Direct Variational Method Research Articles

Variational method and nonlinear oscillations and waves

A direct variational method is developed for studying the asymptotic behavior of a wide class of nonlinear oscillation and wave problems. From some judiciously chosen trial solutions with adjustable parameters, equations governing the change of amplitudes and phases are derived and solved. The method is simple in concept and straightforward in application. Different aspects of the method are illustrated by applications to various examples: the oscillation of a pendulum with changing length; the motion of a charged particle in a strong magnetic field; the linear and nonlinear Klein–Gordon equations; and the linear and nonlinear Korteweg–de Vries equations.

Theory of arbitrary coupling F‐centres in multivalley cubic crystals

AbstractA theory of a large radius polaron bound to a positive Coulomb centre in a multivalley ionic cubic crystal is given for arbitrary values of both the electron–phonon coupling and the Coulomb binding. By means of a direct variational method with highly flexible trial functions, general expressions for the ground state and the first excited state energies are obtained depending on the coupling parameters and on the anisotropy of electron band masses. Under some simplifying assumptions the numerical values of these energies and of the oscillator strengths of optic transitions between the ground and excited states are given.

On a direct variational method for nonlinear heat transfer

In this paper it is shown, that the partial nonlinear differential equation can be reduced to the variational problem. By means of the variational technique based on the Kantorovich method, a nonlinear boundary value problem can be reduced to the set of the ordinary differential equations. The accuracy of the method is estimated by comparing the solutions of problems solved using the variational method and the other method. In this paper the method for the construction of the trial functions is also presented. Three examples are included to illustrate the above method.

Variational methods and nonlinear forced oscillations

An approximate, direct variational method, simple in concept, and straightforward in application is presented to deal with the problem of forced oscillation of nonlinear systems. The general procedure is illustrated in detail by treating a particular example, i.e., the Duffing’s equation. The same procedure is also applied to some other examples in mathematical physics.

Uniform Convergence of Variational Methods for Plane Subsonic Flow

Uniform convergence is proved for certain direct variational methods, assuming the existence of a solution with certain properties. In particular one important property used is that of “subsonicity”. This is then imposed on the approximating functions in order to establish the uniform convergence. This overcomes a crucial assumption made in previous work for Rayleigh–Ritz approximations. further, previous results for bounded subdomains are extented to infinity. The analysis applies mroe generally to plane uniformly elliptic problems when a priori bounds on the first derivatives of the solution are known to exist.

Solution of neutron-diffusion problems in heterogeneous flat reactors by the direct variational method

Solution of neutron-diffusion problems in heterogeneous flat reactors by the direct variational method

Ein direktes variationsverfahren zur behandlung der wärmeÜbertragungsprobleme für erzwungene konvektion

A direct variational method is presented for the determination of a temperature field in forced convection. This method is based on the principle of limited or partial variation first formulated by Rosen. The supplement to this formulation as introduced here satisfies the conditions necessary for the existence of a variational integral. Independently those special conditions are presented which allow the construction of an approximate solution. The advantage of the proposed method is in the possible solution of both linear and non-linear problems. For demonstration three problems are calculated and approximate solutions are compared to exact values.

Nonconvex Structural Optimization Problems

Plastic optimization problems involving convex specific cost functions can be handled by several methods. The earliest technique was based on the uniform energy dissipation principle; a more general approach which could be termed the cost gradiend method was developed later and was extended to multiload, multicomponent systems and to preassigned strength distribution. Some used energy principles in deriving their methods while others used a direct variational method involving slack variables and Lagrangian multipliers. All the foregoing criteria were developed originally as sufficient conditions for various classes of convex cost functions. If the cost function is nonconvex, then the kinematic conditions given by either variational method or by the Prager-Shield approach can be shown to constitute only a necessary condition for a local minimum. In deriving optimal solutions for frames and shells associated with nonconvex cost functions, it has been found that these criteria can still be very useful because they often admit only one (unique) solution. A simple example is presented which demonstrates this method.

Comparison and oscillation theorems for matrix differential inequalitites

Strong comparison theorems of Sturm’s type are established for systems of second order quasilinear elliptic partial differential equations. The technique used leads to new oscillation and nonoscillation criteria for such systems. Some criteria are deduced from a comparison theorem, and others are derived by a direct variational method. Some of our results constitute extensions of known theorems to nonselfadjoint quasilinear systems.

On application of direct variational methods to the solution of parabolic boundary value problems of arbitrary order in the space variables

On application of direct variational methods to the solution of parabolic boundary value problems of arbitrary order in the space variables

Approximate Solutions in Linear, Coupled Thermoelasticity

A method for obtaining approximate solutions to initial-boundary-value problems in the linear theory of coupled thermoelasticity is developed. This procedure is a direct variational method representing an extension of the Ritz method. As an illustration of the procedure, it is applied to a class of one-dimensional, transient problems involving weak thermal shocks. The problems considered are: (a) Rapid heating of a half space through a thermally conducting boundary layer, and (b) gradual heating of the boundary surface of a half space. The solutions generated by the extended Ritz method are compared, for accuracy, to solutions obtained from a numerical inversion scheme for the Laplace transform based on Gaussian quadrature. These comparisons indicate that the variational procedure developed here can yield accurate results.

A variational method for determining eigenvalues of the wave equation applied to tropospheric refraction

A simple and direct variational method is described for finding both complex and real eigenvalues of the wave equation of anomalous propagation in a horizontally stratified atmosphere. It may be looked upon as an extension of Rayleigh's method to complex eigenvalues. In this paper it is illustrated by an example, taken from the duct theory of super-refraction, in which the refractive index of the air varies with height according to a power law. Numerical agreement in the values for the lowest order eigenvalues with those obtained by the differential analyser is better than ½%.