How Can You Solve the Helmholtz Differential Equation?

[Sparky_]

Greetings,

I have recently come across the Helmholtz differential equation.

Is there already a thread here that discusses the mathematic techniques (differential equation techniques) used to solve it?

If not can we discuss how one can solve the equation?

I'm familiar but rusty with linear differential equations with one variable and Laplace transforms but I've been out of school for some time.

I'm wanting the steps used to solve this equation. I do recall from school - things like solving for the roots of the auxillary equation and based upon the roots build the exponential solution - (again from a rusty memory)

Curious (and thanks)
-Sparky_

 

[Chris Hillman]

Suggest some resources

Just to be clear, are we talking about
$$\Delta u + k \, u = \Phi$$
where $\Delta$ is the Laplace operator? If so, almost all books on potential theory or "mathematical methods" discuss methods of solving this equation. You might also try http://eqworld.ipmnet.ru

 

[waht]

Are you familiar with the residue theorem?

 

[Sparky_]

what and Chris,

Unfortunately the residue theorem is familiar in name only. I do recall using it in school (14 years ago now).

I can dig out and search the internet on the subject.

I got out my differential equations book but did not see techniques for the Helmholtz equation.

I can find the solutions for the equation, I just can't find the technique used to solve the equation and obtain these solutions.

Chris - yes that's the equation I'm interested in except I think the Laplacian is a 2nd order.

Any insights on the solution approach?

Thanks again
-Sparky_

 

[Office_Shredder]

Sparky, pointing upwards triangle is just grad squared, so it is second order ;)

(sorry, no idea how to do those symbols in latex, and didn't find it worth the effort to look them up for the post)

 

[waht]

The Helmholtz PDE can be easily solved, and also in what coordinate system?

$$\nabla^2 \phi + k^2 \phi = 0$$I recommend a book "Partial Differential Equations with Fourier and Boundary Value Problems" by Nakhle Asmar. You will find your solutions there. This book is a goldmine.But there is another equation I was talking about,

$$\nabla^2 \phi + k^2 \phi = \delta (r)$$

where you would have to the residue theorem to solve.