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  1. An example to set the scene. [0.5 lecture]
  2. Introducing asymptotic expansions : formal definitions, use of parameters. [1.5 lectures]
  3. Idea of scaling variables. [1 lecture]
  4. Matching Principle and the breakdown of asymptotic expansions. [2 lectures]
  5. Examples and applications, as time permits, selected from: roots of equations, evaluation of integrals, a "regular" ODE, a first order singular ODE, a boundary-layer-type problem, scalings to balance terms, where is the boundary layer?, heat conduction (a PDE example), supersonic flow (another PDE). [3 lectures]
  6. Brief introduction to the method of multiple scales, with applications to oscillatory problems. [2 lectures]


Spring 2010 (Monday, January 11 to Friday, March 19)


  • Tue 11:05 - 11:55



Introduction to Singular Perturbation Theory (MAGIC041)
The Lectures and the Module in Outline
Lecture 1
Some introductory examples to set the scene (without being too careful, at this stage, about the technical details). Introducing the notation: �order� (�big oh� and �little oh�) and �asymptotically equal to� (or �behaves like�).
Lecture 2
Asymptotic sequences and asymptotic expansions, first in one variable and then with respect to a parameter. The concepts of uniformity and of breakdown. Worked examples included.
Lecture 3
The matching principle, introduced via intermediate variables and the overlap region. Worked examples included.
Lecture 4
Some simple applications: roots of equations; integration of functions defined by (matched) asymptotic expansions. Worked examples included.
Lecture 5
Introductory applications to ODEs: simple regular and singular problems. Worked examples included.
Lecture 6
ODEs: some further examples of singular problems; the technique of scaling equations. Worked examples included.
Lecture 7
Boundary-layer problems in ODEs; the position of the boundary layer is discussed for a class of 2nd order ODEs. Worked examples included.
Lecture 8
Applications to PDEs: a regular problem (flow past a distorted circle); singular problems � nonlinear, dispersive wave, and supersonic, thin-aerofoil theory.
Lecture 9
A PDE with a boundary-layer structure (heat transfer to a fluid flowing in a pipe); introduction to the method of multiple scales: nearly linear oscillators. Worked examples included.
Lecture 10
Multiple scales continued, with applications to Mathieu�s equation, a model equation for weakly nonlinear, dispersive waves, and boundary-layer problems.
Copies of the notes, exactly as used on the screen during the lectures (although the pagination is different � for obvious reasons) are available; the former .pdf files are called �Notes�, and those for projection on the screen are named �OH�. There is also available a booklist; a few Appendices that are related to material given in the course, but extend some of the ideas, are also offered.
Associated with each lecture is a short set of exercises, each accessible to the diligent student by the end of the lecture. Additionally, a set of answers is also provided which give, in some cases, relevant intermediate results.


Photo of Joy Allen
Joy Allen
Photo of Chris Bocking
Chris Bocking
(East Anglia)
Photo of Susanne Claus
Susanne Claus
Photo of Samuel Durugo
Samuel Durugo
Photo of Aiman Elragig
Aiman Elragig
Photo of Fred Gent
Fred Gent
Photo of Christopher Marsden
Christopher Marsden
Photo of John Martin
John Martin
Photo of Kieron Rhys Moore
Kieron Rhys Moore
Photo of Shaker Rasheed
Shaker Rasheed
Photo of Moritz Reinhard
Moritz Reinhard
(East Anglia)
Photo of Catherine Saunders
Catherine Saunders
Photo of Thomas Shearer
Thomas Shearer
Photo of Tan Su
Tan Su
Photo of Jorge Vazquez
Jorge Vazquez
Photo of Daniel Wacks
Daniel Wacks
Photo of Tim Yeomans
Tim Yeomans


Perturbation Methods in Fluid MechanicsDyke
Perturbation methods in applied mathematicsCole
Perturbation methods in applied mathematicsKevorkian and Cole
Multiple scale and singular perturbation methodsKevorkian and Cole
Perturbation methodsNayfeh
Introduction to Perturbation TechniquesNayfeh
Matched Asymptotic Expansions: Ideas and TechniquesLagerstrom
Singular Perturbation Theory: Mathematical And Analytical Techniques With Applications To EngineeringJohnson
Perturbation Methods for Engineers and ScientistsBush
Perturbation MethodsHinch
Singular-perturbation theory: an introduction with applicationsSmith
Nonlinear Singular Perturbation Phenomena: Theory and ApplicationsChang and Howes
Asymptotic analysis of singular perturbationsEckhaus
Introduction to perturbation methodsHolmes
Singular Perturbation Methods for Ordinary Differential EquationsO'Malley
Scaling, self-similarity, and intermediate asymptoticsBarenblatt
Fluid mechanics and singular perturbations: a collection of papersKaplun, Lagerstrom, Howard and Liu
Asymptotic Expansions for Ordinary Differential EquationsWasow
Asymptotic ExpansionsCopson
Asymptotic ExpansionsErdelyi
Asymptotics and special functionsOlver
Asymptotic analysisMurray
Asymptotic expansions: their derivation and interpretationDingle
Studies on Divergent Series and Summability and the Asymptotic Development of FunctionsFord
Divergent SeriesHardy


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