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Nonlocal boundary value problems for elliptic-parabolic differential and difference equations. (English) Zbl 1168.35397
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:
35M10 PDEs of mixed type
34G10 Linear differential equations in abstract spaces
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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