文档介绍：Chapter 11
Partial Differential
Equations
A wide variety of partial differential equations occurs in puting. We
cannot begin to cover them all in this book. In this chapter, we limit ourselves to
three model problems for second-order partial differential equations in one or two
space dimensions.
Model Problems
All the problems we consider involve the Laplacian operator, which is
∂2
4 =
∂x2
in one space dimension and
∂2 ∂2
4 = +
∂x2 ∂y2
in two space dimensions. We let ~x denote the single variable x in one dimension
and the pair of variables (x, y) in two dimensions.
The first model problem is the Poisson equation. This elliptic equation does
not involve a time variable, and so describes the steady state, quiescent behavior of
a model variable:
4u = f(~x).
There are no initial conditions.
The second model problem is the heat equation. This parabolic equation occurs
in models involving diffusion and decay:
∂u
= 4u − f(~x).
∂t
The initial condition is
u(~x, 0) = u0(~x).
December 26, 2005
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2 Chapter 11. Partial Differential Equations
The third model problem is the wave equation. This hyperbolic equation de-
scribes how a disturbance travels through matter. If the units are chosen so that
the wave propagation speed is equal to one, the amplitude of a wave satisfies
∂2u
= 4u.
∂t2
Typical initial conditions specify the initial amplitude and take the initial velocity
to be zero:
∂u
u(~x, 0) = u (~x), (~x, 0) = 0.
0 ∂t
In one dimension, all the problems take place on a finite interval on the x-axis. In
more than one space dimension, geometry plays a vital role. In two dimensions, all
the problems take place in a bounded region Ω in the (x, y) plane. In all cases, f(~x)
and u0(~x) are given functions of ~x. All the problems involve boundary conditions
where the value of u or some partial derivative of u is specified on the boundary of
Ω. Unless otherwise specified, we will take the boundary values to be zero.