Application of Analogues of General Kolosov–Muskhelishvili Representations in the Theory of Elastic Mixtures

1999 ◽  
Vol 6 (1) ◽  
pp. 1-18
Author(s):  
M. Basheleishvili
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Elastic Body ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Plane Boundary ◽  
The Third ◽  
Isotropic Elastic Body ◽  
Uniqueness Of A Solution ◽  
Theory Of Elastic Mixtures

Abstract The existence and uniqueness of a solution of the first, the second and the third plane boundary value problem are considered for the basic homogeneous equations of statics in the theory of elastic mixtures. Applying the general Kolosov–Muskhelishvili representations from [Basheleishvili, Georgian Math. J. 4: 223–242, 1997], these problems can be splitted and reduced to the first and the second boundary value problem for an elliptic equation which structurally coincides with the equation of statics of an isotropic elastic body.

scholarly journals ONE-VALUED SOLVABILITY OF A NONLOCAL PROBLEM FOR THE AXISYMMETRIC HELMHOLTZ EQUATION

2017 ◽  
Vol 17 (2) ◽  
pp. 5-14
Author(s):  
A.A. Abashkin
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Problem ◽  
Degenerate Elliptic Equation ◽  
Nonlocal Boundary Value Problem ◽  
Degenerate Elliptic ◽  
Uniqueness Of A Solution

A nonlocal boundary value problem for degenerate elliptic equation is considered. Boundary value of this problem considerably depend on low derivativecoefficient changes. Existence and uniqueness of a solution are proved.

scholarly journals Solving a nonlinear biharmonic boundary value problem

2018 ◽  
Vol 33 (4) ◽  
pp. 308-324
Author(s):  
Dang Quang A ◽  
Truong Ha Hai ◽  
Nguyen Thanh Huong ◽  
Ngo Thi Kim Quy
Keyword(s):  
Iterative Method ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Nonlinear Operator ◽  
Nonlinear Term ◽  
Boundary Value ◽  
New Approach ◽  
Nonlinear Elastic ◽  
Uniqueness Of A Solution ◽  
Theoretical Results

In this paper we study a boundary value problem for a nonlinear biharmonic equation, which models a bending plate on nonlinear elastic foundation. We propose a new approach to existence and uniqueness  and numerical solution of the problem. It is based on the reduction of the problem to finding fixed point of a nonlinear operator for the nonlinear term. The result is that under some easily verified conditions we have established the existence and uniqueness of a solution and the convergence of an iterative method for the solution. The positivity of the solution and the monotony of iterations are also considered. Some examples demonstrate the applicability of the obtained theoretical results and the efficiency of the iterative method.

On One Boundary Value Problem for a Nonlinear Equation with the Iterated Wave Operator in the Principal Part

2008 ◽  
Vol 15 (3) ◽  
pp. 541-554
Author(s):  
Sergo Kharibegashvili ◽  
Bidzina Midodashvili
Keyword(s):  
Nonlinear Equation ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Wave Operator ◽  
Principal Part ◽  
Boundary Value ◽  
Power Nonlinearity ◽  
Spatial Dimensionality ◽  
Uniqueness Of A Solution

Abstract One boundary value problem for a hyperbolic equation with power nonlinearity and the iterated wave operator in the principal part is considered in a conical domain. Depending on the index of nonlinearity and spatial dimensionality of the equation the question on the existence and uniqueness of a solution of a boundary value problem is investigated. The question as to the absence of a solution of this problem is also considered.

scholarly journals On a Nonlocal Boundary Value Problem for Second Order Nonlinear Singular Differential Equations

2000 ◽  
Vol 7 (1) ◽  
pp. 133-154 ◽  
Author(s):  
A. Lomtatidze ◽  
L. Malaguti
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Bounded Variation ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
Carathéodory Conditions ◽  
Uniqueness Of A Solution

Abstract Criteria for the existence and uniqueness of a solution of the boundary value problem are established, where ƒ :]a, b[×R 2 → R satisfies the local Carathéodory conditions, and μ : [a, b] → R is the function of bounded variation. These criteria apply to the case where the function ƒ has nonintegrable singularities in the first argument at the points a and b.

scholarly journals On a nonlocal boundary value problem for the equation fourth-order in partial derivatives

2021 ◽  
pp. 16-23
Author(s):  
О.Ш. Киличов
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Problem ◽  
Order Equation ◽  
Nonlocal Boundary Value Problem ◽  
Fourth Order Equation ◽  
Uniqueness Of A Solution

В данной статье изучается нелокальная задача для уравнения четвертого порядка в которой доказывается существование и единственность решения этой задачи. Решение построено явно в виде ряда Фурье, обоснованы абсолютная и равномерная сходимость полученного ряда и возможность почленного дифференцирования решения по всем переменным. Установлен критерий однозначной разрешимости поставленной краевой задачи. In this article, we study a nonlocal problem for a fourth-order equation in which the existence and uniqueness of a solution to this problem is proved. The solution is constructed explicitly in the form of a Fourier series; the absolute and uniform convergence of the obtained series and the possibility of term-by-term differentiation of the solution with respect to all variables are substantiated. A criterion for the unique solvability of the stated boundary value problem is established.

scholarly journals Existence and uniqueness theorems for a third-order generalized boundary value problem

1989 ◽  
Vol 2 (1) ◽  
pp. 33-51
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Uniqueness Theorems ◽  
Linear Boundary ◽  
Third Order ◽  
The Third ◽  
Special Cases ◽  
Linear Boundary Value Problem ◽  
Natural Way

Let f:[0,1]×ℝ3→ℝ be a function satisfying Caratheodory's conditions, e(x)∈L1[0,1], η∈[0,1], h≥0, k≥0, h+k>0. This paper studies existence and uniqueness questions for the third-order three-point generalized boundary value problem u‴+f(x,u,u′,u″)=e(x),   0<x<1, u(η)=0,   u″(0)−hu′(0)=u″(1)+ku′(1)=0, and the associated special cases corresponding to one or both of h and k equal to infinity. The conditions on the nonlinearity f turn out to be related to the spectrum of the linear boundary value problem u‴=λu′, u(η)=0, u″(0)−hu′(0)=u″(1)+ku′(1)=0, in a natural way.

Convergence Of a Finite Difference Scheme for the Third Boundary-Value Problem for an Elliptic Equation with Variable Coefficients

2001 ◽  
Vol 1 (4) ◽  
pp. 356-366 ◽  
Author(s):  
Boško S. Jovanović ◽  
Branislav Z. Popović
Keyword(s):  
Elliptic Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Boundary Value ◽  
Variable Coefficients ◽  
Convergence Rate Estimate ◽  
The Third ◽  
Third Boundary Value Problem

Abstract In this paper we study the convergence of a finite difference scheme that approximates the third boundary-value problem for an elliptic equation with variable coefficients on a unit square. An ”almost” second-order convergence rate estimate (with additional logarithmic multiplier) is obtained.

scholarly journals On an Inverse Boundary Value Problem for a Fourth Order Elliptic Equation with Integral Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yashar T. Mehraliyev
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Initial Problem ◽  
Order Elliptic Equation ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem ◽  
Fourth Order Elliptic Equation

An inverse boundary value problem for a fourth order elliptic equation is investigated. At first the initial problem is reduced to the equivalent problem for which the existence and uniqueness theorem of the solution is proved. Further, using these facts, the existence and uniqueness of the classic solution of the initial problem are proved.

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