Integral equation

[prometheus]

Are there any standard techniques to solve equations like this?:

\int d \omega K(\omega) g(\omega ,t) = J(t)

where K is the only unknown function. In particular, I want to solve an equation that looks like

\int d \omega K(\omega) \frac{\partial}{\partial t} P_l^{i \omega} (\tanh t) = J(t)

where P_l^{i \omega} is a Legendre function and J is a complicated mess of Legendre functions. What I have so far is, if you expand the derivative for t \to \infty and \omega \to 0 with \omega t fixed we get something like

\int d \omega K(\omega) i e^{i \omega t} = J(t)
i \tilde{K}(t) = J(t)

I don't like this because expanding inside the integral doesn't feel good. Any help or suggestions of books on the subject will be greatly appreciated. :)

[Guest254]

Fredholm theory covers your bases.

I'm afraid I don't understand your comments about "expanding derivatives" - perhaps you could elaborate?

[prometheus]

All I mean by expanding the derivative (perhaps a crap way of putting it) is I rewrite \tanh t = \frac{1-e^{-2t}}{1+e^{-2t}}=\frac{1-x}{1+x} . The limit t \to \infty is now x \to 0 and \frac{d}{dt} P_l^{i \omega} (\tanh t) = -2 x \frac{d}{dx}P_l^{i \omega} (\frac{1-x}{1+x}). This can now be expanded for small x using Taylor's theorem. I'm led to believe that doing this inside an integral is a bad thing. Can you recommend a good beginners book on Fredholm theory?

[Guest254]

Sorry, I was unclear - I meant I don't understand what you mean by the sentence starting with "expanding derivatives...". How can you send \omega\rightarrow 0 -- you're integrating over all \omega!

In general though, if you're integrating an analytic function, then you're absolutely fine to form a power series and integrate termwise as long as your integral stays within the radius of convergence. This is because power series converge uniformly on any compact subset of their domain of convergence.

[Guest254]

Regarding books. I guess you're more interested in the answer (if it exists), rather than the mathematics that proves that there exists an answer, or that there exists a certain answer in a certain function class. All the references I know of are from the functional-analytic standpoint, so might not be of much use!

If you are interested in the more mathsy stuff, then I can probably provide some references.

[prometheus]

Regarding \omega going to 0, the physics here is that this is an equation for a wave. \omega and t are conjugate variables so the physical statement is that taking the limits we are taking corresponds to looking at waves that are of low frequency and long wavelength. This is ok for physics, maybe not for maths ;)

I'm more interested in the answer than how to get the answer (if it exists) but I would like to know how it works. Is there a book that you know of that is entitled something like "Fredholm theory for physicists?"

Cheers for your help. :)

[Guest254]

Sadly, I don't have too much experience with the more hands-on books. However, there seems to be a fair few out there (http://www.amazon.co.uk/s/ref=nb_ss_w_h_/026-8313784-3443658?url=search-alias%3Daps&field-keywords=integral+equations&x=0&y=0). I'd have thought the library would be full of them - possibly in the engineering section for the more relevant texts.

:)

[BenTheMan]

Did you already check Arfken and Weber?

I'm led to believe that doing this inside an integral is a bad thing. Can you recommend a good beginners book on Fredholm theory?

What makes you think this is a bad thing?

[prometheus]

I think the reason is the one that guest has touched on - we're supposed to be integrating over all \omega but we're taking \omega to be small. On physical grounds we're ok, because using this approximation gives the correct result that we've calculated in a different way, but I'm trying to be a bit more rigorous. Cheers for the reference. I'll see if we've got a copy in the library. :)

[temur]

Fredholm theory is about how to show that this equation has a solution. If you want to (at least) approximately compute the solution, you can expand the solution in some basis (like Fourier, various polynomials, Haar etc), and then truncate the expansion.