On a criterion for the solvability of one ill-posed problem for the biharmonic equation

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Tynysbek S. Kal’menov ◽  
Makhmud A. Sadybekov ◽  
Ulzada A. Iskakova
Keyword(s):  
Biharmonic Equation ◽  
Sufficient Conditions ◽  
Adjoint Problem ◽  
Rectangular Domain ◽  
Solution Stability ◽  
Well Posedness ◽  
Local Boundary ◽  
Ill Posed ◽  
Necessary And Sufficient ◽  
Isolated Point

AbstractA local boundary value problem for an inhomogeneous biharmonic equation in a rectangular domain is considered. Boundary conditions are given on the whole boundary of the domain. It is shown that the problem turns out to be self-adjoint. And herewith the problem is ill-posed. An example is constructed demonstrating that the solution stability to the problem is violated. Necessary and sufficient conditions of the existence of a strong solution to the investigated problem are found. The idea of the method is that the solution to the problem is constructed in the form of an expansion on eigenfunctions of this self-adjoint problem. This problem has an isolated point of continuous spectrum in zero. It is shown that there exists a series (a sequence) of eigenvalues converging to zero. Asymptotics of these eigenvalues is found. Namely this asymptotics defines a reason for the ill-posedness of the investigated problem. A space of well-posedness for the investigated problem is constructed.

scholarly journals On an ill-posed problem for a biharmonic equation

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1051-1056 ◽  
Author(s):  
Tynysbek Kal’menov ◽  
Ulzada Iskakova
Keyword(s):  
Boundary Value Problem ◽  
Biharmonic Equation ◽  
Sufficient Conditions ◽  
Rectangular Domain ◽  
Problem Solution ◽  
Local Boundary ◽  
Stability Of Solution ◽  
Ill Posed ◽  
The Stability ◽  
Necessary And Sufficient

A local boundary value problem for the biharmonic equation in a rectangular domain is considered. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found.

scholarly journals Linear Hyperbolic Systems on Networks: well-posedness and qualitative properties

2020 ◽  
Author(s):  
Marjeta Kramar ◽  
Delio Mugnolo ◽  
Serge Nicaise
Keyword(s):  
Boundary Conditions ◽  
Evolution Equations ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Order Reduction ◽  
Well Posedness ◽  
Qualitative Properties ◽  
Local Boundary ◽  
First Order ◽  
Necessary And Sufficient

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first - order reduction . We study forward and backward well - posedness ; furthermore , we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first - order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied . p, li { white-space: pre-wrap; }

IDENTIFICATIONS OF HASLER'S CLASSES OF LINEAR RESISTIVE CIRCUIT STRUCTURES

2004 ◽  
Vol 13 (05) ◽  
pp. 957-980
Author(s):  
J. CEL
Keyword(s):  
Sufficient Conditions ◽  
Input Power ◽  
Maximal Entropy ◽  
Cactus Graph ◽  
Resistance Function ◽  
Total Input ◽  
Feedback Structure ◽  
Ill Posed ◽  
System Of Parameters ◽  
Necessary And Sufficient

Formulae on first and second derivatives of various functions associated with a linear nullator–norator–resistance network such as total input power, driving-point and transfer resistances with respect to parameters are established. As a consequence, the concavity of the driving-point resistance with respect to the system of parameters is obtained which generalizes a scalar result of Schneider. An example is given showing that the driving-point resistance R of a nonreciprocal one-port is not monotone or convex or concave with respect to the system of resistances which shows that the Cohn–Vratsanos and the Shannon–Hagelbarger theorems which characterize R of reciprocal one-port cannot be extended in this way. Next, a simplified variant of the Shannon–Hagelbarger theorem is used to derive separate necessary and sufficient conditions characterizing always well-posed, sometimes ill-posed and always ill-posed classes of linear resistive circuit structures introduced and characterized by Hasler, both new in formulation and proof. This reveals that the form of the second partial derivative of the resistance function is responsible for various kinds of the structural solvability of linear circuits. Alternative "if and only if" criteria for these classes are established. They involve replacements of reciprocal circuit elements by combinations of contractions and removals leading to pairs of complementary directed nullator and directed norator trees with appropriately defined signs, and resemble therefore earlier famous Willson–Nielsen feedback structure and Chua–Nishi cactus graph criteria for circuits containing traditional controlled sources. Finally, the qualitative parts of the Cohn–Vratsanos and the Shannon–Hagelbarger theorems are shown to be simple consequences of much more general principles governing all aspects of life, such as maximal entropy and energy conservation laws.

Levitin-Polyak well-posedness for parametric quasivariational inclusion and disclusion problems

2018 ◽  
Vol 34 (3) ◽  
pp. 295-303
Author(s):  
PANATDA BOONMAN ◽  
◽  
RABIAN WANGKEEREE ◽  
Keyword(s):  
Equilibrium Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Well Posedness ◽  
Quasiequilibrium Problems ◽  
Necessary And Sufficient

In this paper, we aim to suggest the new concept of Levitin-Polyak (for short, LP) well-posedness for the parametric quasivariational inclusion and disclusion problems (for short, (QVIP) (resp. (QVDP))). Necessary and sufficient conditions for LP well-posedness of these problems are proved. As applications, we obtained immediately some results of LP well-posedness for the quasiequilibrium problems and for a scalar equilibrium problem.

scholarly journals Boundary-value problems in a layer for evolutionary pseudo-differential equations with integral conditions

2018 ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Condition ◽  
Well Posedness ◽  
Integral Conditions ◽  
Pseudo Differential Equation ◽  
Necessary And Sufficient ◽  
Well Posed

Boundary-value problems for evolutionary pseudo-differential equations with an integral condition are studied. Necessary and sufficient conditions of well-posedness are obtained for these problems in the Schwartz spaces. Existence of a well-posed boundary-value problem is proved for each evolutionary pseudo-differential equation.

Well-posedness of fractional degenerate differential equations in Banach spaces

2019 ◽  
Vol 22 (2) ◽  
pp. 379-395
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equation ◽  
Fourier Multiplier ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Well Posedness ◽  
Multiplier Theorems ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Closed Linear Operators

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

scholarly journals Iterative Method to Solve a Data Completion Problem for Biharmonic Equation for Rectangular Domain

2017 ◽  
Vol 55 (1) ◽  
pp. 129-147
Author(s):  
Chakir Tajani ◽  
Houda Kajtih ◽  
Ali Daanoun
Keyword(s):  
Iterative Method ◽  
Biharmonic Equation ◽  
Rectangular Domain ◽  
Numerical Results ◽  
Cauchy Problems ◽  
Completion Problem ◽  
Data Completion ◽  
Ill Posed ◽  
Convergent Method ◽  
Direct Problems

AbstractIn this work, we are interested in a class of problems of great importance in many areas of industry and engineering. It is the invese problem for the biharmonic equation. It consists to complete the missing data on the inaccessible part from the measured data on the accessible part of the boundary. To solve this ill-posed problem, we opted for the alternative iterative method developed by Kozlov, Mazya and Fomin which is a convergent method for the elliptical Cauchy problems in general. The numerical implementation of the iterative algorithm is based on the application of the boundary element method (BEM) for a sequence of mixed well-posed direct problems. Numerical results are performed for a square domain showing the effectiveness of the algorithm by BEM to produce accurate and stable numerical results.

scholarly journals DIRICHLET PROBLEM FOR LOADED DEGENERATING EQUATION OF THE MIXED TYPE IN THE RECTANGULAR AREA

2017 ◽  
Vol 19 (6) ◽  
pp. 40-53
Author(s):  
E.P. Melisheva
Keyword(s):  
Dirichlet Problem ◽  
Boundary Problem ◽  
Mixed Type ◽  
Sufficient Conditions ◽  
Rectangular Domain ◽  
One Dimensional ◽  
Rectangular Area ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Uniqueness Of A Solution

In this work necessary and sufficient conditions for uniqueness of a solution to the first boundary problem for Lavrentiev-Bitsadze equation in rectangular domain are established. The solution to the problem is constructed as a sum of series with respect of eigenfunctions of a corresponding one-dimensional Stour-m-Liouviele problem. The stability is shown.

Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent

2019 ◽  
Vol 19 (2) ◽  
pp. 291-316
Author(s):  
Zongming Guo ◽  
Long Wei
Keyword(s):  
Asymptotic Behavior ◽  
Biharmonic Equation ◽  
Sufficient Conditions ◽  
Entire Solution ◽  
Radial Symmetry ◽  
Entire Solutions ◽  
Moving Plane ◽  
Exact Asymptotic Behavior ◽  
Necessary And Sufficient ◽  
Positive Entire Solution

AbstractNecessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation\Delta^{2}u=u^{p}\quad\text{in }\mathbb{R}^{N},\,N\geq 5,\,p>\frac{N+4}{N-4}to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at {\infty} and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution {u(x)=u(|x|)} for any given {u(0)>0}, and the asymptotic behavior of {u(|x|)} at {\infty} is also known. We will see that the behavior similar to that of a radial entire solution of above equation at {\infty}, in turn, determines the radial symmetry of a general positive entire solution {u(x)} of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at {\infty} is obtained.

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