- #1
davidbenari
- 466
- 18
Homework Statement
The temperature distribution of a circ plate with radius ##a##, ##f(r,\theta,t)##, follows the diffusion equation ##\nabla^2f=\frac{1}{\alpha^2}\frac{\partial f}{\partial t}##, with a temperature of zero along the border. The initial temperature is ##f(r,\theta,0)=100rcos\theta##. Find the temperature distribution.
Homework Equations
The Attempt at a Solution
I've found the typical solution that is ##\sum_{m=1}^{\infty} \sum_{n=0}^{\infty} J_n (\frac{k_{mn} r}{a})(A_{mn}cosn\theta+B_{mn}sinn\theta)exp(\frac{-\alpha^2}{a^2} k_{mn} t)## Where ##k_{mn}## is the mth zero of the nth bessel function.
Now I've found this at the initial time. ##\sum_{m=1}^{\infty} \sum_{n=0}^{\infty} J_n (\frac{k_{mn} r}{a})(A_{mn}cosn\theta+B_{mn}sinn\theta)=f(r,\theta,0)=100rcos\theta## To use an approach similar to finding a Fourier series. But I'm stuck here. I haven't a clue what to do here. Integrals are way to tedious here, and I don't see any way I'm getting a simple result.
Any ideas?
thanks
Last edited: