- #1

Leonardo Machado

- 57

- 2

I am studying the heat equation in polar coordinates

$$

u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta})

$$

via separation of variables.

$$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$

which gives the ODEs

$$T''+k \lambda^2 T=0$$

$$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$

$$\Theta''+\mu^2\Theta=0$$

but i can't properly think about the boundary conditions to this problem. I see every where people resolving it with

$$

|u(0,\theta,t)|<\inf \mapsto |R(0)| < \inf

$$

and

$$u(r*,\theta,t)=0 \mapsto R(r*)=0$$

being r* the border of the disc.

But i understand the radial condition as a termal bath at zero temperature and i really want to change it for a finite value but i don't know how to procede without the zeros...

Any suggestions?