Wednesday, 16 January 2008

Welcome to B5b - supplementary lectures (topics in applied differential equations)

This is the website for Maths B5b - supplementary lectures (topics in applied differential equations) for the Hilary 2008 term.

The lecture meets on Thursdays at 11:00 am in SR2 in the Mathematical Institute basement. The plan is to start lectures at 11:05 and finish them at 11:55.

On this blog, I'll be posting links, answering questions that are of general relevance to the course, etc.

The TA for this lecture course is Dave Hewett, who some of you might remember as the TA for C6.3a in the Michaelmas term. He will be marking your assignments. (My plan is to have two assignments: the first will be due on Friday of week 4 and the second will be due on Friday of week 8---though the due date of the second set of problems is still being determined. I will be posting the problems shortly.)

The basic topics for the course will consist of an introduction to the theory of distributions and then an introduction to nonlinear partial differential equations. The course will be divided as follows:

1) Distributions (roughly 3 lectures): This will take what you learned in the past on delta functions, Green's functions, etc. and formalize them mathematically.

2) Further development of hyperbolic equations (roughly 3 lectures): This will be roughly along the lines of what is listed in the official syllabus, which mentiones "the Cauchy-Kovalevskaya theorem, Riemann invariants, shocks and weak solutions, causality." I also hope to discuss (and/or include in the problem sets) rarefaction waves and how this stuff is relevant to a modern, very important numerical method called the level set method. (I can't promise exactly what topics we'll cover, but these are the lines along which I am thinking.)

3) Introduction to dispersive equations (roughly 2 lectures): solitary waves, KdV equation (and how to derive it from the FPU model), NLS equation

I haven't assembled a huge list of references, but here are some that may prove helpful (titles are approximate):

1) Butkov, Mathematical Physics
2) Fritz John, Partial Differential Equations
3) Paul Garabedian, Partial Differential Equations
4) Alwyn Scott, Encyclopedia of Nonlinear Science (selected entries)
5) Gerald Whitham, Linear and Nonlinear Waves
6) Various analysis, pde, introductory nonlinear wave textbooks

I'll try to add more recommendations to the list as we go along. If there are any books (or online resources) you see that you like, please let me know and I'll mention them here.

Note that I do not have any specific online lectures notes for this course.

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