0
$\begingroup$

I'm solving the integral equation with unknown parameter in the kernel,

$g(t)=\int K(t,s,\sigma) f(s) ds$,

where $f(s)$ and $\sigma$ have to be defined. Without the parameter, it is the Fredholm problem of the first kind. But I could not find the name of the problem with the parameter. Thanks!

$\endgroup$
1
$\begingroup$

This would just be a family of Fredholm equations depending on a parameter. Things like this pop up with solutions to the Helmholtz equation when using integral equations, where the Helmholtz equation has a complex parameter $s$ indicating the wavenumber. The behavior of the problem may change depending on $s$, though. For example, if you're looking at the resolvent Laplace equations (forgoing boundary conditions), you have something like

$$ -\Delta u + su = 0 $$

which can be shown to be coercive for $s \neq 0$. But when $s=0$ or is an eigenvalue of the Laplacian, you lose coercivity, and so need a different approach.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.