ON MULTIPOINT NONLOCAL BOUNDARY VALUE PROBLEMS FOR HYPERBOLIC DIFFERENTIAL AND DIFFERENCE EQUATIONS

Abstract

The nonlocal boundary value problem for differential equation \[\left\{ \begin{array}{l} \frac{d^{2}u(t)}{dt^{2}} + Au(t) = f(t) \quad \ (0 \leq t \leq 1), \\ u(0) = \sum\limits_{r=1}^{n} \alpha _{r} u(\lambda_{r}) + \varphi, u_{t}(0) = \sum\limits_{r=1}^{n} \beta _{r} u_{t}(\lambda_{r}) + \psi, \\ 0 \lt \lambda_{1} \leq \lambda_{2} \leq ... \leq \lambda_{n} \leq 1 \end{array} \right. \] 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 \[ \sum_{k=1}^{n} \left\vert \alpha_{k} + \beta_{k} \right\vert + \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert \sum_{\substack{m=1 \\ m \neq k}}^{n} \left\vert \beta_{m} \right\vert \lt |1 + \sum_{k=1}^{n} \alpha_{k} \beta_{k}| \] 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 \[ \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert + \sum_{k=1}^{n} \left\vert \beta_{k} \right\vert + \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert \sum_{k=1}^{n} \left\vert \beta_{k} \right\vert \lt 1 \] 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.