Solving a partial differential equation (Helmholtz equation)

In summary, the latex control on the forum website caused some errors in the equation, which was solved by attaching the details of the problem and what was done so far.
  • #1
Repetit
128
2
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:
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  • #2
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
Perfect, thanks a lot! :)
 
  • #4
Repetit said:
Perfect, thanks a lot! :)
Of course, now you have to know how to solve the radial equation :wink:
 
  • #5
J77 said:
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
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
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.
 

Attachments

  • transient equationdoc.doc
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1. What is a partial differential equation (PDE)?

A PDE is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe complex systems in physics, engineering, and other scientific fields.

2. What is the Helmholtz equation?

The Helmholtz equation is a specific type of PDE that is used to model wave propagation in various physical systems, such as electromagnetic waves and acoustic waves. It is named after German physicist Hermann von Helmholtz.

3. How do you solve a Helmholtz equation?

There are various methods for solving a Helmholtz equation, depending on the specific problem and boundary conditions. Some common techniques include separation of variables, Fourier transform, and numerical methods such as finite difference or finite element methods.

4. What are the applications of solving a Helmholtz equation?

Solving a Helmholtz equation is useful in many fields, such as acoustics, electromagnetics, and fluid dynamics. It can help predict the behavior of waves in different physical systems and aid in the design and optimization of devices and structures.

5. Are there any challenges in solving a Helmholtz equation?

Yes, there are several challenges in solving a Helmholtz equation. One of the main challenges is dealing with the complex boundary conditions and geometry of the problem. Another challenge is the computational cost, as the size of the problem increases, the time and resources needed for solving it also increase.

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