Approximations for a class of volterra integro-differential equations

We compare an internal state method and a direct Runge-Kutta method for solving Volterra integro-differential equations and Volterra delay differential equations. The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. Numerical examples are presented to illustrate the methods and in the cases where both methods are applicable comparisions of the results are included.