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A standard approach to solve ordinary differential equations, when theydescribe dynamical systems, is to adopt a Runge-Kutta or related scheme. Suchschemes, however, are not applicable to the large class of equations which donot constitute dynamical systems. In several physical systems, we encounterintegro-differential equations with memory terms where the time derivative of astate variable at a given time depends on all past states of the system.Secondly, there are equations whose solutions do not have well-defined Taylorseries expansion. The Maxey-Riley-Gatignol equation, which describes thedynamics of an inertial particle in nonuniform and unsteady flow, displays bothchallenges. We use it as a test bed to address the questions we raise, but ourmethod may be applied to all equations of this class. We show that theMaxey-Riley-Gatignol equation can be embedded into an extended Markovian systemwhich is constructed by introducing a new dynamical co-evolving state variablethat encodes memory of past states. We develop a Runge-Kutta algorithm for theresultant Markovian system. The form of the kernels involved in deriving theRunge-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 effortwith simulation time, and the ability to restart a simulation with the finalstate as the new initial condition.

Explicit Runge-Kutta algorithm to solve non-local equations with memory effects: case of the Maxey-Riley-Gatignol equation