数理方程的微分积分形式——一道欧洲奥林匹克物理竞赛实验试题的思考

THE INTEGRAL AND DIFFERENTIAL FORM OF MATHEMATICAL PHYSICS EQUATIONS-REFLECTIONS ON A EUROPEAN OLYMPIAD IN PHYSICS COMPETITION EXPERIMENTAL PROBLEM

数学物理方程是一系列描述客观物理规律的偏微分方程。2021年欧洲奥林匹克物理竞赛(2021EuPhO)中有一道“细杆导热”计算细杆相关参数的实验试题与数学物理方程结合极其紧密。原试题的解答中,命题人利用了Simpson方法,在细杆上平均取五个温度采样点并计算温度的平均值从而近似获得了细杆整体温度随时间的变化。而此题目刚好和数学物理方法中的热扩散问题非常相近,因此我们基于数理方程中的热扩散问题对此题目进行梳理,以此题目为契机,分析数理方程的积分形式、微分形式以及Simpson积分方法的适用性问题。

Mathematical physical equations are a series of partial differential equations that describe objective physical laws.In the 2021 European Olympic Physics Competition(2021EuPhO),there was an experimental question of“heat conduction in a thin rod”which required calculating the relevant parameters of the rod and was closely related to mathematical physics equations.In the original solution to the problem,the proposer used Simpson method to take an average of five temperature sampling points on the thin rod and calculating the average temperature to approximate the overall temperature change of the rod over time.This question is very similar to the thermal diffusion problem in Mathematical Physics Methods,so we sort out this topic based on the thermal diffusion problem in mathematical equations.Taking this question as an opportunity,we analyze the integral form,differential form of mathematical equations and the applicability of Simpson integration method.