On the calculation of nonlinear vibrations of flexible plates and shallow shells by the small parameter method
Small parameter method used for calculating periodic vibrations of flexible shallow shells and plates
PLATE THEORY↗
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Two-Scale Neural Networks for Partial Differential Equations with Small Parameters
We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.
97 MATHEMATICS AND COMPUTING↗
A small parameter model of circulation in a homogeoneous baroclinic ocean
A small parameter model of circulation in a homogeneous baroclinic ocean is presented. The principles common to the construction of small parameter models and certain energetic principles developed in connection with atmospheric processes are made use of. These principles were applied in the study of processes in a baroclinic ocean.
Borisenkov, Y. P.↗
Averaging methods for differential equations with retarded arguments and a small parameter
Differential equations with retarded arguments and small parameter
DIFFERENTIAL EQUATION↗
Periodic solutions of a class of hyperbolic equations containing a small parameter.
Periodic solutions of hyperbolic equations containing small parameter by extending Cesari method for differential equations
PERIODIC FUNCTION↗
Periodic solutions of a class of hyperbolic equations containing a small parameter
Periodic solutions of class of hyperbolic equations containing small parameter - linear and nonlinear wave equations
HYPERBOLIC EQUATION↗
Periodic solutions of differential equations with time lag containing a small parameter
Periodic solutions of differential equations with time lag containing small parameter
DIFFERENTIAL EQUATION↗
Equations of motion and the role of small parameters in the dynamics of nonholonomic systems
Equations of motion and influence of small parameters in dynamics of nonholonomic systems
Neimark, Yu. I.↗
Periodic solutions of second-order nonlinear difference equations containing a small parameter. IV - Multi-discrete time method
It is shown that a discrete multi-time method can be constructed to obtain approximations to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter. Three examples illustrating the method are presented.
Mickens, Ronald E.↗
Averaging methods for differential equations with retarded arguments and a small parameter.
Averaging method for differential equations with retarded arguments
PERTURBATION THEORY↗
Degeneration of the solutions of certain well- posed systems of partial differential equations depending on a small parameter.
Degeneration of solutions of well-posed systems of first order partial differential equations when particular parameter approaches zero
BOUNDARY VALUE↗
On the single first-order partial differential equation with a small parameter.
Singular perturbation problems for partial differential equations
PARTIAL DIFFERENTIAL EQUATION↗
Periodic solutions of differential equations with time lag containing a small parameter.
Periodic solutions for functional differential equations with time lag, developing bifurcation theory representing delay equation extension
Perello, C.↗
Expansion formulae in canonical transformations depending on a small parameter
Perturbation theory based on Lie transforms, reducing Deprit equation to generate general recursion formulas
Kamel, A. A.↗
Periodic solutions of second-order nonlinear difference equations containing a small parameter. II - Equivalent linearization
The classical method of equivalent linearization is extended to a particular class of nonlinear difference equations. It is shown that the method can be used to obtain an approximation of the periodic solutions of these equations. In particular, the parameters of the limit cycle and the limit points can be determined. Three examples illustrating the method are presented.
Mickens, R. E.↗
Periodic solutions of second-order nonlinear difference equations containing a small parameter. III - Perturbation theory
A technique to construct a uniformly valid perturbation series solution to a particular class of nonlinear difference equations is shown. The method allows the determination of approximations to the periodic solutions to these equations. An example illustrating the technique is presented.
Mickens, R. E.↗
On the Use of the Harmonic Linearizaiton Method in the Automatic Control Theory
The method of harmonic linearization (harmonic balance), first proposed by N. M. Krylov and N. N. Bogolyubov for the approximate investigation of nonlinear vibrations, has been developed and received wide practical application to problems in the theory of automatic control. Recently, some doubt has been expressed on the legitimacy of application of the method to these problems, and assertions were made on the absence in them of a small parameter of any kind. Nevertheless, the method gives practical, acceptable results and is a simple and powerful means in engineering computations. Hence, the importance of questions arises as to its justification. The underlying principle of the method is the replacement of the given nonlinear equation by a linear equation. In establishing the method, a small parameter is considered whose presence makes it possible to speak, with some degree of approximation, of the solution of this new equation to the solution of the given nonlinear equation. In an article by the author, certain considerations were given on the presence of the small parameter, but this question has not as yet received a final answer. In the present report, a somewhat different approach to the problem is applied that permits: (a) establishing, in the clearest manner, the form of the presence of the small parameter in nonlinear problems of control theory, solvable by the method of harmonic linearization; (b) connecting it with previous intuitive physical concepts (with the "filter property") and extending the class of problems possessing this property; and (c) discussing various generalizations of the method.
Popov, E. P.↗
Parameters of small scale ionization inhomogeneities in the F region of the ionosphere
The results of the parameters of small-scale ionization inhomogeneities study of the ionosphere's F-region for October-November 1965 and January-February 1966 are presented. It is shown that the most probable values of the inhomogeneity parameters are as follows: degree of anisotropy 1.4-2; vertical dimension 200-400 m; horizontal dimensions along major and minor axes 400-1000 and 200-600 m, respectively; rate of chaotic motions 30-60 m/sec; lifetime 6-9 sec.
Drobzhev, V. I.↗