Euler’s Method: Numerical Solution of First-Order Differential Equations | 欧拉方法：一阶微分方程数值求解

📚 Euler’s Method: Numerical Solution of First-Order Differential Equations | 欧拉方法：一阶微分方程数值求解

When a first-order differential equation cannot be solved neatly by separation or integration, Euler’s method provides a straightforward numerical alternative. It constructs a solution step by step, using only the slope given by the differential equation and a chosen step size.

当一阶微分方程无法通过分离变量或积分直接求解时，欧拉方法提供了一种直接而有效的数值解法。它仅仅利用微分方程给出的斜率和一个选定的步长，逐步构造出近似解。

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1. What Is Euler’s Method? | 什么是欧拉方法？

A first-order ordinary differential equation can be written as y′ = f(x, y), with the initial condition y(x₀) = y₀. Euler’s method treats the derivative as a finite ratio and moves forward in small steps of size h.

一阶常微分方程可以写成 y′ = f(x, y)，并带有初始条件 y(x₀) = y₀。欧拉方法将导数看成一个有限的差比，并以步长 h 向前推进。

yₙ₊₁ = yₙ + h f(xₙ, yₙ)

Here (xₙ, yₙ) is the current point, and f(xₙ, yₙ) is the slope of the tangent at that point.

其中 (xₙ, yₙ) 是当前点，f(xₙ, yₙ) 是这一点处切线的斜率。

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2. The Geometry of Tangent Steps | 切线步进的几何意义

Imagine the true solution as a smooth curve on the x-y plane. At the initial point, the differential equation gives the slope of the curve. Euler’s method follows that tangent line for a short distance h, reaches a new point, then recalculates the tangent slope at that new point.

把真实解想象成 x-y 平面上的一条光滑曲线。在初始点处，微分方程给出曲线的斜率。欧拉方法沿这条切线前进一小段距离 h，到达一个新点，然后在新点处重新计算切线斜率。

Each step is therefore a straight-line approximation, and the approximate solution is a polygonal line that hugs the true curve more closely when h is small.

因此每一步都是直线近似，而近似解就是一条折线。当 h 较小时，这条折线与真实曲线贴合得更紧密。

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3. Deriving Euler’s Formula from Taylor’s Expansion | 由泰勒展开推导欧拉公式

To understand why Euler’s method works, expand the true solution around xₙ using Taylor’s theorem:

为了理解欧拉方法为何有效，我们利用泰勒定理在 xₙ 附近展开真实解：

y(xₙ + h) = y(xₙ) + h y′(xₙ) + O(h²)

Since y′ = f(x, y), we can replace y′(xₙ) by f(xₙ, yₙ). Dropping the O(h²) remainder gives the Euler formula.

因为 y′ = f(x, y)，所以可以用 f(xₙ, yₙ) 替换 y′(xₙ)。忽略 O(h²) 的余项，就得到欧拉公式。

yₙ₊₁ = yₙ + h f(xₙ, yₙ)

The dropped term is the local truncation error, which will be examined later.

被忽略的项就是局部截断误差，后面我们会详细讨论。

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4. Worked Example: Radioactive Decay | 实例：放射性衰变

Consider the differential equation for radioactive decay:

考虑放射性衰变的微分方程：

dy/dt = −λy, y(0) = 100

Let λ = 0.1 s⁻¹ and choose h = 1 s. The Euler recurrence is yₙ₊₁ = yₙ + h(−λyₙ) = yₙ(1 − λh).

取 λ = 0.1 s⁻¹，步长 h = 1 s。欧拉递推式为 yₙ₊₁ = yₙ + h(−λyₙ) = yₙ(1 − λh)。

n	t (s)	Euler y	Exact y = 100e⁻⁰·¹ᵗ
0	0	100.000	100.000
1	1	90.000	90.484
2	2	81.000	81.873
3	3	72.900	74.082
4	4	65.610	67.032
5	5	59.049	60.653

Because the true curve is concave upward in this case, Euler’s tangent steps lie below the curve and underestimate the answer.

在本例中真实曲线是上凹的，因此欧拉的切线步进位于曲线下方，导致结果偏低。

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5. Worked Example: Newton’s Law of Cooling | 实例：牛顿冷却定律

Newton’s law of cooling is a classic first-order equation:

牛顿冷却定律是一阶微分方程的经典例子：

dT/dt = −k(T − T_env), T(0) = 80°C

Take the surrounding temperature T_env = 20°C, k = 0.

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