I'm having trouble solving an integral equation. It appears to me to be a homogeneous Fredholm equation of the second kind. However, I'm being told that this can't be a Fredholm equation, because it is non-linear. Could someone help me in trying to figure out how to classify an integral equation as linear or non-linear. Also, I'll post the equation I need to solve below, and it would be great if anyone could also give me some tips on how to try and solve it. Thank you to all who reply.

The equation is

$$\phi(x) = (x^2 - x)\int_0^1 \mathrm{d}y \frac{\phi(y)}{(y-x)^2}$$

Also, is this by chance related to an eigenvalue problem? I know that might sound like a strange question, but I've seen some people treating these as eigenvalue equations.

By the way, I want to solve the equation for $\phi(x)$

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    $\begingroup$ Looks linear (if $\phi_1$ and $\phi_2$ are solutions so is $\alpha \phi_1+ \beta \phi_2$) $\endgroup$ – Fabian Jun 14 '11 at 22:24
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    $\begingroup$ Are you sure there is a solution? A homogenous Fredholm equation of the second kind has the form $\phi = \lambda K \star \phi$ and typically allows only a solution for certain $\lambda$ called eigenvalues. Can you give more insight as to where this equation arises from. $\endgroup$ – Fabian Jun 14 '11 at 22:42
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    $\begingroup$ On further thought it looks like your integral equation has a rather serious divergence problem for $y=x$. Assuming $\phi(x)$ to be finite leads then to a contradiction (left hand side finite, right hand side infinite -> except possibly for $x=0,1$). Thus, it seems that the only solution is $\phi \equiv 0$. $\endgroup$ – Fabian Jun 14 '11 at 22:52
  • $\begingroup$ Thanks for the feedback. Yes, it is supposed to be a given that $\phi(0) = 0$ and $\phi(1) = 0$. For some more information on where this equation comes from. It comes from an application of the 't Hooft wave equation. I'm trying to solve for the wave function of a pion (where the mass goes to zero). The 't Hooft equation then reduces to the equation above. I don't believe there is any analytical solutions to it, but I'm hoping to find at least a numerical solution to it. $\endgroup$ – adhanlon Jun 15 '11 at 1:40

This is homogenous Fredholm integral equation of the second kind. It certainly is linear in the function $\phi$, as already observed by Fabian (see comments).

An important observation is that the kernel has a singularity in the integration domain, for $0 \le x \le 1$, which makes the equation a singular Fredholm integral equation of the second kind.

I don't know how to solve your equation, but here is a reference that has a chapter dedicated to singular Fredholm integral equations of both the first and second kind:

  • David Porter, David Stirling: "Integral Equations. A practical treatment, from spectral theory to applications".

See chapter 9, "Some singular integral equations".

I'm not quite sure, but I think they don't have examples with a quadratically diverget kernel, but it may be useful to read that chapter nevertheless.

  • $\begingroup$ Thanks for you feedback, I'll check out that book. $\endgroup$ – adhanlon Jun 15 '11 at 17:12

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