Abstract
older 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 H¨ older spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained. Copyright q 2008 A. Ashyralyev and O. Gercek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Highlights
It is known that various problems in fluid mechanics and other areas of engineering, physics, and biological systems lead to partial differential equations of variable types
The proof of Theorem 1.4 is based on the abstract Theorem 1.1 and the symmetry properties of the space operator generated by problem 1.14 and the following theorem on the coercivity inequality for the solution of the elliptic differential problem in L2 Ω
Since the nonlocal boundary value problem 1.1 in the space C 0, 1, H of continuous functions defined on −1, 1 and with values in H is not well-posed for the general positive unbounded operator A and space H, the well-posedness of the difference nonlocal boundary value problem 2.1 in C −1, 1 τ, H norm does not take place uniformly with respect to τ > 0
Summary
The abstract nonlocal boundary value problem −d2u t /dt[2] Au t g t , 0 < t < 1, du t /dt − Au t f t , 1 < t < 0, u 1 u −1 μ 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 Holder 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 Holder spaces is established. Coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained.
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