# Solving a partial differential equation (Helmholtz equation)

1. Apr 19, 2007

### Repetit

Hey!

I am trying to solve this quite nasty (as least I think so : - ) partial differential equation (the Helmholtz equation):

$$\frac{1}{r}\frac{\partial}{\partial r} \left( r \frac{\partial\Psi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \Psi}{\partial \phi^2} + \frac{\partial^2 \Psi}{\partial z^2} + m^2 k^2 \Psi = 0$$

I use separation of variables and write the unkown function $$\Psi(r,\phi,z)$$ as $$\Psi(r,\phi,z) = R(r)\Phi(\phi)Z(z)$$, insert this in the equation and divide by $$R(r)\Phi(\phi)Z(z)$$. This gives me:

$$\frac{1}{r R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) + \frac{1}{r^2 \Phi} \frac{d^2 \Phi}{d \phi^2} + \frac{1}{Z}\frac{d^2 Z}{d z^2} + m^2 k^2 = 0$$

Now, I am not sure how to move on from here because I have 1/r^2 in the $$\Phi$$ term so that I cannot use the usual procedures for solving PDE (equating one term to minus the other terms and setting both equal to some constant). Could someone give me a hint on how to proceed from here?

Last edited: Apr 19, 2007
2. Apr 19, 2007

### HallsofIvy

Staff Emeritus
Do two separations. You have
$$\frac{1}{r R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) + \frac{1}{r^2 \Phi} \frac{d^2 \Phi}{d \phi^2}= \alpha$$
and
$$\frac{1}{Z}\frac{d^2 Z}{d z^2} + m^2 k^2 =-\alpha$$

Now multiply that first equation by r2 to get
$$\frac{r}{ R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) + \frac{1}{ \Phi} \frac{d^2 \Phi}{d \phi^2} = r^2\alpha$$
or
$$\frac{r}{ R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) - r^2\alpha+ \frac{1}{ \Phi} \frac{d^2 \Phi}{d \phi^2} = 0$$
so that
$$\frac{r}{ R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) - r^2\alpha= \beta$$
and
$$\frac{1}{ \Phi} \frac{d^2 \Phi}{d \phi^2} = -\beta$$

3. Apr 20, 2007

### Repetit

Perfect, thanks a lot!! :)

4. Apr 20, 2007

### J77

Of course, now you have to know how to solve the radial equation

5. Apr 20, 2007

### Repetit

Yes that true :) But the radial equation can be rewritten quite easily into Bessels differential equation.

6. Jan 7, 2010

### bhavaji

What if k was not a constant but a function of r and z? How does one proceed now?

7. May 31, 2010

### saurabh5749

Can anybody help me in solving this equation in MATLAB ?? Reply soon...
∂(ΔΨ) /∂t- ∂Ψ/∂x. ∂(ΔΨ)/∂y + ∂Ψ/∂y. ∂(ΔΨ)/∂x = 0

where Ψ = Stream Function
Δ = ∇^2 (laplacian Operator)

8. Aug 19, 2010

### daz71

Solving a transient partial differential equation

hello all,

Could some one help me on this transient heat conduction equation, i had problem with the latex control on the forum website, so i attached the details of the problem and what i did so far as attachement.
thanks.

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