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 On stable high order difference schemes for hyperbolic problems with the Neumann boundary conditions

2019 ◽  
Vol 9 (1) ◽  
pp. 60-72 ◽  
Author(s):  
Ozgur Yildirim
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Nonlocal Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Order Difference

In this paper, third and fourth order of accuracy stable difference schemes for approximately solving multipoint nonlocal boundary value problems for hyperbolic equations with the Neumann boundary conditions are considered. Stability estimates for the solutions of these difference schemes are presented. Finite difference method is used to obtain numerical solutions. Numerical results of errors and CPU times are presented and are analyzed.

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.

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 High order approximation of the inverse elliptic problem with Dirichlet-Neumann conditions

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 947-962 ◽  
Author(s):  
Charyyar Ashyralyyev
Keyword(s):  
Inverse Problem ◽  
Order Approximation ◽  
High Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
High Order Approximation ◽  
Order Of Accuracy ◽  
Accuracy Difference ◽  
Inverse Elliptic Problem ◽  
Coercive Stability

Inverse problem for the multidimensional elliptic equation with Dirichlet-Neumann conditions is considered. High order of accuracy difference schemes for the solution of inverse problem are presented. Stability, almost coercive stability and coercive stability estimates of the third and fourth orders of accuracy difference schemes for this problem are obtained. Numerical results in a two dimensional case are given.

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.

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 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.

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