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Zbl 1168.35397
Ashyralyev, Allaberen; Gercek, Okan
Nonlocal boundary value problems for elliptic-parabolic differential and difference equations.
(English)
[J] Discrete Dyn. Nat. Soc. 2008, Article ID 904824, 16 p. (2008). ISSN 1026-0226; ISSN 1607-887X/e

Summary: The abstract nonlocal boundary value problem $-d^{2}u(t)/dt^{2}+Au(t)=g(t)$, $0<t<1$, $du(t)/dt-Au(t)=f(t)$, $-1<t<0$, $u(1)=u(-1)+\mu$ for differential equations in a Hilbert space $H$ with the self-adjoint positive definite operator $A$ is considered. The well-posedness of this problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of boundary value problems for elliptic-parabolic equations are obtained. The first order of accuracy difference scheme for the approximate solution of this nonlocal boundary value problem is presented. The well-posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained.
MSC 2000:
*35M10 PDE of mixed type
34G10 Linear ODE in abstract spaces
35A05 General existence and uniqueness theorems (PDE)

Keywords: weighted Hölder spaces; Hilbert space; self-adjoint positive definite operator; difference scheme

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