Volterra integro-differential equation

Summary

A Volterra integro-differential equation is an integro-differential equation in which the integral runs only over the past. The integral goes from a fixed initial time up to the current time. So the rate of change at time $t$ depends on the present state and on the history up to $t$, not on the future. Together with an initial condition it is equivalent to a Volterra integral equation, and under Lipschitz conditions it has a unique solution. Such equations model materials with memory, such as viscoelastic bodies, and populations whose growth depends on past sizes.