What Is a PDE?

date2026-07-16document_iddoc_f10ec8080f4d7ec339fcff0ac1a7e625descriptionPDE の未知関数・独立変数・階数・線形性を読み取り、方程式・領域・データ・解のクラスから問題設定を組み立てる。prerequisites多変数関数と偏微分 / 微分方程式とは何かtype講義content_typelecturestatusactive

mathpartial-differential-equationsfoundationlecture

1Introduction

This lecture explains how to interpret a PDE as a relation between an unknown function of several independent variables and its partial derivatives. The independent variables need not be space and time: a static PDE such as Laplace's equation may contain several spatial variables and no time variable.

A {partial differential equation} is an equation or system involving one or more unknown functions of several independent variables and their partial derivatives. This lecture treats a single PDE for a scalar-valued unknown. In $u (x, t)$ , for example, $u$ is the unknown function and $x, t$ are the independent variables.

2Order and linearity

The {order} of a PDE is the highest order of the partial derivatives occurring in it. Thus

u_{t} = κ u_{x x}

is a second-order PDE because it contains $u_{x x}$ , although it is only first order with respect to time.

In a {linear PDE}, the unknown function and its derivatives occur to the first power, no products among them occur, and their coefficients depend only on the independent variables. For the associated operator $L$ ,

L [α u + β v] = α L [u] + β L [v] .

The equations $L [u] = 0$ and $L [u] = f$ are homogeneous and inhomogeneous linear equations, respectively; a zero right-hand side is not a requirement for linearity. By contrast, $u_{t} = u u_{x}$ is nonlinear because it contains a product of the unknown and its derivative.

3Components of a PDE problem

An equation alone does not determine a problem. The following components must be specified separately:

the unknown function and independent variables;
the equation and assumptions on its coefficients;
the {domain} of the unknown function;
data prescribed at an initial time, on a boundary, or on another set;
the regularity class in which a solution is sought.

A {classical solution} has the stated continuity and differentiability, satisfies the equation at every interior point, and satisfies each datum at every point where that datum is prescribed. The required regularity, including coefficient smoothness and compatibility with the data, must be stated for each problem.

4Difference from an ODE

The essential distinction between an ODE and a PDE is the number of independent variables and the kind of derivatives, not the names of the data.

Aspect	ODE	PDE
Unknown function	depends on one independent variable	depends on several independent variables
Derivatives	ordinary derivatives	partial derivatives
Domain	an interval	a region in a plane, in space, or in spacetime
Data	initial or endpoint values, among others	data on an initial surface or part of a boundary, among others

ODEs may have boundary-value problems, while PDEs may be posed on all of space and have no spatial boundary. Because PDE domains have diverse geometries, the admissible location of data must be assessed together with the order and type of the equation.

5Reading the heat equation

Let $κ > 0$ and consider, for $x \in R$ and $t > 0$ ,

u_{t} = κ u_{x x}, u (x, 0) = f (x) .

The unknown is $u (x, t)$ , the PDE is imposed on $R \times (0, \infty)$ , and the initial data are prescribed on $t = 0$ . This is a second-order linear PDE that is first order in time. Since the whole space has no spatial boundary, no spatial boundary condition is imposed. One possible classical-solution class is

u \in C^{2, 1} (R \times (0, \infty)) \cap C (R \times [0, \infty)) .

Here $C^{2, 1}$ denotes the class in which $u, u_{x}, u_{x x}, u_{t}$ are continuous.

If $f (x) = \sin x$ , then

u (x, t) = e^{- κ t} \sin x

satisfies

u_{t} = - κ e^{- κ t} \sin x, u_{x x} = - e^{- κ t} \sin x .

Hence $u_{t} = κ u_{x x}$ and $u (x, 0) = \sin x$ . It is therefore a classical solution of this problem. A candidate formula must be checked against the domain and all data as well as the equation.

6Changing the domain changes the data

To formulate the same heat equation on a bounded interval $0 < x < L$ as a standard uniquely determined initial-boundary-value problem, endpoint data are required in addition to initial data. If a classical solution is continuous up to the closed spacetime boundary, the initial and endpoint data must agree at the corners. The next lecture defines the principal types of initial and boundary conditions.

data/lecture/math/partial-differential-equations/initial-and-boundary-value-problems.lecture.n.md

7Map of representative models

For constant-coefficient models without external forcing or heat sources, with $κ > 0$ and $c > 0$ :

Model	Equation	Time variable
heat	$u_{t} = κ u_{x x}$	present; first order in time
wave	$u_{t t} = c^{2} u_{x x}$	present; second order in time
Laplace	$Δ u = 0$	absent

For $x = (x_{1}, \dots, x_{n}) \in R^{n}$ ,

Δ u = \sum_{j = 1}^{n} u_{x_{j} x_{j}}

is the sum of the second partial derivatives in all spatial directions. Later lectures compare the types, required data, information propagation, conservation, and dissipation of these models under explicit assumptions. Existence, uniqueness, and stability must not be inferred from an equation's name alone.

8Reading checklist

What are the unknown function and independent variables?
What are the overall order and the order with respect to time?
Is the equation linear or nonlinear, and what coefficient assumptions are imposed?
Is the domain the whole space or a bounded region?
Where are the data prescribed, and in which solution class?
Does a candidate satisfy the equation and every datum?

9Subsequent branches

First distinguish initial-value, boundary-value, and initial-boundary-value problems. Then proceed to characteristics for first-order PDEs or principal-part classification for second-order linear PDEs. Compare the representative models only after establishing that classification.

data/lecture/math/partial-differential-equations/method-of-characteristics.lecture.n.md data/lecture/math/partial-differential-equations/classification-of-second-order-linear-pdes.lecture.n.md data/lecture/math/partial-differential-equations/heat-wave-and-laplace-equations.lecture.n.md

10Scope of validity

This lecture used sufficiently smooth classical solutions as its point of entry. To include functions without classical derivatives, one must specify a function space and define satisfaction of the equation through integral identities against test functions or distributional derivatives. Admissible regularity and uniqueness conditions for such weak solutions must be determined separately for each equation.