirichlet problem for incorrectly elliptic equations of the second order was first pointed out by A. V. Bitsadze, having constructed an example of an equation with constant complex obtained a condition for the violation of the uniqueness of the solution to the Dirichlet problem in the unit disc for a hyperbolic equation in the case when the slope angles of the characteristics differ in sign. V. P. Burskii, considering the homogeneous Dirichlet problem in the unit disc for second-order equations with constant complex coefficients and a homogeneous non-degenerate symbol, obtained a criterion for nontrivial solvability in the form of π-irrationality of the angle between the characteristics. In this paper, we investigate the question of violation of the uniqueness of the solution of the homogeneous Dirichlet problem for a system of typeless second-order partial differential equations in a model domain – a circle. The original system is written in the form of an equation with commuting matrix coefficients. The permutability condition allows one to obtain a necessary and sufficient condition for the nontrivial solvability of the problem under consideration in the form of equality to zero of the determinant, the elements of which are expressed in terms of the coefficients of the equation. This form of writing the criterion allows one to construct examples of systems for which the kernel of the Dirichlet problem is nontrivial and infinite-dimensional. The study was based on the integral condition for the connection of associated boundary L-traces, as well as a functional scheme, the application of which reduces the expansion of a matrix function in a Fourier series to a standard expansion of each of its elements. A theorem of nontrivial solvability of the homogeneous Dirichlet problem is proved.

scholarly journals THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS

2020 ◽  
Vol 72 (4) ◽  
pp. 68-72
Author(s):  
Zh.A. Tokibetov ◽  
◽  
N. E. Bashar ◽  
А.К. Pirmanova ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Dirichlet Problem ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Partial Differential ◽  
First Order ◽  
The Cauchy Problem ◽  
Classical Boundary

For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.

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