PDEs and Fourier transforms - is this problem too difficult?

[joriarty]

I have an unusual question, though hopefully someone here can answer it. Apologies if this belongs in the homework forums, not really sure where to put it, as I'm not asking for help with the problems here. I'm currently in the second half of a 12-week third-year University course on PDEs. I have a new lecturer for this half of the course and I think that the homework he is setting is far too difficult for this level of course. Do you think it's too much to expect someone who has only studied PDEs for 8 weeks (at 3 contact hours per week) to be able to solve these questions? (attached)

We have only moved on to non-homogeneous problems two weeks ago and since then I have become almost totally lost :(

Thanks!

 

[Mute]

The questions you posted seem to be of appropriate difficulty for a third year course. They all seem relatively straightforward, once you've figured out what the problem wants. You probably have the skill to solve all these problems, the trick is perhaps figuring out what the problem wants and what to do to solve them. Perhaps you should be specific as to what difficulties you are having with them.

For the first one, for example, I would be surprised if you were expected to actually figure out \alpha from first principles rigorously. I would think you're just supposed to recognize it as a bell curve, so you need a factor of \sqrt{\beta/\pi} to normalize it.

The rest of the questions use the idea that the dirac delta function can be thought of as the limit of a sharply peaked function that gets sharper and sharper as a parameter (beta in this case) tends to infinity (or zero depending on the parameter). Has the professor discussed this in class?

 

[joriarty]

Voila, that bell curve normalisation works! Obvious answer once I read the Wikipedia article on the Gaussian integral (which I had long since forgotten if I had ever learned about it before)

He hasn't talked much about the dirac delta function in class (partly why me and my mates are having such difficulty), though if the problems do seem an appropriate level I think I just need to go and live in the library for a few days with a textbook or two.

Thanks for your help :)

 

[Mute]

For a quick read, see the representations of the delta function section in the wikipedia article:

http://en.wikipedia.org/wiki/Nascent_delta_function#Representations_of_the_delta_function

The basic idea of the problem set is that you're using the Gaussian to approximate the delta function. The problem set is trying to get you to show that using the gaussian approximation you can get the usual defined properties of the delta function. The calculations themselves mostly just involve doing some integrals.