MS324 TMA03

First sorry for not blogging for a month or so been a bit bogged down with MS324 block 2 which is actually quite interesting but unfortunately the TMA does not reflect this.

The main topics are a basic overview of probability, and random walks in sections 1 and 2, An account of the diffusion equation as applied to heat problems and the most interesting part which is not assessed namely the link between microscopic diffusion, random walks and macroscopic diffusion.

The TMA is as they say in the books straightforward but tedious

Question 1 is  a problem based on successive tyre failures of a cyclist where the probability distribution is an exponential one. To solve the question one has to use integration by parts a couple of times

Question 2 is a problem calculating the statistics associated with a random process defined by a recurrence relation

Question 3 concerns heat conduction in a Nuclear core, the diffusion equation reduces to a 1 dimensional form and is relatively easy to solve. Still must confess I couldn't see how to do the last part

Question 4 is a question concerning temperature waves in the earths surface again quite a straightforward question.

Overall then the TMA is quite straightforward but as there are a number of numerical calculations rather tedious. There does seem to be a disconnect between the TMA questions and the course content.
However as I was struggling to motivate myself I decided to cut my losses with only about 3/4 of the TMA done. I provided more or less complete answers to questions 1 and 4 just did the first two parts of question 2 and all but the last part of question three. A bit pathetic I realise but sometimes it's just better to move on.

It would have been more interesting had they asked us to solve the diffusion equation in three dimensions for say a cube or sphere say with the top half heated at one temperature and the other one at a different temperature. For a sphere this would involve setting up the equation in Spherical coordinates separating the variables and solving the resulting differential equations by series resulting in Spherical Harmonics and Legendre polynomials all stuff which should form the core of a third level mathematics course in mathematical methods but is hardly mentioned in this course.

Still as it hasn't then I'll just have to rely on the example sheets from Cambridge to fill the gaps.

http://www.damtp.cam.ac.uk/user/examples/B8b.pdf

Hopefully Block 3 on the calculus of variations and Lagranges equations will be a bit more exciting