SIAM Review, Volume 65, Issue 3, Page 774-805, August 2023.
The beauty and simplicity of compartment modeling makes it a useful approach for simulating dynamics in an amazingly wide range of applications, which are growing rapidly especially in global carbon cycling, hydrological network flows, and epidemiology and population dynamics. These contexts, however, often involve compartment-to-compartment flows that are incongruent with the conventional assumption of complete mixing that results in exponential residence times in linear models. Here we detail a general method for assigning any desired residence time distribution to a given intercompartmental flow, extending compartment modeling capability to transport operations, power-law residence times, diffusions, etc., without resorting to composite compartments, fractional calculus, or partial differential equations (PDEs) for diffusive transport. This is achieved by writing the mass exchange rate coefficients as functions of age-in-compartment as done in one of the first compartment models in 1917, at the cost of converting the usual ordinary differential equations to a system of first-order PDEs. The PDEs are readily converted to a system of integral equations for which a numerical method is devised. Example calculations demonstrate incorporation of advective lags, advective-dispersive transport, power-law residence time distributions, or diffusive domains in compartment models.

中文翻译：

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带记忆的隔室模型

《SIAM 评论》，第 65 卷，第 3 期，第 774-805 页，2023 年 8 月。
区室建模的美观性和简单性使其成为一种在极其广泛的应用中模拟动力学的有用方法，这些应用正在迅速增长，特别是在全球碳循环、水文网络流以及流行病学和人口动态方面。然而，这些情况通常涉及隔室到隔室的流动，这与完全混合的传统假设不一致，完全混合导致线性模型中的指数停留时间。在这里，我们详细介绍了一种通用方法，用于将任何所需的停留时间分布分配给给定的室间流动，将室建模能力扩展到运输操作、幂律停留时间、扩散等，而无需诉诸复合室、分数阶微积分或偏微分扩散传输方程（PDE）。这是通过将质量交换率系数写为房室年龄的函数来实现的，就像 1917 年第一个房室模型中所做的那样，但代价是将通常的常微分方程转换为一阶偏微分方程组。偏微分方程很容易转换为积分方程组，并为其设计了数值方法。示例计算证明了隔室模型中平流滞后、平流弥散传输、幂律停留时间分布或扩散域的结合。偏微分方程很容易转换为积分方程组，并为其设计了数值方法。示例计算证明了隔室模型中平流滞后、平流弥散传输、幂律停留时间分布或扩散域的结合。偏微分方程很容易转换为积分方程组，并为其设计了数值方法。示例计算证明了隔室模型中平流滞后、平流弥散传输、幂律停留时间分布或扩散域的结合。