Journal ArticleDOI
Examples Illustrating the Use of Renormalization Techniques for Singularly Perturbed Differential Equations
Robert E. O'Malley,E. Kirkinis +1 more
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TLDR
In this article, the utility of the Renormalization group approach as a unification of other asymptotic and perturbation methods is illustrated with nine examples, and the utility is discussed.Abstract:
With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.read more
Citations
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The Renormalization Group: A Perturbation Method for the Graduate Curriculum
TL;DR: The renormalization group (RG) method of Chen, Goldenfeld, and Oono is presented in a pedagogical way to increase its visibility in applied mathematics and to argue favorably for its incorporation into the corresponding graduate curriculum.
Journal ArticleDOI
A Combined Renormalization Group‐Multiple Scale Method for Singularly Perturbed Problems
Robert E. O'Malley,E. Kirkinis +1 more
TL;DR: In this article, a straightforward method to asymptotically solve a variety of initial and boundary value problems for singularly perturbed ordinary differential equations whose solution structure can be anticipated is introduced.
Journal ArticleDOI
Variation of parameters and the renormalization group method
TL;DR: In this article, a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters, is presented.
Journal ArticleDOI
Analytical approximation of weakly nonlinear continuous systems using renormalization group method
TL;DR: In this paper, a renormalization group method (RGM) is proposed for determining the analytical approximation of weakly nonlinear continuous systems, and the results are in full agreement with those previously obtained with multiple scales method.
Journal ArticleDOI
An open problem on the optimality of an asymptotic solution to Duffing's nonlinear oscillation problem
Songxin Liang,Sijia Liu +1 more
TL;DR: An affirmative answer to the open problem is given by means of the homotopy analysis method to the asymptotic solution to Duffing's nonlinear oscillation problem.
References
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Journal ArticleDOI
Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory
Lin Yuan Chen,Lin Yuan Chen,Nigel Goldenfeld,Nigel Goldenfeld,Yoshitsugu Oono,Yoshitsugu Oono +5 more
TL;DR: It is shown that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points, deriving both amplitude equations and the center manifold.
Journal ArticleDOI
Simple viscous flows: From boundary layers to the renormalization group
John Veysey,Nigel Goldenfeld +1 more
TL;DR: In this paper, a survey of the literature on the problem of determining the drag on a body moving through a very viscous fluid has been presented, with special emphasis on the logical relationship between different approaches.
Journal ArticleDOI
Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations
TL;DR: In this article, Chen, Goldenfeld, and Oono used the renormalization group (RG) method for singular perturbation problems and showed that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincare-Birkhoff normal forms for these systems up to and including terms of O ( ϵ 2 ), where ϵ is the perturbations parameter.
Journal ArticleDOI
Renormalization-group theoretical reduction
TL;DR: The approach proposed in this paper has considerably eliminated the need for explicit perturbation results, making the RG approach simpler than many conventional singular perturbed approaches, and it may be asserted that reductive extraction of global features of the problem is the essence of singular perturbations methods.
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A Combined Renormalization Group‐Multiple Scale Method for Singularly Perturbed Problems
Robert E. O'Malley,E. Kirkinis +1 more