Volterra integral equation

Summary

A Volterra integral equation is an integral equation in which the integral runs from a fixed point up to the variable $t$. The typical form of the second kind is $x(t) = f(t) + \int_a^t K(t, s) x(s) \, ds$. Every initial value problem for an ordinary differential equation can be written as a Volterra equation of the second kind, in general a nonlinear one. Under mild conditions, the linear equation of the second kind has a unique solution, given by a convergent series of iterated kernels.