Multiple Scale and Singular Perturbation Methods

Kevorkian, J.; Cole, J. D.

doi:10.1007/978-1-4612-3968-0

Cited by 908 publications

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“…It offers a versatile alternative to classical perturbation methods, such as the Poincaré-Lindstedt method [26,34], the method of matched asymptotic expansions [15,18,25,39], the method of multiple time scales [15,18,29,39], the method of averaging [3,41], and the WKBJ method [35] which were each developed for specific types of problems. In numerous examples the results obtained using the RG method are shown to agree [5,6,30,37,49] with those obtained from classical methods.…”

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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“…In step 2, internal forces f t are determined with (6) while the tangent stiffness K t matrix is obtained by finite difference of internal forces. In step 3, the position and velocity of the rider at the end of the time step are determined by (10). Then the vector of external forces p t+∆t is constructed as a sum of the self-weight forces and the force resulting from the rider.…”

Section: Numerical Algorithmmentioning

confidence: 99%

“…In this latter case, which is typically met as the rider travels along the cable, there exists a short zone on each side of the traveling force where the cable inclination varies rapidly. These short boundary layers [10] results in the relative inclination in the cable, between both sides of the force, as seen from a far-field viewpoint. Analytical methods are usually developed in order to avoid the difficult numerical modeling of these boundary layers [3,5]; because this model is mainly devoted to be numerical, this feature is disregarded in this work.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Computation and Design of a World's Record Death-Ride

Denoël¹,

Canor²,

Demanet³

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. A death-ride is a long cable tensioned between two dead ends. Its design should offer a comfortable and/or enjoyable journey to the riders, within the strength limits of the material. Actually, the tensioning operation is usually subtle as a too slack cable with a sagging area would not allow finish the race, while a too taut cable could produce excessive stresses in the cable. The paper presents the design methodology and the dynamic model that were used for the design of a world-record 1.2km long death-ride, with an only 10-mm diameter.

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“…This suggests that the local dynamics can be decomposed into two time-scales: an inner boundary layer where the concentration of agent (A) dominates the concentration of infectious individuals (I), and an outer limit where the concentration of agent is negligible relative to the more slowly decaying concentration of in-fectious individuals. As such, the problem can be solved using the method of multiple scales (Kevorkian and Cole, 1996). By approximating the dynamics on an inner and outer time scale and matching the solutions in the middle, we may obtain a reasonable approximation to the true solution.…”

Section: Asymptotic Susceptibility Levelmentioning

confidence: 99%

A two-phase epidemic driven by diffusion

Reluga

2004

Journal of Theoretical Biology

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Multiple Scale and Singular Perturbation Methods

Cited by 908 publications

References 0 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

Dynamic Computation and Design of a World's Record Death-Ride

A two-phase epidemic driven by diffusion

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