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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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    Looks linear (if $\phi_1$ and $\phi_2$ are solutions so is $\alpha \phi_1+ \beta \phi_2$)2011-06-14
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    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.2011-06-14
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    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 $\phi \equiv 0$.2011-06-14
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    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.2011-06-15

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