Ozgur Yildirim

Abstract: The nonlocal boundary value problem for differential equation\\ \vspace{-0.2cm} $\left\{ \begin{array}{l} {\frac{d^{2}u(t)}{dt^{2}}}+Au(t)=f(t)\quad \ (0\leq t\leq 1),\ \ \\ u(0)=\sum\limits_{r=1}^{n}\alpha _{r}u(\lambda _{r})+\varphi ,u_{t}(0)=\sum\limits_{r=1}^{n}\beta _{r}u_{t}(\lambda _{r})+\psi ,\ \\ 0<\lambda _{1}\leq \lambda _{2}\leq ...\leq \lambda _{n}\leq 1% \end{array}% \right. $ \vspace{0.2cm} oindent in a Hilbert space $H$ with the self-adjoint positive definite operator $A$ is considered. The stability estimates for the solution of the problem under the assumption $\dsum\limits_{k=1}^{n}\left\vert \alpha _{k}+\beta _{k}\right\vert +\dsum\limits_{k=1}^{n}\left\vert \alpha _{k}\right\vert \dsum\limits _{\substack{ m=1 \\ m eq k}}^{n}\left\vert \beta _{m}\right\vert <|1+\ \dsum\limits_{k=1}^{n}\alpha _{k}\beta _{k}|$ oindent are established. The first order of accuracy difference schemes for the approximate solutions of the problem are presented. The stability estimates for the solution of these difference schemes under the assumption $\dsum\limits_{k=1}^{n}\left\vert \alpha _{k}\right\vert +\dsum\limits_{k=1}^{n}\left\vert \beta _{k}\right\vert +\dsum\limits_{k=1}^{n}\left\vert \alpha _{k}\right\vert \dsum\limits_{k=1}^{n}\left\vert \beta _{k}\right\vert <1$ \vspace{0.05cm} oindent are established. In practice, the nonlocal boundary value problems for one dimensional hyperbolic equation with nonlocal boundary conditions in space variable\ and multidimensional hyperbolic equation with Dirichlet condition in space variables are considered. The stability estimates for the solutions of difference schemes for the nonlocal boundary value hyperbolic problems are obtained.

TL;DR: The second order of accuracy absolutely stable difference schemes for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert space H with the self-adjoint positive definite operator A are presented in this article.

TL;DR: In this paper, the abstract Cauchy problem for the hyperbolic equation in a Hilbert space H with self-adjoint positive definite operator A is considered and the third and fourth orders of accuracy difference schemes for the approximate solution of this problem are presented.

TL;DR: Third and fourth order of accuracy stable difference schemes for the approximate solutions of hyperbolic multipoint nonlocal boundary value problem in a Hilbert space H with self-adjoint positive definite operator A are considered.