MAGIC041: An Introduction to Singular Perturbation Theory

Course details

A specialist MAGIC course


Spring 2009
Monday, January 19th to Friday, March 27th; Monday, April 27th to Monday, April 27th


Live lecture hours
Recorded lecture hours
Total advised study hours


09:05 - 09:55


  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]


No prerequisites information is available yet.


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.


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