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

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.

Download Full-text

Numerical solution of a source identification problem: Almost coercivity

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2017-0072 ◽

2019 ◽

Vol 27 (4) ◽

pp. 457-468 ◽

Cited By ~ 1

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.

Download Full-text

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

Discrete Dynamics in Nature and Society ◽

10.1155/ddns.2005.183 ◽

2005 ◽

Vol 2005 (2) ◽

pp. 183-213 ◽

Cited By ~ 32

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.

Download Full-text

High order of accuracy difference schemes for the inverse elliptic problem with Dirichlet condition

Boundary Value Problems ◽

10.1186/1687-2770-2014-5 ◽

2014 ◽

Vol 2014 (1) ◽

Cited By ~ 7

Author(s):

Charyyar Ashyralyyev

Keyword(s):

Elliptic Problem ◽

Dirichlet Condition ◽

High Order ◽

Difference Schemes ◽

Order Of Accuracy ◽

Accuracy Difference ◽

High Order Of Accuracy ◽

Inverse Elliptic Problem

Download Full-text

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

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS ◽

10.31489/2021m2/45-53 ◽

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.

Download Full-text

Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

Abstract and Applied Analysis ◽

10.1155/2013/548017 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 5

Author(s):

Charyyar Ashyralyyev ◽

Mutlu Dedeturk

Keyword(s):

Inverse Problem ◽

Approximate Solution ◽

Elliptic Equation ◽

Finite Difference ◽

Finite Difference Method ◽

Difference Method ◽

Stability Estimates ◽

Two Dimensional ◽

Accuracy Difference ◽

Coercive Stability

Afinite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the first and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem.

Download Full-text

On the Solution of NBVP for Multidimensional Hyperbolic Equations

The Scientific World JOURNAL ◽

10.1155/2014/841602 ◽

2014 ◽

Vol 2014 ◽

pp. 1-22 ◽

Cited By ~ 3

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.

Download Full-text

A note on the difference schemes for hyperbolic-elliptic equations

Abstract and Applied Analysis ◽

10.1155/aaa/2006/14816 ◽

2006 ◽

Vol 2006 ◽

pp. 1-13 ◽

Cited By ~ 10

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.

Download Full-text

A fourth order approximation of the Neumann type overdetermined elliptic problem

Filomat ◽

10.2298/fil1704967a ◽

2017 ◽

Vol 31 (4) ◽

pp. 967-980 ◽

Cited By ~ 2

Author(s):

Charyyar Ashyralyyev

Keyword(s):

Difference Scheme ◽

Fourth Order ◽

Elliptic Problem ◽

Order Approximation ◽

Solution Stability ◽

Stability Estimates ◽

Difference Problem ◽

Neumann Type ◽

Accuracy Difference ◽

Coercive Stability

In this paper, we consider an inverse elliptic problem with Neumann type overdetermination and construct a fourth order of accuracy difference scheme for its solution. Stability, almost coercive stability and coercive stability estimates for the solution of difference problem are proved. Later, we construct a fourth order difference scheme for an inverse problem for multidimensional elliptic equation with Neumann type overdetermination and Dirichlet boundary condition. Finally, we illustrate numerical example with descriptions of numeric realization in a two-dimensional case.

Download Full-text