Techniques of applied mathematics

http://www.maths.ox.ac.uk/~fowler/courses/tech.html

Aims and objectives

This course develops mathematical techniques which are useful in solving `real-world' problems involving differential equations, and is a development of ideas which arise in the second year differential equations course. The course embraces the ethos of mathematical modelling, and aims to show in a practical way how equations `work', and what kinds of solution behaviours can occur.

Connections

The primary material for this course is section A differential equations. The course provides a platform for B6 Fluid mechanics and B8 Topics in applied mathematics, and for the fourth year courses Mathematics and the Environment, Mathematical Physiology, and Perturbation Methods. The introductory course B568 is essential.

Synopsis (B568)

Modelling, conservation and constitutive laws, nondimensionalization.

Perturbation methods, simple boundary layers.

Synopsis (B5a)

Ordinary differential equations: stability, oscillations, Poincaré-Lindstedt method.
Sturm-Liouville systems; integral equations, eigenfunctions.

Partial differential equations: waves and shocks, similarity solutions.

Course materials

Problem sheets numbered 2 to 8 cover material lectured in weeks 2 to 8, and consist (more or less) of some of the exercises in the notes, and are available in pdf format. Problem sheets 2 to 7 (sheet 2 is the first) are now ready.

Specimen finals questions are available in pdf format (as at 27/4/05, for 2004-2005 syllabus). They are actually quite a bit longer and harder than fhs questions. They are also out of date. If they get resurrected for this year's syllabus, it will probably not be till March or April.

Lecture notes are available and can be downloaded in pdf format. They are available at reception.

Reading

A.C. Fowler 2005 Techniques of applied mathematics. Mathematical Institute lecture notes.
Additional reading:
S.D. Howison 2005 Practical applied mathematics: modelling, analysis, approximation. C.U.P.
J.P. Keener 2000 Principles of applied mathematics: transformation and approximation, revised edition. Perseus Books, Cambridge, Mass.
Useful modelling books are:
R. Haberman 1998 Mathematical models. SIAM, Philadelphia.
A.C. Fowler 1997 Mathematical Models in the Applied Sciences. CUP, Cambridge.
A.B. Tayler 1986 Mathematical Models in Applied Mechanics. OUP, Oxford.
Mathematical techniques are in:
E.J. Hinch 1991 Perturbation Methods. CUP, Cambridge.
D.W. Jordan and P. Smith 1999 Nonlinear Ordinary Differential Equations, 3rd ed. OUP, Oxford.
E.A. Coddington and N. Levinson 1972 Theory of ordinary differential equations. Tata McGraw-Hill, New Delhi.
I. Stakgold 2000 Boundary value problems of mathematical physics, vol. I. SIAM, Philadelphia.

Lecture synopsis (2006)

(Page numbers refer to the printed lecture notes, version 2.0.)

Combustion and hysteresis (notes).
Stability and oscillations. (pp. 43-46, 49-54)
Poincaré-Lindstedt method. Van der Pol equation. (pp. 34-40)
Derivation of Sturm-Liouville systems. (pp. 64-70)
Sturm-Liouville theory. (pp. 70-75)Sturm-Liouville theory continued. (pp. 70-75)Comparison methods. (pp. 75-76)
Integral equations. (pp. 79-83)Eigenfunction expansions. (pp. 83-86)
Variational methods. (pp. 86-89)
Waves and shocks. (pp. 92-102)
Burgers' equation, transition layer. (pp. 105-109)
Similarity solutions. The heat equation. (pp. 112-115)
Porous medium equation. (pp. 116-119)

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