Let $h(x)$ be a known well-behaved function, I have to solve for $\sigma(t)$: $$ \phi(x) = \int_a^b\log\left[\left(x-t\right)^2 + \left(h(x) - h(t)\right)^2\right]\sigma(t)dt $$
Where, $b>a>0$, and $\phi(x)$ is known. This is a Fredholm equation of first kind with the kernel: $$ K(x, t) = \log\left[\left(x-t\right)^2 + \left(h(x) - h(t)\right)^2\right] $$
How to do it? Any ideas? Any hints? Anything?
Given the kernel is symmetric, $K(x, t) = K(t, x)$, there exists an orthogonal set of eigenfunctions, in which the solution can be easily found from these eigenfunctions. But, I couldn't find the eigenfunctions itself. Is there a general method to find eigenfunctions for symmetric kernels?
If the Kernel is a displacement kernel, that is, $K(x, t) = K(x-t)$, then this can be easily solved (with laplace/fourier transform, and convolution). But I was unable to transform $K$ into a displacement, by variable substitution, by adding stuff, or multiplying and dividing by things.
I've tried to transform the eigenfunction integral equation into a differential equation, to see what kind of ODE the eigenfunctions solves. But I couldn't find an ODE. By the way, if you can do this, please answer.
I've even tried to transform the whole equation into an ODE, but I couldn't as well. If you can do that too, please answer.
I've tried expanding the logarithm into a series, but wasn't useful.