Lecture 40: Separation of Variables and Heat Equation IVPs
This isn’t very satisfying: there’s no reason to suppose that our initial distribution is a finite sum of sine functions. Physically, such situations are clearly very special. What do we do if we have a more general initial temperature distribution?
Let’s consider what happens if we take an in nite sum of our separated solutions. Then our
solution to (40.1a) and (40.1b) is
u(x, t) =
Now the initial condition specifies that the coefficients must satisfy
This idea is due to the French mathematician Joseph Fourier1, and (40.4) is called the Fourier sine series for f(x).
There are several very important questions raised by this, however. Why should we believe that our initial condition f(x) ought to be able to be written as an infinite sum of sines? Why should we believe that such a sum would converge to anything? We’ll come to these in good time, but keep them in mind.
2.1. Neumann boundary conditions. Now let’s consider a heat equation problem with
homogeneous Neumann conditions,
ut = kuxx
ux(0, t) = ux(l, t)
u(x, 0) = f(x).
We’ll start by again supposing that our solution to (40.5a) is separable, so we have u(x, t) = X(x)T (t) and, as we’re using the same di erential equation as before, we obtain the pair of ODEs (40.2a) and (40.2b). The solution to (40.2b) is still
T (t) = Ae kt.
Now we need to determine the boundary conditions for (40.2a). Our boundary conditions (40.5b) are on ux(0, t) and ux(l, t); thus they are conditions on X0(0) and X0(l), as the t-derivative isn’t controlled at all. So we have the boundary value problem
X00 + λX = 0
X0(0) = 0
X0(l) = 0.
Along the lines of the analogous computation last lecture, this has eigenvalues and eigenfunctions
yn(x) = cos l nπx
for n = 0, 1, 2, . . . So individual separable solutions to (40.5a) and (40.5b) have the form
un(x, t) = Ane
1Fourier (1768-1830) was a big promoter of the French Revolution, and traveled with Napoleon to Egypt, where
he was made governor of Lower Egypt. Fourier has been credited with being one of the first to understand the greenhouse e ect and, more generally, planetary energy balance.
There’s a great (though almost certain apocryphal) story about what led Fourier to study the heat equation. Reportedly, what deeply concerned Fourier was being able to find the ideal depth to build his wine cellar so that the wine would be stored at the perfect temperature year-round, and so he proceeded to try to understand the way heat propagated through the ground.