Linear Methods of Applied Mathematics
Evans M. Harrell II and James V. Herod
(Some remarks for the instructor).
version of 11 May 2000
In this chapter we return to the subject of the heat equation, first
encountered in
Chapter VIII. The
heat equation reads
(20.1)
and was first derived by
Fourier
(see derivation). The same equation
describes the diffusion of a dye or other substance in a still fluid, and
at a microscopic level it results from random processes. We shall use
this physical insight to make a guess at the fundamental solution for the heat equation.
The term fundamental solution is the equivalent of the Green function for a parabolic PDE like the heat equation (20.1). Since the equation is homogeneous, the solution operator will not be an integral involving a forcing function. Rather, the solution responds to the initial and boundary conditions.
In order to limit the complications, we shall begin with the problem of diffusion in one space dimension, and place the boundaries at infinity, so the conditions will not be looked at explicitly. (Implicitly, there is a condition of boundedness; if we allowed the addition of u_{unbounded}(t,x) = x, for example, the problem would be in the second Fredholm alternative.)
XX.1. A well-posed problem for the one-dimensional heat equation would be of the form:
for - < x < and 0 < t < , with given initial data:
Definition XX.2. Suppose that A is a linear operator on a suitable space of functions of x, and that u satisfies
Au usual for Green functions, there is an intuitive physical meaning to the fundamental solution: It is the solution with initial data
There are several ways to derive the fundamental solution of the heat equation in unrestricted space. For example, we could apply either the Fourier or Laplace transforms, in the spatial variables x, to obtain a differential equation involving only t, and then transform back. Instead, here we use the physics of diffusion as a hint to form an ansatz for the solution and then make a calculation.
We take the opportunity at this stage to scale time to make the constant in the heat equation k=1; the mathematically natural time variable is t' = kt.)
The hint is that the underlying mechanism is some sort of random process which is isotropic, unrestricted in space, and invariant in time, then we might expect that the density u(t,x) could be a Gaussian distribution, and once it has this distribution, it should remain a Gaussian, although it will spread out as time goes by.
A Gaussian density on the x-axis, centered at x=0, is of the form
These differ only by the factor a'(t). Gratifyingly, the ansatz works with the very simple choice a(t) = t. We now take a closer look at the fundamental solution we have found:
(See the exercises for some alternative ways to derive the heat kernel.)
We have not yet explained in what sense the heat kernel approaches the delta function as t tends to 0. Notice that the Gaussian distribution of the heat kernel becomes very narrow when t is small, while the height scales so that the integral of the distribution remains one. Hence if we integrate it by any continuous, bounded function f(pix/bfxi.gif) and take the limit, we will in fact get f(x).
In the contrary limit, the distribution becomes very broad, corresponding to our experience that a diffusing substance tends to a widespread, nearly uniform, density.
Model Problem XX.4. Suppose k=1 in (20.2) and the initial condition is
Solution. We perform the integral
Definition XX.5. The function
In addition to helping us solve problems like Model Problem XX.4, the solution of the heat equation with the heat kernel reveals many things about what the solutions can be like. For example, if f(x) is any bounded function, even one with awful discontinuities, we can differentiate the expression in (20.3) under the integral sign. In fact, we can differentiate as often as we like. This means that an initially irregular distribution of temperature or a diffusing substance, is instantaneously smoothed out. Perhaps this is not too surprising when we think that the heat kernel itself solves the heat equation, and changes instantaneously from a delta function to a smooth, Gaussian distribution.
Model Problem XX.6. Again consider the one-dimensionalheat equation with k=1, but include a source of heat which is constant in time:
Solution. Let's try to reduce this problem to the familiar (20.1) solved by the fundamental solution we already know, by subtracting something off to make (20.6) homogeneous.
One way to do this is to write w(t,x) = u(t,x) + F(x), where F''(x) = f(x). That is, F(x) is any second integral of f(x). By substituting, we find that