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.

scholarly journals On the Existence and Uniqueness ofRv-Generalized Solution for Dirichlet Problem with Singularity on All Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
V. Rukavishnikov ◽  
E. Rukavishnikova
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Input Data ◽  
Generalized Solution ◽  
Two Dimensional ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation ◽  
Dimensional Domain

The existence and uniqueness of theRv-generalized solution for the first boundary value problem and a second order elliptic equation with coordinated and uncoordinated degeneracy of input data and with strong singularity solution on all boundary of a two-dimensional domain are established.

A nonlocal problem for a differential operator of even order with involution

2020 ◽  
Vol 26 (2) ◽  
pp. 297-307
Author(s):  
Petro I. Kalenyuk ◽  
Yaroslav O. Baranetskij ◽  
Lubov I. Kolyasa
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

ON THE FIRST BOUNDARY VALUE PROBLEM FOR THE CLASS OF ELLIPTIC SYSTEMS DEGENERATING AT AN INNER POINT

2001 ◽  
Vol 6 (1) ◽  
pp. 147-155 ◽  
Author(s):  
S. Rutkauskas
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Second Order ◽  
Type Problem ◽  
Dirichlet Type ◽  
Holder Class ◽  
Uniqueness Of The Solution ◽  
Dirichlet Type Problem

The Dirichlet type problem for the weakly related elliptic systems of the second order degenerating at an inner point is discussed. Existence and uniqueness of the solution in the Holder class of the vector‐functions is proved.

scholarly journals Nonlocal multipoint problem for a differential equation of $2n$-th order with operator coefficients

2021 ◽  
Vol 13 (2) ◽  
pp. 501-514
Author(s):  
Ya.O. Baranetskij ◽  
I.I. Demkiv ◽  
A.V. Solomko ◽  
O.M. Sus'
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Multipoint Problem ◽  
Root Functions ◽  
Multiple Eigenvalues ◽  
Finite Set ◽  
Uniqueness Of The Solution ◽  
Differential Operator Equation ◽  
Transmutation Operators

In the article, the spectral properties of a multipoint problem for a differential operator equation of order $2n$ are studied. The operator of the problem has an infinite number of multiple eigenvalues. Each multiple eigenvalue corresponds to a finite set of root functions. A commutative group of transmutation operators is constructed. Each element of the group corresponds to the isospectral perturbation of the problem operator with antiperiodic conditions. The conditions for the existence and uniqueness of the solution are established for the selected family of multipoint problems, and this solution is constructed too.

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 DIRCHLET PROBLEM FOR A NONLOCAL WAVE EQUATION WITH RIEMANN-LIOUVILLE DERIVATIVE

2019 ◽  
pp. 6-11
Author(s):  
О.Х. Масаева
Keyword(s):  
Wave Equation ◽  
Dirichlet Problem ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Liouville Derivative ◽  
Uniqueness Of The Solution

Доказано существование и единственность решения задачи Дирихле для уравнения второго порядка с дробной производной. Исследуемое уравнение переходит в волновое уравнение при целом значении порядка дробной производной The existence and uniqueness of the solution to Dirichlet problem for a secondorder equation with a fractional derivative is proved. The equation under study is a wave equation for a integer value of the order of the fractional derivative.

scholarly journals AN INVERSE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER ELLIPTIC EQUATION IN A RECTANGLE

2014 ◽  
Vol 19 (2) ◽  
pp. 241-256 ◽  
Author(s):  
Yashar T. Mehraliyev ◽  
Fatma Kanca
Keyword(s):  
Inverse Problem ◽  
Elliptic Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Inverse Boundary ◽  
Numerical Tests ◽  
Second Order Elliptic Equation

In this paper, the inverse problem of finding a coefficient in a second order elliptic equation is investigated. The conditions for the existence and uniqueness of the classical solution of the problem under consideration are established. Numerical tests using the finite-difference scheme combined with an iteration method is presented and the sensitivity of this scheme with respect to noisy overdetermination data is illustrated.

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