Many physical systems are governed by ordinary or partial differential equations (see, for example, Chapter ''Differential equations'', ''System of Differential Equations''). Typically the solution of such systems are functions of time or of a single space variable (in the case of ODE's), or they depend on multidimensional space coordinates or on space and time (in the case of PDE's). In some cases, the solutions may depend on several time or space scales. An example governed by ODE's is the damped harmonic oscillator, in the two extreme cases of very small or very large damping, the cardiovascular system, where the thickness of the arteries and veins varies from centimeters to microns, shallow water equations, which are valid when water depth is small compared to typical wavelength of surface waves, and sorption kinetics, in which the range of interaction of a surfactant with an air bubble is much smaller than the size of the bubble itself. In all such cases a detailed simulation of the models which resolves all space or time scales is often inefficient or intractable, and usually even unnecessary to provide a reasonable description of the behavior of the system. In the Chapter ''Multiscale modeling with differential equations'' we present examples of systems described by ODE's and PDE's which are intrinsically multiscale, and illustrate how suitable modeling provide an effective way to capture the essential behavior of the solutions of such systems without resolving the small scales.

翻译：许多物理系统由常微分方程或偏微分方程描述（例如，参见“微分方程”章和“微分方程组”章）。通常，这类系统的解在常微分方程情形下是时间或单个空间变量的函数，而在偏微分方程情形下则依赖于多维空间坐标或空间与时间。在某些情况下，解可能依赖于多个时间尺度或空间尺度。受常微分方程支配的示例包括：阻尼极小时或极大时的阻尼谐振子、血管壁厚度从厘米级变化到微米级的心血管系统、水深远小于表面波典型波长时成立的浅水方程，以及表面活性剂与气泡相互作用范围远小于气泡本身的吸附动力学。在所有此类情形中，对模型进行解析所有时空尺度的详细模拟往往效率低下或难以处理，甚至对于合理描述系统行为而言通常也非必要。在“基于微分方程的多尺度建模”一章中，我们将介绍由常微分方程和偏微分方程描述且固有地具有多尺度特性的系统示例，并阐明合适的建模方法如何在不解析小尺度的情况下有效捕捉此类系统解的本质行为。