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Author: Louis B. Rall Publisher: ISBN: Category : Hilbert space Languages : en Pages : 20
Book Description
Variational problems which are formulated in terms of finding maxima or minima of functionals usually lead to only one-sided bounds for approximate solutions. It has been shown recently by B. Noble that a number of variational problems may be formulated in terms of complementary variational problems, the approximate solutions of which give both upper and lower bounds for the desired stationary value. In this paper, complementary variational principles are formulated abstractly in Hilbert space. Necessary and sufficient conditions are given for a system of two equations to represent a variational principle, and sufficient conditions are given for the variational principle to be complementary. (Author).
Author: B. Noble Publisher: ISBN: Category : Languages : en Pages : 33
Book Description
In this series of papers a general variational principle and two complementary variational principles are developed for the simultaneous equations: Tp = partial derivative of W(x, p, u) with respect to u, and Su = partial derivative of W(x, p, u) with respect to p, where T, S are linear operators and S is the adjoint of T. The resulting formulae are of wide applicability and include most of the complementary variational principles which occur in applied mathematics. In this paper the implications of the variational principles are worked out in some detail when T = d/ dx, S = -d/ dx, including a discussion of extra terms which are required to deal with general boundary conditions.