📚 Four Practical Solution Methods for Partial Differential Equations | 偏微分方程的四种常用求解方法
Partial differential equations (PDEs) form the backbone of mathematical physics, engineering, and advanced applied mathematics. At A-Level and early university level, students are expected to recognise standard types of PDEs and apply systematic techniques to solve them. This article presents four essential methods: separation of variables, the method of characteristics, Fourier transforms, and Laplace transforms—each illustrated with typical examples and clear procedural steps.
偏微分方程（PDE）是数学物理、工程学以及高等应用数学的基石。在 A-Level 和大学初级阶段，学生需要能够识别标准类型的 PDE，并运用系统的方法求解。本文将介绍四种必备方法：分离变量法、特征线法、傅里叶变换法和拉普拉斯变换法——每种方法都配有典型例题和清晰的解题步骤。
1. Understanding the PDE Landscape | 认识偏微分方程的基本类型
Before selecting a solution method, you must classify the PDE. The three classical second-order linear PDEs are the wave equation ∂²u/∂t² = c²∂²u/∂x², the heat equation ∂u/∂t = k∂²u/∂x², and Laplace’s equation ∂²u/∂x² + ∂²u/∂y² = 0. Each has distinct physical meaning and demands a different approach.
在选择求解方法之前，必须先对 PDE 进行分类。三类经典的二阶线性 PDE 分别是：波动方程 ∂²u/∂t² = c²∂²u/∂x²、热传导方程 ∂u/∂t = k∂²u/∂x² 以及拉普拉斯方程 ∂²u/∂x² + ∂²u/∂y² = 0。每一种方程都有独特的物理含义，需要采用不同的求解策略。
The order, linearity, and boundary conditions determine which method is most efficient. A first-order PDE like uₓ + uᵧ = 0 can often be solved by characteristics, while a second-order PDE on a finite interval may suit separation of variables. Transform methods excel on infinite domains.
方程的阶数、线性性质以及边界条件决定了哪种方法最有效。一阶 PDE（如 uₓ + uᵧ = 0）通常可用特征线法求解；有限区间上的二阶 PDE 适合分离变量法；而变换方法在半无界或无界区域上表现尤为出色。
2. Separation of Variables: Principle | 分离变量法：基本原理
The core idea of separation of variables is to assume the solution can be written as a product of single-variable functions. For example, in the heat equation, we set u(x,t) = X(x)T(t). Substituting this product into the PDE allows us to separate the variables into two ordinary differential equations connected by a separation constant.
分离变量法的核心思想是假设解可以写成单变量函数的乘积形式。例如，对于热传导方程，我们令 u(x,t) = X(x)T(t)。将这个乘积代入 PDE 后，可以将变量分离为两个由分离常数联系的常微分方程。
Because each side depends on a different variable, both sides must equal a constant, conventionally written as −λ. This produces an eigenvalue problem for X(x) subject to the homogeneous boundary conditions, yielding discrete allowed values of λ.
由于等式两边分别依赖于不同的变量，两边必须都等于一个常数，通常记为 −λ。这便产生了一个关于 X(x) 的特征值问题，结合齐次边界条件，可以得到离散的允许值 λ。
3. Applying Separation of Variables to the Heat Equation | 用分离变量法求解热传导方程
Consider the heat equation uₜ = k uₓₓ for 0 < x < L, with boundary conditions u(0,t) = u(L,t) = 0 and initial condition u(x,0) = f(x). Setting u(x,t) = X(x)T(t) gives X(x)T'(t) = k X''(x)T(t), which rearranges to T'/kT = X''/X = −λ.
考虑热传导方程 uₜ = k uₓₓ，其中 0 < x < L，边界条件为 u(0,t) = u(L,t) = 0，初始条件为 u(x,0) = f(x)。令 u(x,t) = X(x)T(t) 可得 X(x)T'(t) = k X''(x)T(t)，整理后得到 T'/kT = X''/X = −λ。
The spatial equation X” + λX = 0 with X(0) = X(L) = 0 has non-trivial solutions only when λ = n²π²/L², giving Xₙ(x) = sin(nπx/L). The temporal equation gives Tₙ(t) = Cₙ exp(−n²π²kt/L²). By the superposition principle, u(x,t) = Σₙ₌₁^∞ Cₙ sin(nπx/L) exp(−n²π²kt/L²).
空间方程 X” + λX = 0 在 X(0) = X(L) = 0 的条件下，仅当 λ = n²π²/L² 时才有非平凡解，此时 Xₙ(x) = sin(nπx/L)。时间方程给出 Tₙ(t) = Cₙ exp(−n²π²kt/L²)。由叠加原理，u(x,t) = Σₙ₌₁^∞ Cₙ sin(nπx/L) exp(−n²π²kt/L²)。
Finally, apply the initial condition. Set t = 0, so f(x) = Σ Cₙ sin(nπx/L). Using the orthogonality of sine functions ∫₀ᴸ sin(nπx/L) sin(mπx/L) dx = L/2 δₙₘ, we obtain Cₙ = (2/L) ∫₀ᴸ f(x) sin(nπx/L) dx.
最后应用初始条件。令 t = 0，得到 f(x) = Σ Cₙ sin(nπx/L)。利用正弦函数的正交性 ∫₀ᴸ sin(nπx/L) sin(mπx/L) dx = L/2 δₙₘ，可得 Cₙ = (2/L) ∫₀ᴸ f(x) sin(nπx/L) dx。
4. The Method of Characteristics: First-Order PDEs | 特征线法：一阶偏微分方程
The method of characteristics reduces a first-order PDE to a system of ordinary differential equations along curves called characteristic lines. For a quasi-linear PDE of the form a(x,t)uₓ + b(x,t)uₜ = c(x,t,u), the characteristic equations are dx/ds = a, dt/ds = b, and du/ds = c.
特征线法将一阶 PDE 沿着称为“特征线”的曲线化简为一组常微分方程。对于形如 a(x,t)uₓ + b(x,t)uₜ = c(x,t,u) 的拟线性 PDE，其特征方程为 dx/ds = a，dt/ds = b，du/ds = c。
Along a characteristic curve, the PDE becomes an ODE for u. Once the family of characteristic curves is found, the solution is determined by prescribing initial data on a non-characteristic curve.
沿着特征曲线，PDE 变成了关于 u 的 ODE。一旦找到特征曲线族，就可以通过非特征曲线上的初始条件来确定解。
5. Characteristics in Action: The Advection Equation | 特征线法实战：平流方程
Solve the advection equation uₜ + c uₓ = 0, where c is a constant, with initial condition u(x,0) = f(x). The characteristic equations are dx/ds = c, dt/ds = 1, du/ds = 0. From dx/dt = c, we obtain x − ct = constant along each characteristic.
求解平流方程 uₜ + c uₓ = 0，其中 c 为常数，初始条件为 u(x,0) = f(x)。特征方程为 dx/ds = c，dt/ds = 1，du/ds = 0。由 dx/dt = c 可得，沿每一条特征线 x − ct = 常数。
Since du/ds = 0, u is constant along this family of straight lines. Therefore the solution is simply u(x,t) = f(x − ct), which represents a wave profile travelling to the right with speed c.
因为 du/ds = 0，所以 u 在整族直线上保持恒定。因此解就是 u(x,t) = f(x − ct)，表示一个以速度 c 向右传播的波形。
This method extends naturally to systems of first-order equations and to some second-order equations that can be factorised into first-order operators. It is especially useful in wave propagation and traffic flow problems.
这一方法可以自然推广到一阶方程组，也可用于一些可分解为一阶算子的二阶方程。它在波传播和交通流问题中尤其有用。
6. Fourier Transform: Concept and Key Properties | 傅里叶变换：概念与关键性质
For PDEs on infinite or semi-infinite domains, integral transforms often provide the most direct route. The Fourier transform of a function f(x) is defined as F(k) = ∫₋∞^∞ f(x) e^(−ikx) dx, with the inverse f(x) = (1/2π) ∫₋∞^∞ F(k) e^(ikx) dk.
对于无界或半无界区域上的 PDE，积分变换往往是最直接的路径。函数 f(x) 的傅里叶变换定义为 F(k) = ∫₋∞^∞ f(x) e^(−ikx) dx，其逆变换为 f(x) = (1/2π) ∫₋∞^∞ F(k) e^(ikx) dk。
The crucial property is that differentiation becomes multiplication: if F(k) is the transform of f(x), then the transform of f'(x) is ikF(k), and for the second derivative, f”(x) transforms to −k²F(k). This converts PDEs into ODEs (or even algebraic equations) in the transform variable.
关键性质在于微分变为乘法：若 F(k) 是 f(x) 的变换，则 f'(x) 的变换为 ikF(k)，二阶导数 f”(x) 的变换为 −k²F(k)。这可以将 PDE 转化为关于变换变量的 ODE（甚至代数方程）。
7. Fourier Transform Solution of Laplace’s Equation | 傅里叶变换求解拉普拉斯方程
Consider Laplace’s equation on the upper half-plane: uₓₓ + uᵧᵧ = 0, for −∞ < x < ∞, y > 0, with boundary condition u(x,0) = f(x) and the requirement that u → 0 as y → ∞.
考虑上半平面上的拉普拉斯方程：uₓₓ + uᵧᵧ = 0，其中 −∞ < x < ∞，y > 0，边界条件为 u(x,0) = f(x)，并要求当 y → ∞ 时 u → 0。
Applying the Fourier transform in x, we denote û(k,y) = ∫₋∞^∞ u(x,y) e^(−ikx) dx. The PDE becomes −k²û + ∂²û/∂y² = 0, which is an ODE in y. Its general solution is û(k,y) = A(k)e^(−|k|y) + B(k)e^(|k|y). To ensure boundedness as y → ∞, we set B(k) = 0.
对 x 变量应用傅里叶变换，记 û(k,y) = ∫₋∞^∞ u(x,y) e^(−ikx) dx。原 PDE 变为 −k²û + ∂²û/∂y² = 0，这是关于 y 的 ODE。其通解为 û(k,y) = A(k)e^(−|k|y) + B(k)e^(|k|y)。为保证当 y → ∞ 时有界，令 B(k) = 0。
Using the boundary condition, A(k) = F(k). Thus û(k,y) = F(k)e^(−|k|y). Inverting gives u(x,y) = (1/2π) ∫₋∞^∞ F(k)e^(−|k|y)e^(ikx) dk, which can be recognised as a convolution with the Poisson kernel.
利用边界条件可得 A(k) = F(k)。因此 û(k,y) = F(k)e^(−|k|y)。作逆变换得到 u(x,y) = (1/2π) ∫₋∞^∞ F(k)e^(−|k|y)e^(ikx) dk，这可以看作是与泊松核的卷积。
8. Laplace Transform: Tailored for Initial-Value Problems | 拉普拉斯变换：专为初值问题设计
The Laplace transform is defined as U(s) = ∫₀^∞ u(t)e^(−st) dt. It is particularly suited to PDEs where time t appears with first or second derivatives and where initial conditions at t = 0 are specified.
拉普拉斯变换定义为 U(s) = ∫₀^∞ u(t)e^(−st) dt。它特别适合处理时间变量 t 出现一阶或二阶导数且给定 t = 0 处初始条件的 PDE。
The key identities are: transform of u'(t) is sU(s) − u(0), and transform of u”(t) is s²U(s) − su(0) − u'(0). These formulas build the initial conditions directly into the transformed equation.
关键恒等式为：u'(t) 的变换是 sU(s) − u(0)，u”(t) 的变换是 s²U(s) − su(0) − u'(0)。这些公式将初始条件直接内嵌到变换后的方程中。
9. Laplace Transform Solution of the Wave Equation | 拉普拉斯变换求解波动方程
Solve the semi-infinite wave equation uₜₜ = c²uₓₓ for x > 0, t > 0, with u(x,0) = 0, uₜ(x,0) = 0, and the boundary condition u(0,t) = g(t). Taking the Laplace transform with respect to t, let U(x,s) = ℒ{u(x,t)}.
求解半无界波动方程 uₜₜ = c²uₓₓ，其中 x > 0，t > 0，初始条件为 u(x,0) = 0，uₜ(x,0) = 0，边界条件为 u(0,t) = g(t)。对时间变量 t 取拉普拉斯变换，令 U(x,s) = ℒ{u(x,t)}。
s²U(x,s) = c² ∂²U/∂x²
This ODE has the general solution U(x,s) = A(s)e^(−sx/c) + B(s)e^(sx/c). Requiring U to be bounded as x → ∞ forces B(s) = 0. The boundary condition gives U(0,s) = G(s), so A(s) = G(s).
该 ODE 的通解为 U(x,s) = A(s)e^(−sx/c) + B(s)e^(sx/c)。要求当 x → ∞ 时 U 有界可得 B(s) = 0。边界条件给出 U(0,s) = G(s)，因此 A(s) = G(s)。
We now have U(x,s) = G(s)e^(−sx/c). By the shift theorem, multiplying by e^(−sx/c) corresponds to a time delay. Hence u(x,t) = g(t − x/c) for t ≥ x/c and u(x,t) = 0 for t < x/c.
于是得到 U(x,s) = G(s)e^(−sx/c)。由位移定理，乘以 e^(−sx/c) 对应于时间延迟。因此 u(x,t) = g(t − x/c)（当 t ≥ x/c 时），而当 t < x/c 时 u(x,t) = 0。
10. Choosing the Right Method: A Strategic Guide | 选择正确的方法：策略指南
Selecting the appropriate method is a skill in itself. The domain geometry, the type of PDE, and the form of the boundary or initial conditions all influence this choice. The table below summarises the typical scenarios.
选择合适的方法本身是一项技能。区域几何形状、PDE 的类型以及边界或初始条件的形式都会影响选择。下表总结了典型适用场景。
| Method | Best Used For | Key Advantage |
| Separation of Variables | Second-order linear PDEs on finite intervals with homogeneous boundary conditions | Produces eigenfunction series; transparent physical interpretation |
| Characteristics | First-order PDEs, especially advection and wave problems | Reduces PDE to ODE along curves |
| Fourier Transform | PDEs on infinite domains with spatial derivatives | Converts x-derivatives into algebraic factors |
| Laplace Transform | Initial-value problems in time on semi-infinite domains | Builds initial conditions directly into the solution |
Transform methods often overlap; for instance, Laplace transforms can serve as an alternative to Fourier transforms for non-periodic functions on (0, ∞). The choice also depends on whether you prefer solving ODEs in a transform variable or summing infinite series.
变换方法常有重叠；例如，对于 (0, ∞) 上的非周期函数，拉普拉斯变换可以替代傅里叶变换。选择还取决于你更喜欢在变换变量中解 ODE，还是更喜欢对无穷级数求和。
11. Common Pitfalls and Working Accuracy | 常见陷阱与运算准确性
Students frequently make mistakes when applying separation of variables by forgetting to check whether the boundary conditions are homogeneous. If they are not, you must first homogenise them or handle the steady-state part separately.
学生在使用分离变量法时常见的错误是忘记检查边界条件是否为齐次。如果不是齐次边界条件，必须先进行齐次化处理，或者将稳态部分单独处理。
Another common pitfall is losing the absolute value in the Fourier solution of Laplace’s equation: writing e^(−ky) instead of e^(−|k|y) leads to divergent integrals for even initial data. Always check the decay requirement as y → ∞ before discarding the positive exponent term.
另一个常见陷阱是在拉普拉斯方程的傅里叶解中丢掉绝对值：写成 e^(−ky) 而不是 e^(−|k|y) 会导致对于偶初始数据积分发散。在舍弃正指数项之前，务必检查当 y → ∞ 时的衰减条件。
When using Laplace transforms, remember that the inverse transform must be considered piecewise, especially when delays such as e^(−sx/c) appear. The shift theorem applies only to functions that are zero before the shift time.
使用拉普拉斯变换时，记住逆变换必须分段考虑，特别是出现形如 e^(−sx/c) 的延迟项时。位移定理仅适用于在延迟时刻之前为零的函数。
12. Revision Checklist and Final Remarks | 复习清单与结语
To master the four methods, practise identifying the equation type first. For each problem, ask: Is the domain finite or infinite? Is the equation first or second order? Are the initial conditions or boundary conditions more prominent?
要掌握这四种方法，请先练习判断方程类型。每遇到一个问题，问自己：区域是有限还是无限？方程是一阶还是二阶？初始条件和边界条件哪一个更突出？
Separation of Variables suits finite intervals with two homogeneous boundary conditions. Method of Characteristics is the natural choice for first-order PDEs. Fourier Transform is ideal for infinite spatial domains, while Laplace Transform is preferred when time-domain initial conditions dominate.
分离变量法适用于具有两个齐次边界条件的有限区间；特征线法是处理一阶 PDE 的自然选择；傅里叶变换适合无界空间区域；而拉普拉斯变换在时间域初始条件占主导时更为优选。
Work through at least two full examples of each method from the A-Level specification. Focus on clean algebraic manipulation, correct application of orthogonality relations, and careful handling of inverse transforms. With structured practice, PDEs become a rewarding topic rather than an intimidating one.
请完成 A-Level 考纲中每种方法至少两道完整例题。重点关注简洁的代数变形、正交关系的正确应用以及逆变换的细致处理。通过系统训练，偏微分方程将从一个令人畏惧的课题变为一个收获满满的板块。
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