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.

翻译：求解描述动力系统的常微分方程时，标准方法是采用龙格-库塔或相关格式。然而，这类格式不适用于大量不构成动力系统的方程。在许多物理系统中，我们遇到带有记忆项的积分微分方程，其中某时刻状态变量的时间导数依赖于系统所有过去状态。其次，存在解不具有明确泰勒级数展开的方程。描述非均匀非定常流中惯性粒子动力学的Maxey-Riley-Gatignol方程同时呈现了这两类挑战。我们以其作为验证所提出问题的测试基准，但本方法可应用于此类所有方程。我们证明，Maxey-Riley-Gatignol方程可嵌入一个扩展马尔可夫系统，该系统通过引入编码过去状态记忆的动态共进化状态变量构建。我们为所得马尔可夫系统开发了龙格-库塔算法。推导龙格-库塔格式所涉及的核函数形式要求使用$t^{1/2}$的幂次展开。本方法自然继承了标准时间积分器的优势，即恒定内存存储成本、运算量随模拟时间线性增长，以及能够以最终状态作为新初始条件重新启动模拟。