Volterra equation of the first kind

Summary

A Volterra equation of the first kind is an equation $f(t) = \int_a^t K(t, s) \, x(s) \, ds$ in which the unknown function $x$ appears only under the integral. It is the Volterra integral equation analogue of an integral equation of the first kind. Solving it is like inverting an integration, so it behaves like differentiation and is in general ill-posed. When the kernel is smooth and does not vanish on the diagonal, it can be reduced to an equation of the second kind.