The hyperbolic-elliptic equation with the nonlocal condition

Abstract

The nonlocal boundary value problem for a hyperbolic–elliptic equation in a Hilbert space is considered. The stability estimate for the solution of the given problem is obtained. The first and second orders of difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.

Stability of the space identification problem for the elliptic‐telegraph differential equation

“…Similarly, by (14) and the triangle inequality and estimates (3), (4), (5), and (8), we obtain (see Ashyralyev and Sobolevskii 15 )…”

“…there are unique solutions of the initial value problem (19) and boundary value problem (20) and formulas (13) and (14) hold. From nonlocal boundary condition v 0 − v −1 = − , it follows (15).…”

“…and the formula (17) holds. Therefore, for the formal solution of the problem (18) we have the formulas (13), (14), (15), (16) and (17). Formula for p follows from (11) and condition u(1) = .…”

“…Second, we obtain the estimate ‖Av(t)‖ H , for −1 ≤ t ≤ 1. Using formulas (13) and (14) and integrating by parts, we can get formulas…”

“…The nonlocal boundary value problems for hyperbolic-elliptic equation in a Hilbert space were studied; theorems on stability of this problem and the first-and the second-order of accuracy difference schemes for approximate solutions of this problem were proved. 14 Direk and Ashyraliyev 20 studied the initial-value problem for the integral-differential equation of the hyperbolic type in a Hilbert space. Theorems of the uniqueness of solvability of this problem were proved.…”