Ashyralyyev, Charyyar

This paper is devoted to a Bitsadze-Samarskii type overdetermined multipoint nonlocal boundary value problem (NBVP). This inverse problem is reduced to an auxiliary multipoint NBVP. Stability estimates for the solution of the auxiliary NBVP are established. The finite difference method is applied to get the first and second order of accuracy of approximate solutions of the abstract overdetermined problem. Stability estimates for the solution of difference problems are proved. Then the established abstract results are applied to get stability estimates for the solution of the Bitsadze-Samarskii type overdetermined elliptic multidimensional differential and difference problems with multipoint nonlocal boundary conditions (NBVC). Finally, numerical results with explanation on the realization for two dimensional and three dimensional elliptic overdetermined multipoint NBVPs in test examples are presented.

Ashyralyyev, Charyyar

This paper addresses estimation of the solution of inverse problem for the elliptic type equation with second kind boundary conditions and Neumann-type overdetermination at the interior point. We prove various estimates for the solution of this identification problem. Then, the first order accuracy stable difference scheme is studied. The stability, almost coercive stability, and coercive stability inequalities for its solution are obtained. The theoretical results can be applied to investigate identification problems for multidimensional elliptic equation with different kind of boundary conditions.

Ashyralyyev, Charyyar; Dedeturk, Mutlu; Lukashov, A.; Ashyralyev, A.

Approximation of Dirichlet type overdetermined multidimensional elliptic problem with Dirichlet-Neumann boundary conditions are discussed. A third order of accuracy difference scheme for its approximate solution is proposed. The stability, almost coercive stability and coercive stability inequalities for the solution of constructed difference scheme are established. Test example for a two-dimensional elliptic problem is presented.

Ashyralyyev, Charyyar; Akyüz, Gulzipa; Lukashov, A.; Ashyralyev, A.

In this study, we discuss well-posedness of Bitsadze-Samarskii type inverse elliptic problem with Dirichlet conditions. We establish abstract results on stability and coercive stability estimates for the solution of this inverse problem. Then, the abstract results are applied to three overdetermined problems for the multi-dimensional elliptic equation with different boundary conditions. Stability inequalities for solutions of these applications are obtained.

Ashyralyyev, Charyyar; Akkan, Yasar

This article studies the numerical solution of inverse problems for the multidimensional elliptic equation with Dirichlet-Neumann boundary conditions and Neumann type overdetermination. We present first and second order accuracy difference schemes. The stability and almost coercive stability inequalities for the solution are obtained. Numerical examples with explanation on the implementation illustrate the theoretical results.

Ashyralyyev, Charyyar; Dedeturk, Mutlu

In the present study, the inverse problem for a multidimensional elliptic equation with mixed boundary conditions and overdetermination is considered. The first and second orders of accuracy in t and the second order of accuracy in space variables for the approximate solution of this inverse problem are constructed. Stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are established. For the two-dimensional inverse problems with mixed boundary value conditions, numerical results are presented in test examples.

Ashyralyyev, Charyyar; Lukashov, A.; Ashyralyev, A.; Malkowsky, E.; Basar, F.

In the present paper, inverse elliptic problem with Neumann type overdetermination is discussed. A fourth order of accuracy difference scheme for the solution of this identification problem is presented. Stability, almost coercive stability and coercive inequalities for the solution of difference problem are obtained. In application, a fourth order approximation of the inverse problem for multidimensional elliptic equation with Neumann type overdetermination and Dirichlet boundary condition is studied.

Ashyralyyev, Charyyar

The overdetermination problem for elliptic differential equation with Dirichlet boundary condition is considered. The third and fourth orders of accuracy stable difference schemes for the solution of this inverse problem are presented. Stability, almost coercive stability, and coercive inequalities for the solutions of difference problems are established. As a result of the application of established abstract theorems, we get well-posedness of high order difference schemes of the inverse problem for a multidimensional elliptic equation. The theoretical statements are supported by a numerical example.

Ashyralyev, Allaberen; Ashyralyyev, Charyyar

The boundary value problem of determining the parameter of an elliptic equation -u(t) + Au(t) = f(t) + p (0 ≤ t ≤ T), u(0) = φ, u(T) = ψ u(λ) = ξ 0 < λ < T, with a positive operator A in an arbitrary Banach space E is studied. The exact estimates are obtained for the solution of this problem in Hölder norms. Coercive stability estimates for the solution of boundary value problems for multi-dimensional elliptic equations are established.

Ashyralyyev, Charyyar

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.

Ashyralyyev, Charyyar; Cakir, Zafer; Ashyralyev, A; Malkowsky, E

In this paper, some sequences of matrices for matrix pencils are considered. Properties of small singular values of these matrices are investigated. Estimates for singular values are obtained. The result of computer calculations is given.

Ashyralyyev, Charyyar; Cakir, Zafer; Ashyralyev, A; Malkowsky, E

Approximate quadrature formulas for the numerical calculation of the two-dimensional Vekua potential and singular integrals are obtained. The mechanical quadrature method for the two-dimensional quasilinear singular integral equation with Vekua operators is described. The numerical results are compared with the exact solution of the integral equation.

Ashyralyyev, Charyyar; Dedeturk, Mutlu; Ashyralyev, A; Malkowsky, E

The inverse problem for the multidimensional elliptic equation with Neumann-Dirichlet conditions are presented. For the approximate solution of this inverse problem the first and second order of accuracy in t and in space variables difference schemes are constructed. The stability, almost coercive stability and coercive stability estimates for the solution of these difference schemes are obtained. The algorithm for approximate solution is tested in a two-dimensional inverse problem.

Ashyralyyev, Charyyar; Ashyralyev, A; Malkowsky, E

Inverse problem for an elliptic differential equation with Neumann conditions is considered. Stability and coercive stability estimates for the solution of inverse problem with the overdetermination are obtained. The first and second order of accuracy difference schemes are presented. Stability and coercive stability inequalities for these difference schemes are given. In application, inverse problem for the multidimensional elliptic equation is studied. The first and second order of accuracy difference schemes for the multidimensional inverse problem are presented. Well-posedness of both difference problems are established. The results are supported by a numerical example for the two-dimensional elliptic equation.

Ashyralyyev, Charyyar; Dedeturk, Mutlu

Dirichlet şartıyla verilen eliptik diferansiyel denklem için ters problemin iyi tanımlığı incelendi. Ters problemin yaklaşık çözümü için bir sonlu farklar yöntemi uygulandı. Birinci ve ikinci derecede kesinlikte olan fark şemalarının çözümü için kararlılık ve koersiv kararlılık kestirimleri elde edildi. Uygulamada çok boyutlu eliptik denklem için ters problem çalışıldı. Teorik ifadeler iki boyutlu eliptik denklem ile verilen bir sayısal örnekle desteklendi.

Ashyralyyev, Charyyar; Dedeturk, Mutlu

A finite 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.

Ashyralyyev, Charyyar; Demirdag, Oznur

The boundary value problem of determining the parameter p of a parabolic equation ′(t)+A(t)=f(t)+p(0≤t≤1),(0)=,(1)= in an arbitrary Banach space E with the strongly positive operator A is considered. The first order of accuracy stable difference scheme for the approximate solution of this problem is investigated. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained.

Ashyralyyev, Charyyar; Dural, Ayfer; Sozen, Yasar; Ashyralyev, A; Lukashov, A

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Neumann condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example.

Ashyralyyev, Charyyar; Dural, Ayfer; Sozen, Yasar

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example.