Partial Differential Equations: Heat, Wave, and Laplace Equations
Introduction
Partial differential equations (PDEs) describe phenomena involving functions of multiple independent variables and their partial derivatives. They govern diffusion, wave propagation, electrostatic potentials, fluid flow, and quantum mechanical probability amplitudes. Unlike ordinary differential equations that describe systems concentrated at a point, PDEs describe fields extended through space and evolving over time.
The three classic PDEs — the heat equation, wave equation, and Laplace equation — form the foundation of mathematical physics. Each captures a fundamental type of physical behavior: diffusion, propagation, and equilibrium. Mastering their solution reveals patterns that recur throughout applied mathematics. The methods developed for these equations, particularly separation of variables and Fourier series, generalize to countless other PDEs.
Classification of PDEs
PDEs are classified by their order (the highest derivative), linearity, and type. Second-order linear PDEs in two variables fall into three categories: elliptic, parabolic, and hyperbolic. The classification depends on the coefficients of the highest-order terms and mirrors the classification of conic sections.
Elliptic, Parabolic, and Hyperbolic
Elliptic equations, exemplified by Laplace’s equation ∇²u = 0, describe steady-state phenomena. Their solutions are smooth, with no singularities in the interior. Boundary conditions determine the solution uniquely. The temperature distribution in a heated plate at equilibrium satisfies Laplace’s equation.
Parabolic equations, exemplified by the heat equation u_t = α∇²u, describe diffusion and decay. Information propagates infinitely fast but with diminishing amplitude. The solution smooths out initial irregularities exponentially over time. The heat equation appears in thermal conduction, diffusion of chemicals, and option pricing in finance.
Hyperbolic equations, exemplified by the wave equation u_tt = c²∇²u, describe wave propagation at finite speed. Information travels along characteristics at speed c. Solutions preserve initial irregularities, translating and deforming them rather than smoothing them out. The wave equation governs sound, light, and seismic waves.
Boundary and Initial Conditions
PDEs require appropriate conditions to determine a unique solution. Initial conditions specify the state at t = 0 for time-dependent problems. Boundary conditions specify behavior on the domain boundary. Dirichlet conditions prescribe the function value on the boundary. Neumann conditions prescribe the normal derivative. Mixed conditions combine both types.
Choosing the right conditions is essential for a well-posed problem. Hadamard’s criteria require existence, uniqueness, and continuous dependence on data. Ill-posed problems — like solving the heat equation backward in time — violate these criteria and require special regularization techniques.
Separation of Variables
Separation of variables reduces a PDE to a system of ODEs by assuming the solution factors into functions of each independent variable. This method works when the PDE and boundary conditions are linear and homogeneous, and the domain has a regular shape.
The Heat Equation on a Finite Rod
The one-dimensional heat equation u_t = αu_xx on a rod of length L with zero-temperature ends is solved by assuming u(x,t) = X(x)T(t). Substituting and dividing by αXT gives X″/X = T′/(αT) = −λ, where λ is a separation constant. The X equation becomes X″ + λX = 0 with X(0) = X(L) = 0, yielding eigenvalues λ_n = (nπ/L)² and eigenfunctions X_n = sin(nπx/L).
The T equation T′ = −αλT gives exponential decay T_n = e^{−α(nπ/L)²t}. The full solution is an infinite series of these product solutions, with coefficients determined by the initial temperature distribution using Fourier sine series. The series converges rapidly because higher modes decay faster — after short times, only the fundamental mode remains significant.
The Wave Equation on a Finite String
The one-dimensional wave equation u_tt = c²u_xx for a plucked string with fixed ends is solved similarly. Separation gives X″ + λX = 0 with the same eigenvalues and eigenfunctions as the heat equation, but the time equation T″ + λc²T = 0 produces sinusoidal oscillations T_n = A_n cos(nπct/L) + B_n sin(nπct/L).
The general solution represents the string’s vibration as a superposition of normal modes. Each mode oscillates at its characteristic frequency f_n = nc/(2L). The fundamental frequency determines the pitch, while higher harmonics determine the timbre. D’Alembert’s solution u(x,t) = f(x+ct) + g(x−ct) reveals that waves travel in opposite directions at speed c without changing shape.
Laplace’s Equation in a Rectangle
Laplace’s equation u_xx + u_yy = 0 in a rectangle with boundary conditions on all four sides is solved by first reducing to homogeneous boundary conditions on three sides, then applying separation of variables. The solution takes the form of a Fourier series involving hyperbolic sines and sines, with coefficients determined by the remaining boundary condition.
The solution exhibits the maximum principle: the maximum of u occurs on the boundary. This property makes Laplace’s equation well-suited for interpolation — the interior values are a weighted average of the boundary values, a fact exploited in numerical solution methods like numerical analysis.
Fourier Series and Transforms
Fourier series represent periodic functions as infinite sums of sines and cosines. They provide the coefficients for separation of variables solutions and reveal the frequency content of signals. The Fourier transform extends this representation to nonperiodic functions on infinite domains.
Fourier Sine and Cosine Series
Any sufficiently nice function on [0,L] can be expanded in a Fourier sine series f(x) = Σ b_n sin(nπx/L) with coefficients b_n = (2/L)∫f(x) sin(nπx/L)dx. The cosine series uses cosines and includes a constant term. The choice depends on the boundary conditions of the problem.
Convergence of Fourier Series
Fourier series converge to the function at points of continuity. At jump discontinuities, the series exhibits the Gibbs phenomenon — overshooting the function value by about 9% of the jump magnitude. This overshoot persists no matter how many terms are included, though its spatial extent shrinks as more terms are added.
The Dirichlet and Jordan conditions provide sufficient conditions for convergence: piecewise smoothness with finitely many discontinuities guarantees pointwise convergence. Uniform convergence requires stronger conditions but ensures the series can be integrated and differentiated term by term.
Discrete Fourier Transform
The discrete Fourier transform (DFT) operates on sampled data, converting N time-domain samples into N frequency-domain coefficients. The fast Fourier transform (FFT) computes the DFT in O(N log N) operations, making practical the spectral analysis of large datasets. The FFT revolutionized signal processing, enabling real-time audio analysis, image compression, and numerical solution of PDEs.
The DFT assumes the signal is periodic with period N. Windowing functions like Hamming and Blackman windows reduce spectral leakage caused by discontinuities at the period boundaries. Zero-padding increases frequency resolution by interpolating between DFT bins.
The Fourier Transform
The Fourier transform converts a function of space or time into a function of frequency. Its inverse transform reconstructs the original function. The transform turns derivatives into algebraic factors, converting PDEs into ODEs or algebraic equations. The heat equation u_t = αu_xx becomes dû/dt = −αk²û, which solves immediately to û(k,t) = û(k,0)e^{−αk²t}.
The Fourier transform reveals that the heat equation strongly damps high frequencies — short-wavelength features disappear quickly. The wave equation preserves all frequencies exactly, merely changing their phase. The Laplace equation forbids oscillations in the interior entirely. These insights from fourier analysis unify the qualitative behavior of all three classic PDEs.
Advanced Topics
Green’s functions provide the response of a PDE to a point source. The solution to any forcing is then the convolution of the Green’s function with the forcing term. The method of characteristics reduces first-order PDEs to systems of ODEs along characteristic curves. Numerical methods like finite differences, finite elements, and spectral methods approximate PDEs on computers.
The method of characteristics applies to first-order PDEs by finding curves along which the PDE reduces to an ODE. For the transport equation u_t + cu_x = 0, characteristics are lines x − ct = constant, along which u is constant. The solution is any function of x − ct, representing a wave traveling at speed c without distortion.
Numerical Methods for PDEs
Finite difference methods discretize the spatial domain into a grid and approximate derivatives by difference quotients. The explicit Euler method for the heat equation is conditionally stable, requiring the time step to satisfy Δt ≤ (Δx)²/(2α). Implicit methods are unconditionally stable but require solving a linear system at each time step.
Finite element methods divide the domain into elements and approximate the solution as a combination of basis functions. They handle complex geometries and natural boundary conditions naturally. Spectral methods use global basis functions (Fourier series or Chebyshev polynomials), achieving exponential convergence for smooth solutions on simple domains.
What makes a PDE elliptic, parabolic, or hyperbolic? The classification depends on the coefficients of the highest-order derivatives. Elliptic equations (like Laplace) have no real characteristics, parabolic (like heat) have one, and hyperbolic (like wave) have two.
Why does the heat equation smooth out initial conditions? The exponential damping of high-frequency Fourier modes means sharp features decay quickly, leaving only smooth, slowly varying components.
What is the maximum principle for Laplace’s equation? The maximum of a solution to Laplace’s equation occurs on the boundary, never in the interior. This property underlies stability and uniqueness proofs.
How does the wave equation differ from the heat equation? The wave equation preserves amplitude and propagates signals at finite speed. The heat equation diffuses signals instantly (infinite propagation speed) with exponential decay.
Fourier Analysis — Numerical Analysis — Ordinary Differential Equations