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Solving a partial differential equation (Helmholtz equation)

  1. Apr 19, 2007 #1
    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: Apr 19, 2007
  2. jcsd
  3. Apr 19, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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]
     
  4. Apr 20, 2007 #3
    Perfect, thanks a lot!! :)
     
  5. Apr 20, 2007 #4

    J77

    User Avatar

    Of course, now you have to know how to solve the radial equation :wink:
     
  6. Apr 20, 2007 #5
    Yes that true :) But the radial equation can be rewritten quite easily into Bessels differential equation.
     
  7. Jan 7, 2010 #6
    What if k was not a constant but a function of r and z? How does one proceed now?
     
  8. May 31, 2010 #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)
     
  9. Aug 19, 2010 #8
    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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