Integral equation of the third kind

Summary

An integral equation of the third kind is a linear integral equation $a_0(x) \varphi(x) = f(x) + \lambda \int K(x, s) \varphi(s) \, ds$ whose coefficient $a_0$ vanishes at some points but not everywhere. It sits between the first kind, where $a_0 = 0$, and the second kind, where $a_0 = 1$. Near the zeros of $a_0$ the equation behaves like one of the first kind, so its solutions may be singular there. Such equations occur in transport theory and in some problems of mathematical physics.