The Use of an lnvariance Condition in the Solution of Multiple‐Scale Singular Perturbation Problems: Ordinary Differential Equations

Woodruff, S.

doi:10.1002/sapm1993903225

Cited by 21 publications

(15 citation statements)

References 11 publications

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“…Independently and concurrently to [5,6], Woodruff in [45,46] developed a method based on an infinitesimal invariance condition that is analogous to the RG condition. This method has many similar features to the RG method, and it yields similar results to the RG method.…”

Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

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For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O( 2 ), where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ ).

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Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

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“…We note that [27] and [28] solve (3.2) by an invariance condition method, while [29] shows that the Navier-Stokes equation for a slightly compressible fluid can be considered as an abstract equation of this form. A more general standard form than (3.1) is needed, however, to encounter the classic small-divisor problem [30,Chapter 5], [31,.…”

Section: Vector Systemsmentioning

confidence: 99%

Working with multiscale asymptotics

2005

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This paper surveys, compares and updates techniques to obtain the asymptotic solution of the weakly nonlinear oscillator equationÿ + y + f (y,ẏ) = 0 as → 0 and for corresponding first-order vector systems. The solutions found by the regular perturbation method generally feature resonance and so break down as t → ∞. The classical methods of averaging and multiple scales eliminate such secular behavior and provide asymptotic solutions valid for time intervals of length t = O( −1 ). The renormalization group method proposed by Chen et al. [Phys. Rev. E 54 (1996) 376-394] gives equivalent results. Several well-known examples are solved with these methods to demonstrate the respective techniques and the equivalency of the approximations produced. Finally, an amplitude-equation method is derived that incorporates the best features of all these techniques. This method is both straightforward to automate with a computer-algebra system and flexible enough to allow the forcing f to depend on the small parameter.This paper is dedicated to Jerry Kevorkian with sincere appreciation for his long career of dedicated teaching at the University of Washington and for his substantial contributions to multiscale asymptotics.

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“…(2) the short-wave wavelength is 0 (17) relative to the gravity-capillary length scale; (3) the short -wave surface elevation amplitude is O( E); (4) the band-width spectral distribution for the short wave is O( JL).…”

Section: Construction Of the Small-scale Expansionmentioning

confidence: 99%

“…A new perturbation method [16,17] for multiple-scale problems is used in this paper. The method is based on an invariance condition and consists of two main steps.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Perturbation Analysis of an Interaction between Long and Short Surface Waves

Woodruff

Messiter

1994

Stud Appl Math

Self Cite

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The interaction of finite-amplitude long gravity waves with a small-amplitude packet of short capillary waves is studied by a multiple-scale method based on the invariance of the perturbation expansion under certain translations. The result of the analysis is a set of equations coupling the complex amplitude of the packet of short waves with.the long-wave velocity potential and surface elevation. The short wave is described by a Ginzburg-Landau equation with coefficients that depend on properties of the long wave. The long-wave potential and surface elevation satisfy the usual free-surface conditions augmented by forcing terms representing effects of the short waves. The derivation removes some of the restrictions imposed in earlier studies.

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The Use of an lnvariance Condition in the Solution of Multiple‐Scale Singular Perturbation Problems: Ordinary Differential Equations

Cited by 21 publications

References 11 publications

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Working with multiscale asymptotics

A Perturbation Analysis of an Interaction between Long and Short Surface Waves

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