Singular Perturbation Methods for Ordinary Differential Equations

O’Malley, Robert E.

doi:10.1007/978-1-4612-0977-5

Cited by 520 publications

(435 citation statements)

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“…(This observation is important because the lower-order terms have already been determined correctly in the preceding applications.) We illustrate Theorem 3.1 with an example from the Michaelis-Menten-Henri mechanism of enzyme kinetics [4], [10], [26], [27], [28]. Similar results (for q = 1, 2) have been obtained by Valorani, Goussis, and Najm [33] for a model equation due to Davis and Skodje [3].…”

Section: )supporting

confidence: 66%

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

2004

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Summary. This article is concerned with the asymptotic accuracy of the Computational Singular Perturbation (CSP) method developed by Lam and Goussis [The CSP method for simplifying kinetics, Int. J. Chem. Kin. 26 (1994) to reduce the dimensionality of a system of chemical kinetics equations. The method, which is generally applicable to multiple-time scale problems arising in a broad array of scientific disciplines, exploits the presence of disparate time scales to model the dynamics by an evolution equation on a lower-dimensional slow manifold. In this article it is shown that the successive applications of the CSP algorithm generate, order by order, the asymptotic expansion of a slow manifold. The results are illustrated on the Michaelis-Menten-Henri equations of enzyme kinetics. PAC numbers. 82.33.Vx, 87.15.Rn, 82.33.Tb, 02.60.Lj, 02.30.Yy

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Section: )supporting

confidence: 66%

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

2004

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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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“…The expansion for w, w (x,e) = w0 (x) + ewi (x) + e2 w2 (x) + ..., (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) implies that…”

Section: Model Problemmentioning

confidence: 99%

“…We notice that since expansion (4.11) is our hypothetical p, the outer expansion u must be contained in w when both written in terms of x = r]~1^. Consequently we should expect to find unknown ^ 's from Equations (4.16) while searching for unknown tjk s. The sequence of equations for tjk s has the following form (see Equations .., (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) with the successive equations containing other Aj's which enter the formulas for PjkS. At first glance, the task of finding Aj's and tjk s using Equations (4.19) seems to be daunting.…”

Section: Modelmentioning

confidence: 99%

Untitled

2003

Quart. Appl. Math.

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Singular Perturbation Methods for Ordinary Differential Equations

Cited by 520 publications

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Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

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

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