The renormalization group: A perturbation method for the graduate curriculum

In this paper the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. Lett., 73 (1994), pp.1311-1315; Phys. Rev. E, 54 (1996), pp.376-394] 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.The method is illustrated by some linear and nonlinear singular perturbation problems. Key word. © 2012 Society for Industrial and Applied Mathematics.

A survey in mathematics for industry: Two-timing and matched asymptotic expansions for singular perturbation problems

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383-410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations. © 2011 Cambridge University Press.

Amplitude equations and asymptotic expansions for multi-scale problems

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. © 2010 - IOS Press and the authors. All rights reserved.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

Examples illustrating the use of renormalization techniques for singularly perturbed differential equations

With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Feasibility studies on using microwave-heating techniques for recycling lead waste

Lead is present everywhere in the environment and has been defined as one of the greatest threats to the human health. In this paper, attempts have been made to study a way of recycling the lead produced from waste usage and disposed of in such a way as to avoid degrading the surrounding environment. In order to contain the waste, recycled asphalt material is mixed with the lead and then heated with microwave energy. This is an attempt to solidify and reduce the lead contaminants and use the final product as sub-base material in road pavement construction. The microwave heating of the specimens is carried out with 30%, 50%, 80% and 100% of power at 800W. The optimum power mode is used to compare with the conventional heating of asphalt with sulfur additive. The results are characterized by compact density, permeability, and subjected to toxicity test with regards to lead concentration. A mechanical test to evaluate the stability is also performed on the three methods of solidification and to prove that microwave zapping method allow to convert into an environmentally stable material for recycling without having to be deposited in a landfill site.