An Introduction to Singular Perturbation Theory (MAGIC041)
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Spring 2010 (Monday, January 11 to Friday, March 19)
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: â��orderâ�� (â��big ohâ�� and â��little ohâ��) and â��asymptotically equal toâ�� (or â��behaves likeâ��). 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 â�� 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 Mathieuâ��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 â�� for obvious reasons) are available; the former .pdf files are called â��Notesâ��, and those for projection on the screen are named â��OHâ��. 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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