scholarly journals A fourth order approximation of the Neumann type overdetermined elliptic problem

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 967-980 ◽  
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.

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

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.

scholarly journals Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination

2020 ◽  
Vol 99 (3) ◽  
pp. 5-17
Author(s):  
C. Ashyralyyev ◽  
◽  
A Cay ◽  
◽  
Keyword(s):  
Inverse Problem ◽  
Difference Scheme ◽  
Nonlocal Boundary ◽  
Nonlocal Problem ◽  
Integral Condition ◽  
Identification Problem ◽  
Stability Estimates ◽  
Difference Problem ◽  
Integral Boundary ◽  
Neumann Type

In modeling various real processes, an important role is played by methods of solution source identification problem for partial differential equation. The current paper is devoted to approximate of elliptic over determined problem with integral condition for derivatives. In the beginning, inverse problem is reduced to some auxiliary nonlocal boundary value problem with integral boundary condition for derivatives. The parameter of equation is defined after solving that auxiliary nonlocal problem. The second order of accuracy difference scheme for approximately solving abstract elliptic overdetermined problem is proposed. By using operator approach existence of solution difference problem is proved. For solution of constructed difference scheme stability and coercive stability estimates are established. Later, obtained abstract results are applied to get stability estimates for solution Neumann-type overdetermined elliptic multidimensional difference problems with integral conditions. Finally, by using MATLAB program, we present numerical results for two dimensional and three dimensional test examples with short explanation on realization on computer.

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.

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 A Difference Scheme and Its Error Analysis for a Poisson Equation with Nonlocal Boundary Conditions

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Chunsheng Feng ◽  
Cunyun Nie ◽  
Haiyuan Yu ◽  
Liping Zhou ◽  
Karthikeyan Rajagopal
Keyword(s):  
Difference Scheme ◽  
Elliptic Problem ◽  
Nonlocal Condition ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Discrete Fourier Transformation ◽  
Optimal Error Estimate ◽  
Taylor Formula ◽  
Accuracy Difference

The elliptic problem with a nonlocal boundary condition is widely applied in the field of science and engineering, such as the chaotic system. Firstly, we construct one high-accuracy difference scheme for a kind of elliptic problem by tactfully introducing an equivalent relation for one nonlocal condition. Then, we obtain the local truncation error equation by the Taylor formula and, initially, prove that the new scheme can reach the asymptotic optimal error estimate O h 2 ln   h in the maximum norm through ingeniously transforming a two-dimensional problem to a one-dimensional one through bringing in the discrete Fourier transformation. Numerical experiments demonstrate the correctness of theoretical results.

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