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Introduction to Singular Perturbation Theory (MAGIC041)
The Lectures and the Module in Outline
Some introductory examples to set the scene (without being too careful, at this stage, about the technical details). Introducing the notation: Ã¢ï¿½ï¿½orderÃ¢ï¿½ï¿½ (Ã¢ï¿½ï¿½big ohÃ¢ï¿½ï¿½ and Ã¢ï¿½ï¿½little ohÃ¢ï¿½ï¿½) and Ã¢ï¿½ï¿½asymptotically equal toÃ¢ï¿½ï¿½ (or Ã¢ï¿½ï¿½behaves likeÃ¢ï¿½ï¿½).
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.
The matching principle, introduced via intermediate variables and the overlap region. Worked examples included.
Some simple applications: roots of equations; integration of functions defined by (matched) asymptotic expansions. Worked examples included.
Introductory applications to ODEs: simple regular and singular problems. Worked examples included.
ODEs: some further examples of singular problems; the technique of scaling equations. Worked examples included.
Boundary-layer problems in ODEs; the position of the boundary layer is discussed for a class of 2nd order ODEs. Worked examples included.
Applications to PDEs: a regular problem (flow past a distorted circle); singular problems Ã¢ï¿½ï¿½ nonlinear, dispersive wave, and supersonic, thin-aerofoil theory.
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.
Multiple scales continued, with applications to MathieuÃ¢ï¿½ï¿½s 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 Ã¢ï¿½ï¿½ for obvious reasons) are available; the former .pdf files are called Ã¢ï¿½ï¿½NotesÃ¢ï¿½ï¿½, and those for projection on the screen are named Ã¢ï¿½ï¿½OHÃ¢ï¿½ï¿½. 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.