scholarly journals Convergence of Hill's Method for Nonselfadjoint Operators

2012 ◽  
Vol 50 (1) ◽  
pp. 64-78 ◽  
Author(s):  
Mathew A. Johnson ◽  
Kevin Zumbrun
Keyword(s):  
Hill’S Method

A Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Solutions of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Xuefeng Wang ◽  
Weidong Zhu
Keyword(s):  
Boundary Condition ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Jacobian Matrix ◽  
Additional Function ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Hill’S Method ◽  
Sine Functions ◽  
Nonlinear String

Abstract A spatial and temporal harmonic balance (STHB) method is demonstrated in this work by solving periodic solutions of a nonlinear string equation with a linear complex boundary condition, and stability analysis of the solutions is conducted by using the Hill’s method. In the STHB method, sine functions are used as basis functions in the space coordinate of the solutions, so that the spatial harmonic balance procedure can be implemented by the fast discrete sine transform. A trial function of a solution is formed by truncated sine functions and an additional function to satisfy the boundary conditions. In order to use sine functions as test functions, the method derives a relationship between the additional coordinate associated with the additional function and generalized coordinates associated with the sine functions. An analytical method to derive the Jacobian matrix of the harmonic balanced residual is also developed, and the matrix can be used in the Newton method to solve periodic solutions. The STHB procedures and analytical derivation of the Jacobian matrix make solutions of the nonlinear string equation with the linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of linearized ordinary differential equations (ODEs) that are associated with the governing partial differential equation is also developed, so that one can directly use the Hill’s method to analyze the stability of the periodic solutions without deriving the linearized ODEs. The frequency-response curve of the periodic solutions is obtained and their stability is examined.

A Generalized Hill’s Method for the Stability Analysis of Parametrically Excited Dynamic Systems

1982 ◽  
Vol 49 (1) ◽  
pp. 217-223 ◽  
Author(s):  
S. T. Noah ◽  
G. R. Hopkins
Keyword(s):  
Stability Analysis ◽  
Dynamic Systems ◽  
Absolute Convergence ◽  
General System ◽  
Closed Form Expression ◽  
Form Expression ◽  
Null Solution ◽  
Hill’S Method ◽  
The Stability ◽  
Pulsating Fluid

A method is described for investigating the stability of the null solution for a general system of linear second-order differential equations with periodic coefficients. The method is based on a generalization of Hill’s analysis and leads to a generalized Hill’s infinite determinant. Following a proof of its absolute convergence, a closed-form expression for the characteristic infinite determinant is obtained. Methods for the stability analysis utilizing different forms of the characteristic determinant are discussed. For cases where the instabilities are of the simple parametric type, a truncated form of the determinant may be used directly to locate the boundaries of the resonance regions in terms of appropriate system parameters. The present generalized Hill’s method is applied to a multidegree-of-freedom discretized system describing pipes conveying pulsating fluid. It is demonstrated that the method is a flexible and efficient computational tool for the stability analysis of general periodic systems.

The limit T?? in Hill's method for quasienergy states

1980 ◽  
Vol 45 (1) ◽  
pp. 894-896
Author(s):  
A. O. Melikyan ◽  
K. Kh. Simonyan
Keyword(s):  
Hill’S Method

PREPARATION OF ALPHAs1- AND BETA-CASEIN RELEASED DURING PREPARATION OF KAPPA-CASEIN FROM BUFFALO MILK

1972 ◽  
Vol 35 (1) ◽  
pp. 56-58
Author(s):  
M. A. Khorshid ◽  
M. Bhimasena Rao
Keyword(s):  
Gel Electrophoresis ◽  
Buffalo Milk ◽  
Starch Gel Electrophoresis ◽  
Beta Casein ◽  
Starch Gel ◽  
Hill’S Method

Methods were modified to improve preparation of αs1- and β-casein from the precipitate obtained when k-casein was prepared by Hill's method. A concentration of 3.3 M urea was used instead of 4.6 M urea to precipitate all the αs1-casein. β-Casein was precipitated from the second-cycle casein fraction P at pH 4.9 as described by others, the gummy precipitate was washed with alcohol, and then air dried. Starch-gel electrophoresis showed that αs1- and β-casein prepared by this method followed by further purification through redissolving and reprecipitating, was essentially free from specific fractions.

scholarly journals On the convergence of Hill’s method

2010 ◽  
Vol 79 (269) ◽  
pp. 169-169 ◽  
Author(s):  
Christopher W. Curtis ◽  
Bernard Deconinck
Keyword(s):  
Hill’S Method

scholarly journals Stability of the Solar System

1979 ◽  
Vol 81 ◽  
pp. 7-15
Author(s):  
Victor Szebehely
Keyword(s):  
Dynamical System ◽  
Solar System ◽  
Quantitative Measure ◽  
Marginal Stability ◽  
Actual Behavior ◽  
The Sun ◽  
Dynamical Models ◽  
Restricted Problems ◽  
Hill’S Method ◽  
The Stability

This paper reviews the present status of research on the problem of stability of satellite and planetary systems in general. In addition new results concerning the stability of the solar system are described. Hill's method is generalized and related to bifurcation (or catastrophe) theory. The general and the restricted problems of three bodies are used as dynamical models. A quantitative measure of stability is introduced by establishing the differences between the actual behavior of the dynamical system as given today and its critical state. The marginal stability of the lunar orbit is discussed as well as the behavior of the Sun-Jupiter-Saturn system. Numerical values representing the measure of stability of several components of the solar system are given, indicating in the majority of cases bounded behavior.

Transverse instabilities of deep-water solitary waves

2006 ◽  
Vol 462 (2071) ◽  
pp. 2039-2061 ◽  
Author(s):  
Bernard Deconinck ◽  
Dmitry E Pelinovsky ◽  
John D Carter
Keyword(s):  
Growth Rate ◽  
Deep Water ◽  
Wave Train ◽  
Two Dimensional ◽  
Nls Equation ◽  
One Dimensional ◽  
Hill’S Method ◽  
Real Growth ◽  
Dimensional Soliton ◽  
Complex Growth Rate

The dynamics of a one-dimensional slowly modulated, nearly monochromatic localized wave train in deep water is described by a one-dimensional soliton solution of a two-dimensional nonlinear Schrödinger (NLS) equation. In this paper, the instability of such a wave train with respect to transverse perturbations is examined numerically in the context of the NLS equation, using Hill's method. A variety of instabilities are obtained and discussed. Among these, we show that the solitary wave is susceptible to an oscillatory instability (complex growth rate) due to perturbations with arbitrarily short wavelength. Further, there is a cut-off on the instability with real growth rates. We show analytically that the nature of this cut-off is different from what is claimed in previous works.

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