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I'm trying to solve with Mathematica an integral equation. I found this excellent answer (How to solve a non-linear integral equation?) solving with a collocation method a problem which can be restated as:

$$ \int_a^b f \left( \phi \left(x \right) \right) \mathrm{d} x= 1 $$

for some function $f$, $\phi(x)$ being the unknown. My problem is slightly more complicated:

$$ \phi(x) = \int_a^b K \left( x, y \right) f \left( \phi \left( y \right) \right) \mathrm{d} y $$

and the kernel $K(x,y)$ is singular for $x=y$, it behaves as $\sim \frac{1}{\left| x - y \right|}$. I would like to know if it is possible to extend the collocation method to this case, or, alternatively, which other methods can be used to numerically solve my integral equation.


Some more details:

$$ f(\phi(x)) = \frac{\phi(x)}{\sqrt{\phi^2(x) + C^2}} $$ $$ K(x,y) = y \frac{\mathrm{e}^{-\left| x - y \right|}}{\left| x - y \right|} $$

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closed as off-topic by Louis, MarcoB, Yves Klett, Jens, m_goldberg Feb 19 at 0:57

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – MarcoB, Jens
If this question can be reworded to fit the rules in the help center, please edit the question.

I do not have an answer offhand (would require, at minimum, a specific K,f pair), but here are some links that might contain helpful approaches. Check Moe, Larry, and Curly (I really need some pointers on naming my links). – Daniel Lichtblau Feb 28 '14 at 16:32
I edited the question to include some more details! Thank you for your links! – zakk Feb 28 '14 at 16:47
At a glance I'm having a hard time seeing how that integral could possibly converge for a<x<b. Assuming i'm wrong and it does converge your challenge is to develop a quadrature rule that is accurate at the singular point. Have fun..! (I think this really is a math.stackexchange question in any case) (BTW The right most "=1" is a typo right..?) – george2079 Feb 28 '14 at 19:53
@george2079: it should converge, at least it is eq. 5 in this paper: The authors refer to an unpublished paper for the numerical part, commenting as follows: letting $x=\tan(\beta)$, we set up a Gaussian-quadrature grid for $\beta$ and convert the above equations into a matrix form which can be solved iteratively. The logarithmic singularities are treated separately. – zakk Mar 1 '14 at 1:54
Will this ultimately become a Mathematica question? Shouldn't the OP include Mathematica code defining the relevant functions and of the attempts so far? Some focus on the programmatic tools that allow the integral to be resolved? This is already been nominated to close on ground of "unclear what you're asking". – rhermans Feb 18 at 9:21