scholarly journals A high order of accuracy of difference schemes for the nonlocal boundary value Schrödinger problem

2021 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ali Sirma
Keyword(s):  
Boundary Value ◽  
Nonlocal Boundary ◽  
High Order ◽  
Difference Schemes ◽  
Order Of Accuracy ◽  
High Order Of Accuracy

scholarly journals Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskiĭ
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Continuous Functions ◽  
High Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Smooth Functions ◽  
Order Of Accuracy ◽  
High Order Of Accuracy ◽  
The Difference

It is well known the differential equation−u″(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the positive operatorAis ill-posed in the Banach spaceC(E)=C((−∞,∞),E)of the bounded continuous functionsϕ(t)defined on the whole real line with norm‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions inC(τ,E)of these difference schemes is established.

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 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.

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