scholarly journals Stability of a Second Order of Accuracy Difference Scheme for Hyperbolic Equation in a Hilbert Space

2007 ◽  
Vol 2007 ◽  
pp. 1-25 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Mehmet Emir Koksal
Keyword(s):  
Hilbert Space ◽  
Difference Scheme ◽  
Hyperbolic Equation ◽  
Initial Value Problem ◽  
Second Order ◽  
Stability Estimates ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Positive Definite Operators ◽  
Accuracy Difference

The initial-value problem for hyperbolic equation d2u(t)/dt2+A(t)u(t)=f(t)(0≤t≤T), u(0)=ϕ,u′(0)=ψ in a Hilbert space H with the self-adjoint positive definite operators A(t) is considered. The second order of accuracy difference scheme for the approximately solving this initial-value problem is presented. The stability estimates for the solution of this difference scheme are established.

scholarly journals A second order of accuracy difference scheme for Schrödinger equations with an unknown parameter

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 981-993 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Mesut Urun
Keyword(s):  
Difference Scheme ◽  
Unknown Parameter ◽  
Second Order ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
Accuracy Difference ◽  
The Stability ◽  
Parameter Problem

In the present study, the second order of accuracy difference scheme for numerical solution of the boundary value problem for the differential equation with an unknown parameter p {idu(t)/dt + Au(t) + iu(t) = f (t) + p, 0 < t < T, u(0) = ? u(T) = ? in a Hilbert space H with self-adjoint positive definite operator A is presented. Theorem on the stability of this difference scheme is established. The stability estimates for the solution of difference schemes for two determination of an unknown parameter problem for Schr?dinger equations are given.

scholarly journals A note on the difference schemes for hyperbolic equations

2001 ◽  
Vol 6 (2) ◽  
pp. 63-70 ◽  
Author(s):  
A. Ashyralyev ◽  
P. E. Sobolevskii
Keyword(s):  
Hilbert Space ◽  
Initial Value Problem ◽  
Hyperbolic Equations ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Initial Value ◽  
Order Accuracy ◽  
Accuracy Difference ◽  
The Difference ◽  
The Stability

The initial value problem for hyperbolic equationsd 2u(t)/dt 2+A u(t)=f(t)(0≤t≤1),u(0)=φ,u′(0)=ψ, in a Hilbert spaceHis considered. The first and second order accuracy difference schemes generated by the integer power ofAapproximately solving this initial value problem are presented. The stability estimates for the solution of these difference schemes are obtained.

Numerical solution of a source identification problem: Almost coercivity

2019 ◽  
Vol 27 (4) ◽  
pp. 457-468 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Abdullah Said Erdogan ◽  
Ali Ugur Sazaklioglu
Keyword(s):  
Numerical Solution ◽  
Source Identification ◽  
Identification Problem ◽  
Second Order ◽  
Numerical Results ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Accuracy Difference ◽  
Coercive Stability

Abstract The present paper is devoted to the investigation of a source identification problem that describes the flow in capillaries in the case when an unknown pressure acts on the system. First and second order of accuracy difference schemes are presented for the numerical solution of this problem. Almost coercive stability estimates for these difference schemes are established. Additionally, some numerical results are provided by testing the proposed methods on an example.

scholarly journals On fourth order accuracy stable difference scheme for a multi-point overdetermined elliptic problem

2021 ◽  
Vol 102 (2) ◽  
pp. 45-53
Author(s):  
C. Ashyralyyev ◽  
◽  
G. Akyuz ◽  
◽  
Keyword(s):  
Approximate Solution ◽  
Difference Scheme ◽  
Fourth Order ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Uniqueness Of The Solution ◽  
Accuracy Difference ◽  
The Difference ◽  
Overdetermined Boundary Value Problem ◽  
Coercive Stability

In this paper fourth order of accuracy difference scheme for approximate solution of a multi-point elliptic overdetermined problem in a Hilbert space is proposed. The existence and uniqueness of the solution of the difference scheme are obtained by using the functional operator approach. Stability, almost coercive stability, and coercive stability estimates for the solution of difference scheme are established. These theoretical results can be applied to construct a stable highly accurate difference scheme for approximate solution of multi-point overdetermined boundary value problem for multidimensional elliptic partial differential equations.

scholarly journals On the stability of the Schrödinger equation with time delay

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 759-766 ◽  
Author(s):  
Deniz Agirseven
Keyword(s):  
Hilbert Space ◽  
Time Delay ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Stability Estimates ◽  
Initial Value ◽  
Dinger Equation ◽  
The Stability

In the present paper, the initial value problem for the Schr?dinger equation with time delay in a Hilbert space is investigated. Theorems on stability estimates for the solution of the problem are established. The applications of theorems for three types of Schr?dinger problems are provided.

scholarly journals A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ozgur Yildirim
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Hyperbolic Problem ◽  
Definite Operator ◽  
The Stability

The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert spaceHwith the self-adjoint positive definite operatorA. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.

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