- #1
yungman
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Homework Statement
This is a question in the book to solve Heat Problem
[tex]\frac{\partial \;u}{\partial\; t}=\frac{\partial^2 u}{\partial\; r^2}+\frac{1}{r}\frac{\partial\; u}{\partial\;r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}[/tex]
With 0<r<1, [itex]0<\theta<2\pi[/itex], t>0. And [itex]u(1,\theta,t)=\sin(3\theta),\;u(r,\theta,0)=0[/itex]
The solution manual gave this which I don't agree:
165824[/ATTACH]"]
What the solution manual did is for [itex]u_1[/itex], it has to assume [itex]\frac{\partial \;u}{\partial\; t}=0[/itex] in order using Dirichlet problem to get (1a) shown in the scanned note.
I disagree.
Homework Equations
I think it should use the complete solution shown in (2a), then let t=0 where
[tex]u_{1}(r,\theta,0)=\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}J_{m}(\lambda_{mn}r)[a_{mn}\cos (m\theta)+b_{mn}\sin (m\theta)][/tex]I don't agree with the first part, you cannot assume [itex]\frac{\partial u}{\partial t}=0[/itex]. Please explain to me.
Thanks