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 < $$\chi$$ < $$\pi$$, $$-\frac{H}{2} < z < \frac{H}{2}$$

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

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