On the solvability of a nonlocal boundary problem for the Helmholtz operator in a half-disk

2012 ◽  
Vol 2 (2) ◽  
Author(s):  
Makhmud Sadybekov ◽  
Gulnar Dildabek ◽  
Aizhan Tengayeva
Keyword(s):  
Boundary Problem ◽  
Nonlocal Boundary ◽  
Helmholtz Operator ◽  
Half Disk ◽  
Nonlocal Boundary Problem

scholarly journals ONE DIMENSIONAL BROWNIAN MOTION WITH HOLDING AND JUMPING BOUNDARY

2022 ◽  
Author(s):  
Yuk Leung
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Probability Density ◽  
Boundary Problem ◽  
Exponential Distribution ◽  
Boundary Point ◽  
Nonlocal Boundary ◽  
Unit Interval ◽  
One Dimensional ◽  
Nonlocal Boundary Problem

Let a particle start at some point in the unit interval I := [0, 1] and undergo Brownian motion in I until it hits one of the end points. At this instant the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I with a probability density μ0 or μ1 parametrized by the boundary point it was from. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper [10]. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process.

ON POSITIVE EIGENFUNCTIONS OF STURM-LIOUVILLE PROBLEM WITH NONLOCAL TWO‐POINT BOUNDARY CONDITION

2007 ◽  
Vol 12 (2) ◽  
pp. 215-226 ◽  
Author(s):  
Sigita Pečiulytė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Liouville Problem ◽  
Point Boundary ◽  
Existence Domain ◽  
Classical Boundary ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Nonlocal Boundary Problem

Positive eigenvalues and corresponding eigenfunctions of the linear Sturm‐Liouville problem with one classical boundary condition and another nonlocal two‐point boundary condition are considered in this paper. Four cases of nonlocal two‐point boundary conditions are analysed. We get positive eigenfunctions existence domain for each case of these problems. This domain depends on the parameters of the nonlocal boundary problem and it gives necessary and sufficient conditions for existing positive eigenvalues with positive eigenfunctions.

scholarly journals INVERSE PROBLEM WITH INTEGRAL IN TIME OVERDETERMINATION AND NONLOCAL BOUNDARY CONDITIONS FOR HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (1-2) ◽  
pp. 27-32
Author(s):  
A. V. Duzheva
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Boundary Problem ◽  
Existence Result ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Nonlocal Boundary Conditions ◽  
Time Variable ◽  
Initial Boundary Problem

In this article, we consider a question of sovability of an inverse problem for a linear hyprbolic equation. Properties of the solution of an associated nonlocal initial-boundary problem with displacement in boundary conditions are used to develop an existence result for the identification of the unknown source. Overdetermination is represented as integral with respect to time-variable.

scholarly journals Boundary problem with integral displacement for the model equation of elliptic type

2020 ◽  
pp. 65-74
Author(s):  
З.А. Нахушева
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problems ◽  
Boundary Problem ◽  
Model Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Elliptic Type ◽  
Order Elliptic Equation ◽  
Local Boundary ◽  
Second Order Elliptic Equation

Для модельного эллиптического уравнения второго порядка рассматривается метод редукции нелокальных краевых задач с интегральным смещением к локальным краевым задачам для уравнения более высокого порядка составного типа. Исследуется разрешимость поставленных задач. For a model second order elliptic equation is considered the method of reduction of nonlocal boundary value problems with integral offset to the local boundary value problems for equations of higher order composite type. The solvability of tasks is investigated.

scholarly journals The nonlocal boundary problem with perturbations of antiperiodicity conditions for the eliptic equation with constant coefficients

2018 ◽  
Vol 10 (2) ◽  
pp. 215-234
Author(s):  
Ya.O. Baranetskij ◽  
I.Ya. Ivasiuk ◽  
P.I. Kalenyuk ◽  
A.V. Solomko
Keyword(s):  
Riesz Basis ◽  
Nonlocal Boundary ◽  
Nonlocal Problem ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Transformation Operator ◽  
Constant Coefficients ◽  
Adjoint Functions ◽  
Nonlocal Boundary Problem

In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$ Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. Also we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G)$. In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$

scholarly journals Optimization Related to Some Nonlocal Problems of Kirchhoff Type

2016 ◽  
Vol 68 (3) ◽  
pp. 521-540 ◽  
Author(s):  
Behrouz Emamizadeh ◽  
Amin Farjudian ◽  
Mohsen Zivari-Rezapour
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Minimization Problem ◽  
Optimization Problems ◽  
Nonlocal Boundary ◽  
Energy Functional ◽  
The Other ◽  
Maximization Problem ◽  
Kirchhoff Type

AbstractIn this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton, we are able to show that both problems are solvable and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions.The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type that is stable. Some numerical results are included to conûrm the theory.

Solvability of some nonlocal boundary value problems for the Helmholtz equation in a half-disk

2010 ◽  
Vol 82 (1) ◽  
pp. 621-624 ◽  
Author(s):  
E. I. Moiseev ◽  
V. E. Ambartsumyan
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problems ◽  
Half Disk
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