Asymptotic behavior of ratio between the components of solutions of a linear system of ordinary differential equation and its applications to the determination of Lyapunov characteristic numbers

1970 ◽  
Vol 85 (1) ◽  
pp. 259-276 ◽  
Author(s):  
A. F. Izé
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Linear System ◽  
Characteristic Numbers ◽  
Lyapunov Characteristic Numbers

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

On the Determination of Joint Reactions in Multibody Mechanisms

2004 ◽  
Vol 126 (2) ◽  
pp. 341-350 ◽  
Author(s):  
Wojciech Blajer
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Algebraic Equation ◽  
Matrix Inversion ◽  
Differential Algebraic Equation ◽  
Joint Coordinates ◽  
Reduced Dimension ◽  
Absolute Coordinates ◽  
Coordinate Partitioning

In this paper some existing codes for the determination of joint reactions in multibody mechanisms are first reviewed. The codes relate to the DAE (differential-algebraic equation) dynamics formulations in absolute coordinates and in relative joint coordinates, and to the ODE (ordinary differential equation) formulations obtained by applying the coordinate partitioning method to these both coordinate types. On this background a novel efficient approach to the determination of joint reactions is presented, naturally associated with the reduced-dimension formulations of mechanism dynamics. By introducing open-constraint coordinates to specify the prohibited relative motions in the joints, pseudoinverse matrices to the constraint Jacobian matrices are derived in an automatic way. The involvement of the pseudo-inverses leads to schemes in which the joint reactions are obtained directly in resolved forms—no matrix inversion is needed as it is required in the classical codes. This makes the developed schemes especially well suited for both symbolic manipulators and computer implementations. Illustrative examples are provided.

scholarly journals Positive decreasing solutions of second order quasilinear ordinary differential equations in the framework of regular variation

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1995-2010 ◽  
Author(s):  
Jelena Milosevic ◽  
Jelena Manojlovic
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Analysis ◽  
Ordinary Differential Equations ◽  
Regular Variation ◽  
Second Order ◽  
Precise Information ◽  
Varying Coefficients

This paper is concerned with asymptotic analysis of positive decreasing solutions of the secondorder quasilinear ordinary differential equation (E) (p(t)?(|x'(t)|))'=q(t)?(x(t)), with the regularly varying coefficients p, q, ?, ?. An application of the theory of regular variation gives the possibility of determining the precise information about asymptotic behavior at infinity of solutions of equation (E) such that lim t?? x(t)=0, lim t?? p(t)?(-x'(t))=?.

XXVII.—Expansions of Lamé Functions into Series of Legendre Functions

1948 ◽  
Vol 62 (3) ◽  
pp. 247-267
Author(s):  
A. Erdélyi
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Infinite Series ◽  
Recurrence Relations ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Jacobi Series ◽  
Characteristic Numbers ◽  
Transcendental Equations

Summary28. This paper contains the investigation of certain properties of periodic solutions of Lamé's differential equation by means of representation of these solutions by (in general infinite) series of associated Legendre functions. Terminating series of associated Legendre functions representing Lamé polynomials have been used by E. Heine and G. H. Darwin. The latter used them also for numerical computation of Lamé polynomials. It appears that infinite series of Legendre functions representing transcendental Lamé functions have not been discussed previously. In two respects these series seem to be superior to the generally used power-series and Fourier-Jacobi series, (i) They are convergent in some parts of the complex plane of the variable where both power-series and Fourier-Jacobi series diverge, (ii) They permit by simply replacing Legendre functions of first kind by those of second kind, to deal with Lamé functions of second kind as well as Lamé functions of first kind (§ 15).In §§ 2 and 8 of the present paper the series are heuristically deduced from the integral equations satisfied by periodic Lamé functions. Inserting the series found heuristically, with unknown coefficients, into Lamé's differential equation, recurrence relations for the coefficients are obtained (§§ 9–12). These recurrence relations yield the (in general transcendental) equations in form of (in general infinite) continued fractions for the determination of the characteristic numbers. The convergence of the series can be discussed completely.There are altogether forty-eight different series. Every one of the eight types of Lamé polynomials is represented by six different series (see table in § 13). There are interesting relations (§ 14) between series representing the same function.Next infinite series representing transcendental Lamé functions are discussed. It is seen that transcendental Lamé functions are only simply-periodic (§§ 18, 19). Lamé functions of real (§§ 20–22) and imaginary (§§ 23-24) period are represented by series of Legendre functions the variables of which are different in both cases.The paper concludes with a brief discussion of the most important limiting cases, and a short mention of other types of series of Legendre functions representing Lamé functions.

scholarly journals Some comparison criteria in oscillation theory

1984 ◽  
Vol 36 (2) ◽  
pp. 176-186 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Arbitrary Order ◽  
Oscillation Theory ◽  
Linear Ordinary Differential Equation ◽  
Order Linear ◽  
Comparison Criteria

AbstractThe purpose of this paper is to establish comparison criteria, by which the oscillatory and asymptotic behavior of linear retarded differential equations of arbitrary order is inherited from the oscillation of an associated second order linear ordinary differential equation. These criteria are new even in the case of ordinary differential equations.

scholarly journals A canonical form of the equation of motion of linear dynamical systems

2018 ◽  
Vol 474 (2211) ◽  
pp. 20170809
Author(s):  
Daniel T. Kawano ◽  
Rubens Goncalves Salsa ◽  
Fai Ma ◽  
Matthias Morzfeld
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamical Systems ◽  
Linear System ◽  
Canonical Form ◽  
Equation Of Motion ◽  
Positive Definite ◽  
Order Ordinary Differential Equation ◽  
Linear Dynamical Systems ◽  
Almost All

The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.

The mixing of ground water and sea water in permeable subsoils

1958 ◽  
Vol 4 (5) ◽  
pp. 479-488 ◽  
Author(s):  
G. F. Carrier
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Ground Water ◽  
Sea Water ◽  
Original Model ◽  
Linear Ordinary Differential Equation ◽  
Mixing Process ◽  
Salinity Distribution ◽  
Permeable Media

The subterranean mixing in permeable media of sea water and ground water is studied. The model for this mixing process which was suggested by C. K. Wentworth is adopted, but is soon discarded in favour of a more tractable formulation whose equivalence to the original model is established. The analysis is carried to the point where the determination of the salinity distribution of the ground water in a given subsoil requires only the solution of an elementary linear ordinary differential equation.

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