Complementary variational principles and the Bubnov-Galerkin method

Milan Journal of Mathematics - An account is given of the simple correspondence that exists between the classical variational method for solving linear integral equations, and the Bubnov-Galerkin...     

Complementary variational principles for poiseuille flow of an Oldroyd fluid

The complementary variational principles are given for the poiseuille flow of an Oldroyd fluid by taking the pressure gradient to be exponentially increasing with time and the bounds on the flux are obtained.     

A note on variational principles for complex equations

This paper presents variational principles for equations Lψ = f(ψ), where f is a complex function of the complex vector ψ and L is a linear complex operator.     

On variational principles for linear initial value problems

Euler-Lagrange and Euler-Hamilton variational principles are presented for a class of linear initial value problems.     

Complementary variational principles for radiation heat transfer in an absorbing medium

The radiation heat flux in a plane slab of an absorbing medium is evaluated by determining upper and lower bounds via complementary variational principles. The results obtained are in excellent agreement with those corresponding to other more cumbersome solution procedures (for small values of the spacing parameterd).

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Riassunto Si valuta il trasporto di energia raggiante in una lastra piana di un mezzo assorbente, determinandone limiti superiori ed inferiori, mediante principi variazionali complementari. I risultati ottenuti sono in eccellente accordo con quelli ottenuti mediante tecniche risolutive più complesse (per piccoli valori del parametrod).

     

On the existence of variational principles for differential-difference evolution equations

The problem of existence of variational principles for wide classes of generally nonlinear differential-difference equations with nonpotential operators is investigated.     

A variational principle for the nonstationary linear navier-stokes equations

A time-nonlocal boundary value problem for the linear evolution Navier-Stokes equations is considered, with time-averaged data being prescribed instead of initial ones. The solving of the problem is reduced to the minimizing of a quadratic functional.     

Variational principles for a class of coupled linear equations

This paper presents variational and extremum principles for pairs of coupled linear equations. The results are illustrated by several examples arising in mathematical physics.     

Consistent solutions of linear equations in variational derivatives

In this paper, we consider the classical existence problem for consistent and uniformly consistent solutions of linear equations in variational derivatives. In particular, we generalize several results obtained by V. V. Zhikov and B. A. Shcherbakov concerning the existence of almost-periodic, consistent, and uniformly consistent solutions of ordinary differential equations.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 176–187.Original Russian Text Copyright © 2005 by A. I. Gerko.This revised version was published online in April 2005 with a corrected issue number.     

Consistent solutions of linear equations in variational derivatives

In this paper, we consider the classical existence problem for consistent and uniformly consistent solutions of linear equations in variational derivatives. In particular, we generalize several results obtained by V. V. Zhikov and B. A. Shcherbakov concerning the existence of almost-periodic, consistent, and uniformly consistent solutions of ordinary differential equations.     

Variational principles for variational inequalities

This paper describes, and analyzes, variational principles for the solutions of finite dimensional variational inequalities on closed convex sets. No symmetry conditions are required for the function. A saddle point characterization of the solutions is also described. The relevant functionals are explicitly described for some particular examples.     

Non-differentiable variational principles

We develop a calculus of variations for functionals which are defined on a set of non-differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non-differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler-Lagrange equation. We finally prove that solutions of the Schrödinger equation can be obtained as extremals of a non-differentiable variational principle, leading to an extended Hamilton's principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space-time.     

Duality algorithms for nonconvex variational principles

This paper describes, and analyzes, a method of successive approximations for finding critical points of a function which can be written as the difference of two convex functions. The method is based on using a non-convex duality theory. At each iteration one solves a convex, optimization problem. This alternates between the primal and the dual variables. Under very general structural conditions on the problem, we prove that the resulting sequence is a descent sequence, which converges to a critical point of the problem. To illustrate the method, it is applied to some weighted eigenvalue problems, to a problem from astrophysics, and to some semilinear elliptic equations.     

Transitivity and variational principles

We apply an order reasoning to mappings satisfying the triangle inequality. This general approach yields the Ekeland’s variational principle as one of the consequences. In addition we obtain an extension of the Brøndsted variational principle and of the Takahashi fixed point theorem.     

Reflection principles for linear elliptic second order partial differential equations with constant coefflcients

Summary Reflection principles, analogous to the classicalSchwarz reflection principle for harmonic functions, are obtained for solutions of linear elliptic second order partial differential equations with constant coefficients. The boundary conditions employed are supposed to be satisfied in a limiting sense only, and do not require (a priori) the existence of derivatives on the boundary. To Mauro Picone on his 70^th birth day. This research was supported in part by the United States Air Force under Contract N^o. AF(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command. The work of this author was sponsored by the Office of Ordnance Research, U.S. Army, under the Contract DA-36-034-ORD-1486.