This is a question in the book to solve Heat Problem

$\displaystyle \frac{\partial \;u}{\partial\; t}=\frac{\partial^2 u}{\partial\; r^2}+\frac{1}{r}\frac{\partial\; u}{\partial\;r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}$

With 0<r<1, $\displaystyle 0<\theta<2\pi$, t>0. And $\displaystyle u(1,\theta,t)=\sin(3\theta),\;u(r,\theta,0)=0$

The solution manual gave this which I don't agree:

What the solution manual did is for $\displaystyle u_1$, it has to assume $\displaystyle \frac{\partial \;u}{\partial\; t}=0$ in order using Dirichlet problem to get (1a) shown in the scanned note.

I disagree.

I think it should use the complete solution shown in (2a), then let t=0 where

$\displaystyle u_{1}(r,\theta,0)=\sum_{m=0}^{\infty}\sum_{n=1}^{\ infty}J_{m}(\lambda_{mn}r)[a_{mn}\cos (m\theta)+b_{mn}\sin (m\theta)]$

I don't agree with the first part as I don't think you can assume $\displaystyle \frac{\partial u}{\partial t}=0$. Please explain to me.

Thanks