scholarly journals An eigenvalue problem for the differential operator with an integral condition

2012 ◽  
Vol 53 ◽  
Author(s):  
Kristina Jakubėlienė
Keyword(s):  
Parabolic Equation ◽  
Differential Operator ◽  
Eigenvalue Problem ◽  
Integral Condition ◽  
Stability Conditions ◽  
Two Dimensional ◽  
One Dimensional ◽  
Nonlocal Integral Condition

We analyze solution of a two-dimensional parabolic equation with a nonlocal integral condition by a locally one-dimensional method. The main aim of the paper is to deduce stability conditions of a system of one-dimensional equations with one integral condition. To this end, we analyze the structure of the spectrum of the differential operator with an integral condition.

On the Eigenvalue Problem for One-Dimensional Differential Operator with Nonlocal Integral Condition

2004 ◽  
Vol 9 (2) ◽  
pp. 109-116 ◽  
Author(s):  
R. Čiupaila ◽  
Ž. Jesevičiūtė ◽  
M. Sapagovas
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Integral Condition ◽  
Positive Eigenvalue ◽  
One Dimensional ◽  
Multiple Eigenvalues ◽  
Nonlocal Integral Condition

The article investigates the eigenvalue problem for ordinary onedimensional differential operator with nonlocal integral condition. Such a problem is met in the literature quite rarely and is considerably less investigated. Also the conditions for existence of non-positive eigenvalue or multiple eigenvalues are obtained.

scholarly journals Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition

2012 ◽  
Vol 17 (1) ◽  
pp. 91-98 ◽  
Author(s):  
Mifodijus Sapagovas ◽  
Kristina Jakubėlienė
Keyword(s):  
Parabolic Equation ◽  
Nonlocal Condition ◽  
Integral Condition ◽  
Rectangular Domain ◽  
Alternating Direction Method ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Alternating Direction ◽  
Nonlocal Integral Condition ◽  
Dimensional Domain

Two-dimensional parabolic equation with nonlocal condition is solved by alternating direction method in the rectangular domain. Values of the solution on the boundary points are bind with the integral of the solution in whole two-dimensional domain. Because of this nonlocal condition, the classical alternating direction method is complemented by the solution of low dimension system of algebraic equations. The peculiarities of the method are considered.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

scholarly journals Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Necmettin Aggez
Keyword(s):  
Space Variable ◽  
Hyperbolic Equations ◽  
Nonlocal Boundary ◽  
Integral Condition ◽  
Variable Coefficients ◽  
Difference Schemes ◽  
Order Difference ◽  
One Dimensional ◽  
Nonlocal Integral Condition ◽  
Multidimensional Hyperbolic Equations

The stable difference schemes for the approximate solution of the nonlocal boundary value problem for multidimensional hyperbolic equations with dependent in space variable coefficients are presented. Stability of these difference schemes and of the first- and second-order difference derivatives is obtained. The theoretical statements for the solution of these difference schemes for one-dimensional hyperbolic equations are supported by numerical examples.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Penghe Ge ◽  
Hongjun Cao
Keyword(s):  
Neuron Model ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Fast Subsystem ◽  
One Dimensional ◽  
Sensitive Dependence ◽  
The Stability ◽  
Slow Subsystem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

On a Nonlocal Boundary-value Problem for Two-dimensional Elliptic Equation

2001 ◽  
Vol 3 (1) ◽  
pp. 62-71
Author(s):  
Givi Berikelashvili ◽  
Nikolai I. Ionkin ◽  
Valentina A. Morozova
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Weighted Sobolev Space ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Condition ◽  
Two Dimensional ◽  
Averaging Operators ◽  
Nonlocal Integral Condition ◽  
Constant Coefficients

AbstractA boundary-value problem with a nonlocal integral condition is considered for a two-dimensional elliptic equation with constant coefficients and a mixed derivative. The existence and uniqueness of a weak solution of this problem are proved in a weighted Sobolev space. A difference scheme is constructed using the Steklov averaging operators.

scholarly journals Numerical Analysis of the Eigenvalue Problem for One-Dimensional Differential Operator with Nonlocal Integral Conditions

2009 ◽  
Vol 14 (1) ◽  
pp. 115-122 ◽  
Author(s):  
S. Sajavičius ◽  
M. Sapagovas
Keyword(s):  
Numerical Analysis ◽  
Differential Operator ◽  
Eigenvalue Problem ◽  
General Problem ◽  
Nonlocal Conditions ◽  
Spectral Structure ◽  
Integral Conditions ◽  
One Dimensional ◽  
Special Cases ◽  
Nonlocal Integral Conditions

In this paper the eigenvalue problem for one-dimensional differential operator with nonlocal integral conditions is investigated numerically. The special cases of general problem are analyzed and hypothesis about the dependence of the spectral structure of that problem on the coefficient of differential operator and the parameters of nonlocal conditions are formulated.

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