The Heat, Wave, and Laplace Equations

date2026-07-16document_iddoc_36f16a47d0b53d03c6cd2539d28947e0descriptionheat・wave・Laplace 方程式を、必要データ、情報依存、保存・散逸、境界支配の共通軸で比較する。prerequisitesPDE の初期値問題と境界値問題 / 二階線形 PDE の分類 / Laplacian と物理応用type講義content_typelecturestatusactive

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1Introduction

This lecture compares three representative partial differential equations according to the data they require, the regions on which information depends, conservation and dissipation, and the role of boundaries. The preceding classification identified the heat equation as parabolic, the wave equation as hyperbolic, and the Laplace equation as elliptic. We now explain how these distinctions affect standard constant-coefficient models.

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2Standard Setting

Let $κ > 0$ and $c > 0$ . We consider sufficiently regular classical solutions of constant-coefficient equations without forcing, heat sources, or damping. The heat and wave equations are written in one spatial dimension $(x, t)$ , whereas the Laplace equation is posed on $Ω \subset R^{n}$ .

Model	Type	Equation	Time order
Heat	Parabolic	$u_{t} = κ u_{x x}$	1
Wave	Hyperbolic	$u_{t t} = c^{2} u_{x x}$	2
Laplace	Elliptic	$Δ u = 0$	No time variable

3An Equation Alone Does Not Define a Problem

A PDE problem combines an equation, a domain, and data. On the whole line, the heat and wave equations have no spatial boundary. On a bounded interval $0 < x < L$ , both require endpoint conditions such as Dirichlet, Neumann, or Robin conditions.

Because the heat equation is first order in time, one prescribes one initial temperature, $u (x, 0) = f (x)$ .
Because the wave equation is second order in time, one prescribes both displacement $u (x, 0) = f (x)$ and velocity $u_{t} (x, 0) = g (x)$ .
The Laplace equation has no time evolution and is posed as a boundary-value problem, for example with $u |_{\partial Ω} = h$ .

Classical solutions of bounded-domain evolution problems also require compatibility where the initial surface meets the spatial boundary. A pure Neumann problem for the Laplace equation has a compatibility condition and is unique only up to an additive constant.

The preceding lecture on initial- and boundary-value problems develops these aspects of the problem specification.

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4Heat: Infinite Propagation and Smoothing

On $R$ , let $f \in C_{c} (R)$ , where $C_{c} (R)$ denotes the continuous functions that vanish outside a bounded interval. The solution is expressed using the heat kernel

G_{t} (x) = \frac{1}{\sqrt{4 π κ t}} \exp (- \frac{x^{2}}{4 κ t}), t > 0,

u (x, t) = \int_{R} G_{t} (x - y) f (y) d y .

For $t > 0$ this is smooth, and it converges uniformly to $f$ as $t ↓ 0$ . Since $G_{t} (x - y) > 0$ , if $f [PARSE ERROR: Undefined("Command(\"ge\")")] 0$ and $f [PARSE ERROR: Undefined("Command(\"not\")")] \equiv 0$ , then $u (x, t) > 0$ at every $x$ for every $t > 0$ . Thus a disturbance influences the entire space immediately in this model; this property is called infinite propagation speed.

The lecture on Fourier transforms and PDEs derives the heat kernel.

data/lecture/math/partial-differential-equations/fourier-transforms-and-pdes.lecture.n.md

On $0 < x < L$ with homogeneous Dirichlet conditions,

\frac{d}{d t} \frac{1}{2} \int_{0}^{L} u^{2} d x = - κ \int_{0}^{L} u_{x^{2}} d x [PARSE ERROR: Undefined("Command(\"le\")")] 0,

so the squared $L^{2}$ norm is nonincreasing. Under homogeneous Neumann conditions,

\frac{d}{d t} \int_{0}^{L} u d x = κ [u_{x}]_{0^{L}} = 0,

so the mean temperature is conserved. The same integration by parts shows that the squared $L^{2}$ norm is nonincreasing under homogeneous Neumann conditions because its boundary term also vanishes. Boundary conditions therefore affect both conserved quantities and long-time behavior.

A mode is a spatial eigenfunction compatible with the boundary conditions whose time amplitude evolves independently under the linear PDE. For homogeneous Dirichlet conditions, $\sin (n π x / L)$ is such a spatial mode, and

u_{n} (x, t) = e^{- κ (n π / L)^{2} t} \sin (n π x / L) .

Direct substitution verifies both the equation and the boundary conditions. Higher-frequency modes decay faster, which explains the smoothing mechanism. The next lecture develops the selection and superposition of these modes by separation of variables.

5Wave: Finite Propagation and Energy

For $x \in R$ , $t [PARSE ERROR: Undefined("Command(\"ge\")")] 0$ , $f \in C^{2} (R)$ , and $g \in C^{1} (R)$ , d'Alembert's formula is

u (x, t) = \frac{f (x - c t) + f (x + c t)}{2} + \frac{1}{2 c} \int_{x - c t}^{x + c t} g (s) d s .

The value at $(x, t)$ depends only on initial data in $[x - c t, x + c t]$ . This domain of dependence expresses finite propagation with speed $c$ . The factorization

\partial_{t^{2}} - c^{2} \partial_{x^{2}} = (\partial_{t} - c \partial_{x}) (\partial_{t} + c \partial_{x})

and the characteristic coordinates $x - c t$ and $x + c t$ underlie the formula. On a bounded interval, reflected waves must instead be treated according to the boundary conditions.

Define

E (t) = \frac{1}{2} \int_{0}^{L} (u_{t^{2}} + c^{2} u_{x^{2}}) d x .

Without forcing,

E^{'} (t) = c^{2} [u_{t} u_{x}]_{0^{L}} .

For fixed endpoints $u = 0$ , one has $u_{t} = 0$ at the endpoints; for free endpoints $u_{x} = 0$ . In either case the boundary flux vanishes and $E (t) = E (0)$ . Boundary input, forcing, or damping generally changes the energy.

data/lecture/math/partial-differential-equations/introduction-to-energy-methods.lecture.n.md

6Laplace: Global Boundary Dependence without Time Evolution

A $C^{2}$ solution of $Δ u = 0$ is harmonic and satisfies the mean-value property. Unlike the preceding evolution equations, the Laplace equation describes a static global relation between boundary and interior values.

For the unit disk, let $h$ be continuous and $2 π$ -periodic, and write $e^{i α} = (\cos α, \sin α)$ . With boundary data $u (e^{i φ}) = h (φ)$ , the Poisson integral formula is

u (r e^{i θ}) = \frac{1}{2 π} \int_{0}^{2 π} \frac{1 - r^{2}}{1 - 2 r \cos (θ - φ) + r^{2}} h (φ) d φ, 0 [PARSE ERROR: Undefined("Command(\"le\")")] r < 1.

It shows that one interior value depends on data from the entire boundary. This is static global dependence, not finite or infinite propagation in time.

If $Ω$ is bounded and connected and $u \in C^{2} (Ω) \cap C (\overline{Ω})$ is harmonic, the maximum principle places the maximum and minimum on $\partial Ω$ . It implies uniqueness for harmonic functions with the same Dirichlet data. Existence requires a separate argument, and a pure Neumann problem remains indeterminate up to an additive constant.

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7Comparison on Common Axes

Axis	Heat	Wave	Laplace
Basic data	One initial value	Two initial values	Boundary values such as Dirichlet data
Bounded domain	Also requires lateral boundary conditions	Also requires lateral boundary conditions	Requires spatial boundary conditions
Information dependence	Reaches all space at every positive time	Domain of dependence with speed $c$	No temporal causality; globally coupled to the boundary
Integral quantity	Squared norm is nonincreasing for homogeneous Dirichlet or Neumann data; mean is conserved for homogeneous Neumann data	Energy is conserved in a closed system	Static equilibrium rather than temporal conservation
Pointwise control	Maximum principle on the initial surface and lateral boundary	No analogous maximum principle	Maximum principle controlled by the spatial boundary
Typical long-time behavior	Relaxes to zero, the initial mean, or a stationary distribution according to boundary data	Undamped fixed-end solutions generally continue to oscillate	No time evolution

For the heat equation, the parabolic boundary consists of the initial surface and the spatial boundary, not the upper surface at the final time. A later lecture proves the maximum principles for the heat and Laplace equations.

The long-time row describes typical behavior for sufficiently regular solutions in the one-dimensional, constant-coefficient, source-free setting of this lecture, with homogeneous or fixed boundary data that are independent of time.

8Dimensional Check

If $u$ is temperature, $t$ is time, and $x$ is length, then $κ$ has dimensions length squared per time. In the wave equation, $c$ has dimensions length per time and is the propagation speed. Accordingly, the diffusion length is $\sqrt{κ t}$ , whereas the wave travel distance is $c t$ .

9Scope of Validity

The formulas and identities above depend on constant coefficients, absence of sources, and sufficient regularity. For nonlinear diffusion, damping, forcing, inhomogeneous media, or weak data, conservation laws, propagation speeds, and maximum principles require separate analysis. This lecture does not complete the solution methods; it identifies the roles developed by separation of variables, Fourier analysis, maximum principles, and energy methods.

10Related Lectures

data/lecture/math/partial-differential-equations/separation-of-variables-and-fourier-series.lecture.n.md data/lecture/math/vector-calculus/laplacian-and-physical-applications.lecture.n.md