scholarly journals Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ali Sirma
Keyword(s):  
Schrödinger Equation ◽  
Boundary Value Problem ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Nonlocal Boundary Value Problem ◽  
The Stability ◽  
Crank Nicolson

The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy -modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.

scholarly journals Well-Posedness of the Boundary Value Problem for Parabolic Equations in Difference Analogues of Spaces of Smooth Functions

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
A. Ashyralyev
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem ◽  
Smooth Functions ◽  
Accuracy Difference

The first and second orders of accuracy difference schemes for the approximate solutions of the nonlocal boundary value problemv′(t)+Av(t)=f(t)(0≤t≤1),v(0)=v(λ)+μ,0<λ≤1, for differential equation in an arbitrary Banach spaceEwith the strongly positive operatorAare considered. The well-posedness of these difference schemes in difference analogues of spaces of smooth functions is established. In applications, the coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary value problem for parabolic equation 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 difference schemes for Bitsadze-Samarskii type nonlocal boundary value problem involving integral condition

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 1027-1047 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Elif Ozturk
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Condition ◽  
Difference Schemes ◽  
Nonlocal Boundary Value Problem ◽  
Stability Of Difference Schemes ◽  
Gauss Elimination ◽  
Accuracy Difference ◽  
Gauss Elimination Method

In this study, the stable difference schemes for the numerical solution of Bitsadze-Samarskii type nonlocal boundary-value problem involving integral condition for the elliptic equations are studied. The second and fourth orders of the accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples.

scholarly journals On well-posedness of the nonlocal boundary value problem for parabolic difference equations

2004 ◽  
Vol 2004 (2) ◽  
pp. 273-286 ◽  
Author(s):  
A. Ashyralyev ◽  
I. Karatay ◽  
P. E. Sobolevskii
Keyword(s):  
Boundary Value Problem ◽  
Difference Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem ◽  
The Difference ◽  
Type Boundary ◽  
The Stability

We consider the nonlocal boundary value problem for difference equations(uk−uk−1)/τ+Auk=φk,1≤k≤N,Nτ=1, andu0=u[λ/τ]+φ,0<λ≤1, in an arbitrary Banach spaceEwith the strongly positive operatorA. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given.

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