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Chapter 7

The Diffusion Equation

In this section we begin a study of the one-dimensional diffusion equation u t = ku xx .
Diffusions are very different from waves, and this is reflected in the mathematical properties
of the equations.

7.1 Diffusion on the whole line

Our purpose in this section is to solve the problemu

u t = ku xx (−∞ < x < ∞, 0 < t < ∞), (7.1)
u(x, 0) = φ(x). (7.2)

As with the wave equation, the problem on the infinite line has a certain ”purity,” which
makes it easier to solve than the finite-interval problem. Also as with the wave equation,
we will end up with an explicit formula. But it will be derived by a method very different
from the methods used before. (The characteristics for the diffusion equation are just the
lines t = constant and play no major role in the analysis.) Because the solution of (7.1)
is not easy to derive, we first set the stage by making some general comments.
Our method is to solve it for a particular φ(x) and then build the general solution from
this particular one. We’ll use five basic invariance properties of the diffusion equation (7.1).
1) The translate u(x − y, t) of any solution u(x, t) is another solution, for any fixed y.

2) Any derivative (u x or u t or u xx , etc.) of a solution is again a solution.

3) A linear combination of solutions of (7.1) is again a solution of (7.1). (This is just
linearity.)

4) An integral of solutions is again a solution. Thus if S(x, t) is a solution of (7.1),
then so is S(x − y, t) and so is

Z ∞
v(x, t) = S(x − y, t)g(y)dy
−∞
for any function g(y), as long as this improper integral converges appropriately.

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