  A standard approach to solve ordinary differential equations, when they
describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such
schemes, however, are not applicable to the large class of equations which do
not constitute dynamical systems. In several physical systems, we encounter
integro-differential equations with memory terms where the time derivative of a
state variable at a given time depends on all past states of the system.
Secondly, there are equations whose solutions do not have well-defined Taylor
series expansion. The Maxey-Riley-Gatignol equation, which describes the
dynamics of an inertial particle in nonuniform and unsteady flow, displays both
challenges. We use it as a test bed to address the questions we raise, but our
method may be applied to all equations of this class. We show that the
Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system
which is constructed by introducing a new dynamical co-evolving state variable
that encodes memory of past states. We develop a Runge-Kutta algorithm for the
resultant Markovian system. The form of the kernels involved in deriving the
Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$.
Our approach naturally inherits the benefits of standard time-integrators,
namely a constant memory storage cost, a linear growth of operational effort
with simulation time, and the ability to restart a simulation with the final
state as the new initial condition.
