Solution of heat equation

Question

I have been asked to prove the below function satisfies heat equation.

u(r,t)=2Uπ∑n=1∞ {(−1)^n+1}(1/r)sin(nπr)e^−n^2π^2t

I have tried to equate ∂u/∂t = ∂^2u/∂x^2 however both let hand side and right hand side are not tallying.

Can you please help. Is this the correct function?

Answer 1

Hi Rebeca,
I think I see your problem, you have written the heat conduction (partial differential) equation in rectangular coordinates but the function u(r,t) is in cylindrical or polar coordinates. The equation you should be using is:

∂u/∂t=∂²u/∂r² + 1/r(∂u/∂r) if you test u(r,t) using this equation I think you will have a better result. Just test a general term in the series if that works then the equation will be satisfied for all terms.
Hope this helps, let me know if you have any questions.

Jim