Numerical Methods for First Order Differential Equations | 一阶微分方程数值解法

📚 Numerical Methods for First Order Differential Equations | 一阶微分方程数值解法

Many first order differential equations cannot be solved analytically using standard integration or exact methods. Numerical methods allow us to approximate solution curves by stepping forward from an initial condition. At AQA A-Level, the key method is Euler’s method, which replaces the curve with a sequence of short straight-line segments.

许多一阶微分方程无法用标准积分或精确方法求出解析解。数值方法让我们从初始条件出发，通过逐步推进来近似解曲线。在 AQA A-Level 中，核心方法是欧拉方法，它用一系列短直线段代替真实曲线。

1. First Order Differential Equations and Numerical Solutions | 一阶微分方程与数值解

In AQA A-Level Mathematics, you will work with differential equations of the form dy/dx = f(x,y) where f is a given function. Some can be solved exactly using separation of variables, but in pure and applied contexts we often meet equations that have no closed-form solution.

在 AQA A-Level 数学中，你会接触到 dy/dx=f(x,y) 形式的微分方程，其中 f 是已知函数。有些方程可以用分离变量法精确求解，但在纯数学和应用情境中，我们常遇到没有闭合形式解的方程。

Numerical methods fill this gap by producing approximate values of y at chosen x-values. Instead of finding a formula for y(x), we construct a sequence of points that follow the direction field of the differential equation.

数值方法通过生成所选 x 值处 y 的近似值来填补这一空白。我们不是求出 y(x) 的表达式，而是构造一系列点来追踪微分方程的方向场。

2. Euler’s Method: Core Idea | 欧拉方法：核心思想

Euler’s method is the simplest numerical scheme for an initial value problem. It starts from the known point (x0,y0) and uses the gradient f(x0,y0) to predict the value of y after one step of length h.

欧拉方法是求解初值问题最简单的数值格式。它从已知点 (x0,y0) 出发，利用梯度 f(x0,y0) 预测步长为 h 的一步之后 y 的值。

Geometrically, you follow the tangent line at the current point. Because the actual solution curve curves away from the tangent, this introduces an error, but over a short step the tangent is a reasonable approximation.

几何上，你沿着当前点的切线移动。由于真实解曲线会偏离切线，这会引入误差，但在短步长内切线是合理的近似。

3. Step Size and Local Linear Approximation | 步长与局部线性近似

The local linear approximation is the foundation of Euler’s method. If h is small, y(x0+h) ≈ y(x0) + h f(x0,y0). The error in this one-step approximation is proportional to h².

局部线性近似是欧拉方法的基础。如果 h 较小，则 y(x0+h) ≈ y(x0) + h f(x0,y0)。这一单步近似的误差与 h² 成正比。

However, the global error after many steps behaves like a constant times h. This means reducing h from 0.1 to 0.05 will roughly halve the total error, but it doubles the amount of work.

然而，许多步之后的全局误差表现为常数乘以 h。这意味着将 h 从 0.1 减小到 0.05 大约会使总误差减半，但计算量会翻倍。

4. Euler’s Method Algorithm and Formula | 欧拉方法的算法与公式

To apply Euler’s method, keep a table with columns for n, xₙ, yₙ, f(xₙ,yₙ) and h f(xₙ,yₙ). The recurrence is:

应用欧拉方法时，可维护一个包含 n、xₙ、yₙ、f(xₙ,yₙ) 和 h f(xₙ,yₙ) 列的表格。递推式为：

xₙ₊₁ = xₙ + h

yₙ₊₁ = yₙ + h f(xₙ, yₙ)

Note that f(xₙ,yₙ) is evaluated using the current approximate yₙ, not the true solution. This is what makes the method iterative and approximate.

注意 f(xₙ,yₙ) 是使用当前近似值 yₙ 计算的，而不是真实解。这正是该方法具有迭代性和近似性的原因。

5. Worked Example: y’ = x + y | 例题：y’ = x + y

Let us apply Euler’s method to dy/dx = x + y, y(0)=1 with h=0.1. We need y(0.3). At each stage compute f(x,y)=x+y then new y.

让我们用欧拉方法求解 dy/dx=x+y，y(0)=1，h=0.1。求 y(0.3)。每一步计算 f(x,y)=x+y，然后求新 y。

The iteration table is:

迭代表为：