Partial Differentiation Problem

[Lucky mkhonza]

Hi to all,

I have been given the following problem as an assignment.

\frac{\partial ^2 \phi}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial \phi}{\partial \rho} + \frac{1}{\rho^2}\frac{\partial \phi}{\partial \chi^2} + \frac{\partial ^2 \phi}{\partial Z^2}+B^2\phi = 0

Here is my attempt to the problem:
Assuming \phi = S(\rho,\chi)Z(z)

\frac{1}{S(\rho,\chi)}\frac{\partial ^2 S(\rho,\chi)}{\partial \rho^2} + \frac{1}{S(\rho,\chi) \rho }\frac{\partial S(\rho,\chi)}{\partial \rho} + \frac{1}{S(\rho,\chi) \rho^2}\frac{\partial S(\rho,\chi)}{\partial \chi^2} + \frac {1}{Z}\frac{\partial ^2 Z}{\partial Z^2} + B^2 = 0

Separating the variables we get

\frac{\partial ^2 z}{\partial Z^2} + B^2 Z = 0

\frac{1}{S(\rho,\chi)}\frac{\partial ^2 S(\rho,\chi)}{\partial \rho^2} + \frac{1}{S(\rho,\chi) \rho }\frac{\partial S(\rho,\chi)}{\partial \rho} + \frac{1}{S(\rho,\chi) \rho^2}\frac{\partial S(\rho,\chi)}{\partial \chi^2} + B^2 = 0

Assuming S(\rho, \chi) = \rho(\rho) \chi(\chi)

\frac{1}{\rho}\frac{\partial ^2 \rho}{\partial \rho^2} + \frac{1}{\rho^2 }\frac{\partial \rho}{\partial \rho} + \frac{1}{\rho^2 \chi}\frac{\partial ^2 \chi}{\partial \chi^2} + B^2 = 0

How can I solve this last PDE?

Thank you in advance

 

[Hyperreality]

Okay, I haven't actually tried the problem, but you could try first to classify the pde (hyperbolic, elliptic, parabolic). In this case, the class of the pde depends on your choice of \rho. Have you tried it?

 

[Dr Transport]

if you multiply by \rho^{2} you'll be able to separate the variables completely.

 

[Lucky mkhonza]

[Hyperreality] Okay, I haven't actually tried the problem, but you could try first to classify the pde (hyperbolic, elliptic, parabolic). In this case, the class of the pde depends on your choice of \rho
. Have you tried it? [/Q]

I don't know as to which class the PDE falls to. Let me state the whole problem so that it becomes clear to everyone.
Solve the following

\frac{\partial ^2 \phi}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial \phi}{\partial \rho} + \frac{1}{\rho^2}\frac{\partial ^2 \phi}{\partial \chi^2} + \frac{\partial ^2 \phi}{\partial Z^2}+B^2\phi = 0

Where: 0 < \rho < R, 0 < \chi < \pi, -\frac{H}{2} &lt; z &lt; \frac{H}{2}

The Boundary Conditions are

\phi(R,\chi,z) = 0
\phi(\rho,0,z) = \phi(\rho,\pi,z) = 0
\phi(\rho,\chi, \pm \frac{H}{2}) = 0


[Dr Transport] if you multiply by \rho^{2} you'll be able to separate the variables completely. [/Q]

As you suggested to multiply the last PDE by \rho^{2}, when separating the PDE's involving both \rho and \chi I still have \rho^{2} on one of the PDE's involving \chi. See below

\frac{1}{\chi}\frac{\partial ^2 \chi}{\partial \chi^2} + B^2 \rho^2 = 0

And

\rho \frac{\partial ^2 \rho}{\partial \rho^2} + \frac{\partial \rho}{\partial \rho} + B^2 \rho^2 = 0