- #1
yungman
- 5,708
- 240
[tex]\frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{r}^2}+ \frac {2}{r} \frac {\partial{u}}{\partial{r}}+\frac{1}{r^2}\left[\frac{\partial^2{u}}{\partial{\theta^2}}+\cot\theta \frac{\partial{u}}{\partial {\theta}} +\csc\theta\frac{\partial^2{u}}{\partial{\phi}^2}\right]+q(r,\theta,t)[/tex]
Where [itex]0<r<a,\;0<\theta<\pi,\;0<\phi<2\pi,\;t>0[/itex]
with Boundary condition [itex]u(a,\theta,\phi,t)=0[/itex] and initial condition [itex]u(r,\theta,\phi,0)=f(r,\theta,\phi)[/itex].
I understand how to get to
[tex]u(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} B_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}[/tex]
What I don't understand is the next step, the book assume
[tex]g(r,\theta,\phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} g_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}[/tex]
Where [itex]g_{jnm}[/itex] is another constant.
How do you justify to assume [itex]g(r,\theta,\phi)[/itex]?
Where [itex]0<r<a,\;0<\theta<\pi,\;0<\phi<2\pi,\;t>0[/itex]
with Boundary condition [itex]u(a,\theta,\phi,t)=0[/itex] and initial condition [itex]u(r,\theta,\phi,0)=f(r,\theta,\phi)[/itex].
I understand how to get to
[tex]u(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} B_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}[/tex]
What I don't understand is the next step, the book assume
[tex]g(r,\theta,\phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} g_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}[/tex]
Where [itex]g_{jnm}[/itex] is another constant.
How do you justify to assume [itex]g(r,\theta,\phi)[/itex]?