Ashyralyyev, C.; Akyuz, G.

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.

Ashyralyyev, C.; Akyuz, G.

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 multipoint overdetermined boundary value problem for multidimensional elliptic partial differential equations.

Ashyralyev, A.; Ashyralyyev, C.; Zvyagin, V.G.

We study the source identification problem for an elliptic differential equation in a Banach space. The exact estimates for the solution of source identification problem in Hölder norms are obtained. In applications, four elliptic source identification problems are investigated. Stability and coercive stability estimates for solution of source identification problems for elliptic equations are obtained.

Ashyralyyev, C.; Cay, A.

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.

Ashyralyyev, Ch; Ozturk, B.

In this paper, we generalize the well known finite difference method to compute derivatives of real valued function to approximate of second order complex derivatives w(zz) and w((zz) over bar) for complex-valued function w. Exploring different combinations of terms, we derive several approximations to compute the second order derivatives of complex-valued function. Several second order of accuracy finite differences to calculate derivatives are proposed. Error analysis in test examples is carried out by using Matlab program.