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.

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 On the stability of the linear delay differential and difference equations

2001 ◽  
Vol 6 (5) ◽  
pp. 267-297 ◽  
Author(s):  
A. Ashyralyev ◽  
P. E. Sobolevskii
Keyword(s):  
Difference Equations ◽  
Initial Value Problem ◽  
Parabolic Type ◽  
Stability Estimates ◽  
Initial Value ◽  
Linear Delay ◽  
Accuracy Difference ◽  
The Stability ◽  
Delay Differential ◽  
Delay Partial Differential Equations

We consider the initial-value problem for linear delay partial differential equations of the parabolic type. We give a sufficient condition for the stability of the solution of this initial-value problem. We present the stability estimates for the solutions of the first and second order accuracy difference schemes for approximately solving this initial-value problem. We obtain the stability estimates in Hölder norms for the solutions of the initial-value problem of the delay differential and difference equations of the parabolic type.

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 Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations

2005 ◽  
Vol 2005 (2) ◽  
pp. 183-213 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskii
Keyword(s):  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
High Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
Accuracy Difference ◽  
High Order Of Accuracy ◽  
The Stability

We consider the abstract Cauchy problem for differential equation of the hyperbolic typev″(t)+Av(t)=f(t)(0≤t≤T),v(0)=v0,v′(0)=v′0in an arbitrary Hilbert spaceHwith the selfadjoint positive definite operatorA. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of this problem are presented. The stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.

scholarly journals A note on the difference schemes for hyperbolic-elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
A. Ashyralyev ◽  
G. Judakova ◽  
P. E. Sobolevskii
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Accuracy Difference ◽  
The Difference ◽  
The Stability

The nonlocal boundary value problem for hyperbolic-elliptic equationd2u(t)/dt2+Au(t)=f(t),(0≤t≤1),−d2u(t)/dt2+Au(t)=g(t),(−1≤t≤0),u(0)=ϕ,u(1)=u(−1)in a Hilbert spaceHis considered. The second order of accuracy difference schemes for approximate solutions of this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established.

scholarly journals Stability of Finite Difference Schemes on Irregular Meshes for Von Foerster-Type 1-D Equations

2007 ◽  
Vol 14 (4) ◽  
pp. 793-805
Author(s):  
Piotr Zwierkowski
Keyword(s):  
Finite Difference ◽  
Initial Value Problem ◽  
Difference Schemes ◽  
Spatial Variable ◽  
Finite Difference Schemes ◽  
Initial Value ◽  
One Dimensional ◽  
Irregular Meshes ◽  
The Stability

Abstract We consider a generalized von Foerster equation in one dimensional spatial variable and construct finite difference schemes for the initial value problem. The stability of finite difference schemes on irregular meshes generated by characteristics is studied.

scholarly journals Stability and Monotonicity of Difference Schemes for Nonlinear Scalar Conservation Laws and Multidimensional Quasi-linear Parabolic Equations

2009 ◽  
Vol 9 (3) ◽  
pp. 253-280 ◽  
Author(s):  
P. Matus ◽  
S. Lemeshevsky
Keyword(s):  
Conservation Laws ◽  
Parabolic Equations ◽  
Input Data ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Linear Parabolic Equations ◽  
The Difference ◽  
The Stability

Abstract We have proved the difference analogue of a Bihari-type inequality. Using this inequality, we study the stability in C and monotonicity of the difference schemes approximating initial-boundary value problems for nonlinear conservation laws and multi-dimensional parabolic equations. It has been shown that in the nonlinear case the stability and monotonicity are determined not only by the behavior of the approximate solution but also by its difference derivatives appearing in the nonlinear terms of the equation. The stability estimates are obtained without any assumptions about the properties of the solution and nonlinear coefficients of the differential problem. Here we use restrictions only on input data (initial and boundary conditions and the right-hand side). The sufficient conditions of the shock wave generation is formulated for input data. For the Riemann problem two exact and stable difference schemes are analyzed.

scholarly journals On the Solution of NBVP for Multidimensional Hyperbolic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Necmettin Aggez
Keyword(s):  
Approximate Solution ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Hyperbolic Problems ◽  
Order Of Accuracy ◽  
Numerical Examples ◽  
Accuracy Difference ◽  
Multidimensional Hyperbolic Equations

We are interested in studying multidimensional hyperbolic equations with nonlocal integral and Neumann or nonclassical conditions. For the approximate solution of this problem first and second order of accuracy difference schemes are presented. Stability estimates for the solution of these difference schemes are established. Some numerical examples illustrating applicability of these methods to hyperbolic problems are given.

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 The influence of circulation on the stability of vortices to mode-one disturbances

1995 ◽  
Vol 451 (1943) ◽  
pp. 747-755 ◽  
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Initial Value Problem ◽  
Large Time ◽  
Asymptotic Methods ◽  
Two Dimensional ◽  
Initial Value ◽  
The Stability ◽  
Ultimate Fate ◽  
Solution Behaviour

The initial value problem for the two-dimensional inviscid vorticity equation, linearized about an azimuthal basic velocity field with monotonic angular velocity, is solved exactly for mode-one disturbances. The solution behaviour is investigated for large time using asymptotic methods. The circulation of the basic state is found to govern the ultimate fate of the disturbance: for basic state vorticity distributions with non-zero circulation, the perturbation tends to the steady solution first mentioned in Michalke & Timme (1967), while for zero circulation, the perturbation grows without bound. The latter case has potentially important implications for the stability of isolated eddies in geophysics.

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