Solving a partial differential equation (Helmholtz equation)

  • Thread starter Repetit
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  • #1
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Main Question or Discussion Point

Hey!

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

[tex]
\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
[/tex]

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

[tex]
\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
[/tex]

Now, I am not sure how to move on from here because I have 1/r^2 in the [tex]\Phi[/tex] 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?

Thanks in advance
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
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Do two separations. You have
[tex]\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[/tex]
and
[tex]\frac{1}{Z}\frac{d^2 Z}{d z^2} + m^2 k^2 =-\alpha[/tex]

Now multiply that first equation by r2 to get
[tex]\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[/tex]
or
[tex]\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[/tex]
so that
[tex]\frac{r}{ R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) - r^2\alpha= \beta[/tex]
and
[tex]\frac{1}{ \Phi} \frac{d^2 \Phi}{d \phi^2} = -\beta[/tex]
 
  • #3
128
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Perfect, thanks a lot!! :)
 
  • #4
J77
1,076
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Perfect, thanks a lot!! :)
Of course, now you have to know how to solve the radial equation :wink:
 
  • #5
128
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Of course, now you have to know how to solve the radial equation :wink:
Yes that true :) But the radial equation can be rewritten quite easily into Bessels differential equation.
 
  • #6
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What if k was not a constant but a function of r and z? How does one proceed now?
 
  • #7
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
13
0
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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