Workbook guide +problem in Math. Phys.

[blade86]

Hey I have a degree robotics and moved to theoretical physics so am still struggling to keep up with the mathematics. I was wondering if there are any good workbooks out there where I can just practice a lot of problems, mainly (special functions: Bessel, Legendre, Laplace, Integral transforms, ODEs PDEs and complex variables)?
Also some help with the following problem would be helpful:

If a cube of side length a originally at temp. T$_{0}$ is placed in a reservoir at temp. T=0 K, show the subsequent temp. is :

$T(x,t)$ = $T_0$ $\Sigma_{l,m,n}$ $64 \over lmn \pi^3$ $sin[ {{l \pi} \over{a}} x]$ $sin[ {{m \pi} \over{a}} y]$ $sin[ {{n \pi} \over{a}} z]$ $e^{-(l^2 + m^2 + n^2)({\pi \over a})^2 \kappa t}$

where $\kappa$ is the heat conductivity.

Any help or guidance on how to approach the solution would be much appreciated.

 

[Mandelbroth]

Hey I have a degree robotics and moved to theoretical physics so am still struggling to keep up with the mathematics. I was wondering if there are any good workbooks out there where I can just practice a lot of problems, mainly (special functions: Bessel, Legendre, Laplace, Integral transforms, ODEs PDEs and complex variables)?
Also some help with the following problem would be helpful:

If a cube of side length a originally at temp. T$_{0}$ is placed in a reservoir at temp. T=0 K, show the subsequent temp. is :

$T(x,t)$ = $T_0$ $\Sigma_{l,m,n}$ $64 \over lmn \pi^3$ $sin[ {{l \pi} \over{a}} x]$ $sin[ {{m \pi} \over{a}} y]$ $sin[ {{n \pi} \over{a}} z]$ $e^{-(l^2 + m^2 + n^2)({\pi \over a})^2 \kappa t}$

where $\kappa$ is the heat conductivity.

Any help or guidance on how to approach the solution would be much appreciated.

I'm not really a physics guy, but my initial reaction is to start with the heat equation.


$$\frac{\partial T}{\partial t}-\frac{\kappa}{\rho c_p}\nabla^2T=0.$$

As an aside, am I the only one here who sees that mess and automatically imagines Steve Irwin yelling "Crikey! Look at the size of that thing!" ? :rofl:

Edit: If you really want to get good at the math, what I've done for practice is stalk the forums and try to answer whatever questions I deem worth answering. It's actually rather effective.

 

[jasonRF]

Are you just looking for a bunch of good problems with some answers to check your work? Many university classes post homework assignments and solutions. Try
ocw.mit.edu

jason

 

[blade86]

Thanks for the tips, yeah answering physics forums seems like a good idea. I tried the solution.. maybe its something like this (attached pdf). but not sure about the subsequent temperature.

 

[blade86]

@jasonRF yes indeed I am just looking to solve a lot of problems to become more fluent in the mathematics.