📚 Mathematical Modelling: Applications of Mathematical Models in Science and Engineering | 数学建模：科学与工程中的数学模型应用
Mathematical modelling is the art of translating real-world problems into mathematical language, solving the resulting equations, and interpreting the results to gain insight into the original system. It is a powerful tool used across physics, biology, economics, and engineering, enabling us to predict, optimise, and control complex processes.
数学建模是将现实世界问题转化为数学语言、求解所得方程并解释结果以洞悉原始系统的艺术。它是物理、生物、经济和工程等领域中极为强大的工具，使我们能够预测、优化和控制复杂过程。
1. What Is a Mathematical Model? | 什么是数学模型？
A mathematical model is a description of a system using mathematical concepts and language. Models can be deterministic or stochastic, linear or nonlinear, continuous or discrete. They involve variables, parameters, functions, equations, and inequalities.
数学模型是用数学概念和语言对系统进行的描述。模型可以是确定性的或随机的、线性的或非线性的、连续的或离散的。它们涉及变量、参数、函数、方程和不等式。
The modelling process typically follows a cycle: identify the problem, make assumptions, formulate the model, solve it, validate the results, and refine if necessary. This iterative approach ensures the model remains useful and accurate.
建模过程通常遵循一个循环：识别问题、作出假设、建立模型、求解、验证结果，并在必要时修正。这种迭代方法确保模型保持实用和准确。
2. Types of Models | 模型的类型
Models can be classified in several ways. Analytical models have closed-form solutions; numerical models require computational methods. Empirical models are based on data, while mechanistic models are based on underlying physical principles.
模型可按多种方式分类。解析模型具有闭式解；数值模型需要计算方法。经验模型基于数据，而机理模型基于物理原理。
- Deterministic models: produce the same output for a given input; e.g. Newton’s second law.
- 随机（确定）模型：给定输入产生相同输出，例如牛顿第二定律。
- Stochastic models: include randomness; e.g. population growth with random fluctuations.
- 随机（概率）模型：包含随机性，例如具有随机波动的种群增长。
- Continuous models: use differential equations; e.g. radioactive decay.
- 连续模型：使用微分方程，例如放射性衰变。
- Discrete models: use difference equations; e.g. compound interest.
- 离散模型：使用差分方程，例如复利计算。
3. Building a Model: Assumptions and Variables | 建模：假设与变量
Every model begins with simplifying assumptions. For example, a projectile motion model often neglects air resistance, assuming gravity is constant and the Earth is flat over the range of motion. These assumptions make the mathematics tractable while still capturing essential behaviour.
每个模型都从简化假设开始。例如，抛体运动模型常常忽略空气阻力，假设重力恒定且运动范围内地球为平面。这些假设使数学处理易于操作，同时仍能捕捉基本行为。
Variables are quantities that change, such as time, position, temperature, or population size. Parameters are fixed values within the model, like the gravitational constant or the rate of decay. Distinguishing between them is crucial for correct formulation.
变量是随时间推移而变化的量，如时间、位置、温度或种群数量。参数是模型中的固定值，如引力常数或衰变速率。区分变量与参数对正确建模至关重要。
4. Exponential Growth and Decay | 指数增长与衰减
Many natural processes follow the law that the rate of change of a quantity is proportional to the quantity itself. If y(t) is the amount at time t, then
许多自然过程遵循“变化率与数量本身成正比”的规律。如果 y(t) 是时间 t 时的数量，那么
dy/dt = k y
where k is a constant. If k > 0, the solution is exponential growth; if k < 0, it is exponential decay. The solution is
其中 k 为常数。若 k > 0，则解为指数增长；若 k < 0，则为指数衰减。解为
y(t) = y₀ e^(kt)
where y₀ is the initial amount. This model applies to radioactive decay (with k negative), population growth under unlimited resources, and continuous compound interest.
其中 y₀ 为初始数量。该模型适用于放射性衰变（k 为负）、资源无限时的种群增长以及连续复利。
5. Newton’s Law of Cooling | 牛顿冷却定律
A classic engineering model is Newton’s law of cooling, which states that the rate of heat loss of an object is proportional to the difference between its temperature and the ambient temperature. If T(t) is the object’s temperature and Tₐ the surrounding temperature, then
工程中的经典模型是牛顿冷却定律，它指出物体热量损失速率与物体温度和环境温度之差成正比。设 T(t) 为物体温度，Tₐ 为周围环境温度，则
dT/dt = -k (T – Tₐ)
This is a separable differential equation. Its solution is
这是一个可分离变量的微分方程，其解为
T(t) = Tₐ + (T₀ – Tₐ) e^(-kt)
where T₀ is the initial temperature. This model is used in forensic science to estimate time of death and in engineering to design cooling systems.
其中 T₀ 是初始温度。该模型用于法医学中估算死亡时间，也用于工程中设计冷却系统。
6. Population Models: Logistic Growth | 种群模型：逻辑斯蒂增长
The simple exponential population model predicts unlimited growth, which is unrealistic. The logistic model introduces a carrying capacity K, the maximum population the environment can sustain. Its differential equation is
简单的指数种群模型预测无限增长，这不符合实际。逻辑斯蒂模型引入了环境容纳量 K，即环境可维持的最大种群数量。其微分方程为
dP/dt = r P (1 – P/K)
where r is the intrinsic growth rate. When P is small, growth is nearly exponential; as P approaches K, growth slows and eventually stops. The analytic solution is the logistic function
其中 r 为固有增长率。当 P 很小时，增长近似指数；当 P 趋近 K 时，增长减慢并最终停止。解析解为逻辑斯蒂函数
P(t) = K / (1 + ((K – P₀)/P₀) e^(-rt))
This model is widely used in ecology, epidemiology, and even in the adoption of new technologies.
该模型广泛应用于生态学、流行病学，甚至新技术的推广中。
7. Modelling with Differential Equations in Mechanics | 力学中的微分方程建模
Newton’s second law, F = ma, is itself a mathematical model. For a mass m attached to a spring, Hooke’s law gives F = -kx, where x is displacement from equilibrium. Combining these yields the second-order differential equation
牛顿第二定律 F = ma 本身就是数学模型。对于连接在弹簧上的质量 m，胡克定律给出 F = -kx，其中 x 是偏离平衡位置的位移。两者结合产生二阶微分方程
m d²x/dt² + k x = 0
The solution is sinusoidal, x(t) = A cos(ωt + φ), where ω = √(k/m) is the natural angular frequency. This model describes simple harmonic motion and forms the basis for understanding vibrations, oscillations, and even electrical circuits via analogy.
其解为正弦形式，x(t) = A cos(ωt + φ)，其中 ω = √(k/m) 为固有角频率。该模型描述简谐运动，是理解振动、振荡以及类比电路的基础。
8. Modelling in Circuit Analysis | 电路分析中的建模
Electrical circuits are modelled using differential equations derived from Kirchhoff’s laws. For a series RLC circuit (resistor R, inductor L, capacitor C), the charge q on the capacitor satisfies
电路使用由基尔霍夫定律导出的微分方程建模。对于串联 RLC 电路（电阻 R、电感 L、电容 C），电容器上的电荷 q 满足
L d²q/dt² + R dq/dt + q/C = E(t)
where E(t) is the applied voltage. This equation is analogous to the damped driven harmonic oscillator in mechanics. The solution reveals how current and voltage vary over time, which is essential for filter design and signal processing.
其中 E(t) 为外加电压。该方程与力学中的阻尼受迫谐振子类似。解揭示了电流和电压随时间变化的方式，对滤波器设计和信号处理至关重要。
9. Numerical Methods: When Analytical Solutions Fail | 数值方法：当解析解失效时
Many real-world models involve nonlinear equations that cannot be solved analytically. Numerical methods such as Euler’s method, Runge-Kutta methods, and finite difference approximations are used to approximate solutions on a computer. Euler’s method approximates the solution of dy/dx = f(x, y) using the recurrence
许多现实模型涉及无法解析求解的非线性方程。欧拉方法、龙格-库塔方法、有限差分近似等数值方法用于在计算机上近似解。欧拉方法通过递推式近似 dy/dx = f(x, y) 的解
yₙ₊₁ = yₙ + h f(xₙ, yₙ)
where h is the step size. Smaller h gives better accuracy at higher computational cost. Numerical modelling allows engineers to simulate complex systems from weather forecasting to aerodynamic design.
其中 h 为步长。h 越小精度越高，但计算成本也越高。数值建模使工程师能够模拟从天气预报到空气动力学设计等复杂系统。
10. Validation and Sensitivity Analysis | 模型验证与灵敏度分析
A model is only useful if it is validated against real data. Validation involves comparing model predictions with observations and assessing error. If the model fails, assumptions must be revisited. Sensitivity analysis tests how changes in parameters affect outputs, helping identify which parameters are most influential.
模型只有经过真实数据验证才有用。验证涉及将模型预测与观测结果进行比较并评估误差。如果模型失败，必须重新审视假设。灵敏度分析测试参数变化如何影响输出，帮助识别哪些参数影响最大。
| Aspect | Question |
| Validity | Does the model represent the system? |
| Accuracy | How close are predictions to observations? |
| Robustness | Does it work under varying conditions? |
| Sensitivity | Which parameters matter most? |
Validation and sensitivity analysis are essential steps in the modelling cycle.
验证与灵敏度分析是建模周期中的关键步骤。
11. Example: Modelling a Falling Object with Air Resistance | 示例：考虑空气阻力的落体模型
Consider an object of mass m falling vertically with air resistance proportional to velocity. The forces are gravity (mg downward) and drag (kv upward). Newton’s second law gives
考虑质量为 m 的物体垂直下落，空气阻力与速度成正比。力包括重力（mg 向下）和阻力（kv 向上）。牛顿第二定律给出
m dv/dt = mg – k v
This first-order linear differential equation has the solution
这个一阶线性微分方程的解为
v(t) = (mg/k)(1 – e^(-kt/m))
As t → ∞, the velocity approaches the terminal velocity v_T = mg/k. This model is essential for skydiving physics, parachute design, and predicting the motion of falling particles in fluids.
当 t → ∞ 时，速度趋近极限速度 v_T = mg/k。该模型对跳伞物理、降落伞设计以及预测流体中颗粒下落运动至关重要。
12. The Role of Mathematical Modelling in Modern Engineering | 数学建模在现代工程中的作用
Mathematical modelling is at the heart of modern engineering. It is used in designing aircraft, optimising supply chains, predicting climate change, modelling the spread of diseases, and improving medical imaging. With the rise of machine learning and data science, hybrid models combining physical principles with data-driven techniques are becoming increasingly powerful.
数学建模是现代工程的核心。它用于设计飞机、优化供应链、预测气候变化、模拟疾病传播和改善医学成像。随着机器学习和数据科学的兴起，结合物理原理与数据驱动技术的混合模型正变得越来越强大。
Mastering mathematical modelling requires practice in translating problems, solving equations, and critically interpreting results. It is not just a mathematical skill; it is a way of thinking that bridges theory and reality.
掌握数学建模需要练习问题转化、方程求解和结果批判性解释。它不仅是数学技能，更是一种连接理论与现实的思维方式。
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