MST324 TMA02 (Nearly finished my pudding)

Well after finally eating my meat for M338 I've nearly finished my pudding for MST324. What a joy (says he optimistically) The questions were on the wave equation and Fourier transforms.

Question 1 Involved using D'Alembert's solution to the wave equation to solve a particular boundary problem.

Question 2 (The best)  involved solving the inhomogeneous damped wave equation for a particular function using seperation of variables. This is a showcase question consolidating all the techniques we have covered so far. The steps are

1) Write the function u(x,t)  as a product  of two functions f(x)g(t)
2) Substitute these into the partial differential equation and this gives two separate ordinary differential equations in both f(x) and g(t) in terms of a constant .
3) Solve the homeogeneous  differential equations for f(x) and g(t) using the appropriate boundary conditons
4) Use the inhomogeneous function on the RHS of the partial differential equation to obtain particular integrals for f(x) and g(t).
5) The resulting solution is then u(x,t) = f(x)g(t)

This is maths at it's best, some of the steps might be a bit tricky but I think it's really cool how it all hangs together.

Question 3 Another question on a solution of a partial differential equation by separation of variables. This was another nice question and a bit simpler than question 2

Question 4 Testing your knowledge of the properties of Fourier transforms based around a question involving the recurrence relationships of Hermite Polynomials. This was OK but a bit tricky in parts and I still have to do the last bit.

So overall I'm quite pleased and this TMA has been a joy to do, there was none of the head scratching associated with M338, but I can see the last bit of ice cream in the tub and it will soon be back to struggling with the many definitions of continuity for Topological spaces and so forth. Anyway it looks like I'm not the only one to have struggled with the TMA for M338. According to my friend Duncan who is doing M208 this year, Alan my tutor for M338 has said that it is taking him 2 hours per question per person to mark the scripts. So if the tutors find it difficult it's not surprising that we do.