Variational formulations for transmission problems of the electromagnetic field

The subject of this article is a review of all possible transmission problems for electromagnetic phenomena. In particular, we study the case of a perfect dielectric and a perfect conductor via a (formal) limit with conductivity approaching zero or infinity and discuss the expected regularity of the involved unknowns. Finally, we formulate equivalent variational formulations for each considered problem.     

On an iterative method for the solution of discretized elliptic problems with imperfect contact condition

Iterative method for the solution of discretized elliptic boundary value problem with imperfect contact condition on an internal boundary is constructed. It is proved that the method is robust with respect to the coefficient K appearing in the imperfect contact condition, i.e., its convergence rate is independent of the coefficient K. Numerical tests confirming the theoretical result are presented.     

Variational technique and principal solution in half-linear oscillation criteria

We investigate oscillatory properties of half-linear differential equations. The investigated half-linear equation is viewed as a perturbation of another (nonoscillatory) equation of the same form and perturbation is allowed both in the absolute term and in the derivative term. First, the Sturmian type theorem for solutions of the associated Riccati equation is established. In the second part of the paper we prove new half-linear oscillation criteria using the variational technique.     

Variational approach to the solution of linear multiparameter eigenvalue problems

We associate a multiparameter spectral problem in a real Euclidean space with a variational problem of finding a minimum of a certain functional. We establish the equivalence of the spectralproblem and the variational problem. On the basis of the gradient procedure, we propose a numerical algorithm for the determination of its eigenvalues and eigenvectors. The local convergence of the algorithm is proved.     

Variational iterative method and initial-value problems

The variational iterative method is revisited for initial-value problems in ordinary or partial differential equation. A distributional characterization of the Lagrange multiplier - the keystone of the method - is proposed, that may be interpreted as a retarded Green function. Such a formulation makes possible the simplification of the iteration formula into a Picard iterative scheme, and facilitates the convergence analysis. The approximate analytical solution of a nonlinear Klein-Gordon equation with inhomogeneous initial data is proposed.     

Variational iteration method for solving sixth-order boundary value problems

In this paper, we have shown that sixth-order boundary value problems can be transformed into a system of integral equations, which can be solved by using variational iteration method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. It is observed that the proposed technique is more useful and is easier to implement because one does not need to calculate the Adomian’s polynomials which is itself a difficult task.     

Variational iteration method for solving two-point boundary value problems

Variational iteration method is introduced to solve two-point boundary value problems. Numerical results demonstrate that the method is promising and may overcome the difficulty arising in Adomian decomposition method.     

Variational method for precise asymptotic formulas for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem motivated by the perturbed elliptic sine-Gordon equation where p >1 is a constant and λ ∈ R is an eigenvalue parameter. Our aim is to clarify the asymptotic relationship between L ^p+1-norm, L ^∞-norm of the solutions and λ when these norms are large. To this end, we consider an associated variational problem with this equation on a manifold new parameter), and obtain a solution pair . We establish the precise asymptotic formulas for .     

Variational method in the stability analysis of nonconservative problems

Zusammenfassung Das Stabilitätsproblem eines Voigt-Kelvin-Körpers, welcher nichtkonservativen und gyroskopischen Kräften unterworfen ist, wird — ähnlich der bekannten klassischen Methode — als Variationsaufgabe formuliert. Zwei Variationsprinzipien, basierend auf beschränkten zulässigen Verschiebungen, werden aufgestellt und in einem Näherungsverfahren verwendet, das als Verallgemeinerung des vielverwendeten Ritzschen Verfahrens als konservatives Problem gelten kann. Zur Instruktion wird ein Beispiel gegeben.     

Variational problems with lags

In this paper, we study some questions concerning the minima of the functional $$J\left( y \right) = \int_{x_1 }^{x_2 } {f\left( {x,y\left( x \right),y\left( {x - r} \right),\dot y\left( x \right),\dot y\left( {x - r} \right)} \right)dx} $$ In Section 2, we obtain an analogue to the Jacobi condition to add to the list of previously obtained necessary conditions. A transversality condition is developed in Section 3. In Section 4, we obtain an existence theorem. The techniques used are modifications of those used in the classical problems. In Section 5, we show that the theory of fields for the classical problem fails to work for our problem.     

Variational problems with constraints

Summary The problem considered is that of maximizing dt subject to x=G(x, y), x(0)=c, and0 ≤y≤x. An essentially new feature is determining in what regions y=x,0<y<x, and y=0.     

Convergence of the method of partition of regions for second-order elliptic conjugation problems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 8, pp. 1137–1141, August, 1989.     

Direct shooting method for the solution of boundary-value problems

A new iterative method is developed to solve the boundary-value problems for ordinary nonlinear differential equations. The method requires that only the original system of differential equations is solved once in each iteration. The initial conditions for a new iteration are evaluated directly from the given boundary conditions and the initial and boundary conditions obtained in the previous iteration, thus avoiding the necessity of solving a system of algebraic equations. The convergence proofs of the method are given. Examples of the application of the method are presented and discussed.     

A quasilinearization method for elliptic problems with a nonlinear boundary condition

We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.     

Variational problems with free interfaces

Summary A large class of problems arise in the material sciences involving free interfaces. To establish the existence and regularity (including the regularity of free interfaces) of solutions has been an important and interesting issue. Here we were able to do so in a model case accounted in optimal designs.The method developed in this paper is rather general and may be useful for many other related problems.     

Variational problems with topological constraints

We introduce a decomposition that captures much of the topology of level sets for functions in certain Sobolev spaces, and allows the definition of an analog of the decreasing rearrangement of a function which respects the topology of level sets. In a variety of settings this decomposition is preserved under weak limits, and so is useful in establishing existence of minimizers of various variational problems with prescribed topological properties. These include variational problems in which ‘topological rearrangements’ are fixed, and ones in which the functional depends on derivatives of rearrangements. Received: December 16, 1998 / Accepted: July 16, 1999     

The domain decomposition method for parabolic transmission problems

An estimate of the rate of convergence is given for the domain decomposition method for the second-order parabolic transmission problem. A brief discussion of the method and some of its applications are presented.     

Ritz method for approximate solution of optimization problems

The article investigates convergence of the minimum points of special classes of functionals defined on reflexive Banach spaces.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 171–178, 2002.