本文主要研究偏微分方程在交通流中的应用,并将该案例应用到具体的教学中去。首先设计并分析了全新的单车道与多车道的交通流模型,此模型在车辆数量守恒的基础上,将问题抽象为关于车流量的一阶微分方程,通过求解方程,集中讨论了在有(无)流出情况下单车道的变化情况与抑制激波的产生等性质。通过模型的建立以及求解的方程的分析方法,可以培养学生的数学建模能力以及解决实际问题的能力。This paper primarily investigates the application of partial differential equations in traffic flow and applies this case to specific teaching practices. Initially, a novel single-lane and multi-lane traffic flow model is designed and analyzed. Based on the conservation of vehicle numbers, the problem is abstracted into a first-order differential equation concerning traffic flow. By solving the equation, the paper focuses on discussing the changes in single-lane conditions with (or without) outflow and the suppression of shock wave generation, among other properties. Through the establishment of the model and the analysis of the solved equations, students’ abilities in mathematical modeling and problem-solving in real-world scenarios can be cultivated.
目的探讨和分析拉格朗日(Joseph Louis Lagrange,1736—1813)重新定义一阶偏微分方程完全积分概念的原因和背景。方法历史分析和文献考证。结果拉格朗日从欧拉的完全积分定义出发,在用常数变易法探讨一阶偏微分方程积分的过程中受到启发,萌生了关于积分"完全性"的新思想。随后,他把这种新思想运用于常微分方程,成功解释了奇解现象,受此驱动,提出了一阶偏微分方程完全积分的新定义。结论拉格朗日的完全积分新定义是他追求方程一般性解法的体现和产物。
