Partial Differential Equations with MATLAB

10 An Introduction to Partial Differential Equations with MATLAB©R

1.3 Initial and Boundary Conditions

In some of the exercises in the previous sections, we were asked to solve a
PDE and then to find the subset of those solutions which also satisfied an
additional requirement. Theseside conditionsare part and parcel of the
study of PDEs. As these conditions arise naturally in physical settings, let’s
introduce them by way of a specific physical problem.
In Section 1.1 we mentioned theheat equation

ut=α^2 uxx. (1.4)

Here,αis a constant andu=u(x, t) represents the temperature at any point
xalong a narrow piece of material, at any timet. (See Figure 1.1—we will
havemuchmore to say about this equation in Chapter 2 and beyond.) We
are asked to find the temperature function, that is, to solve the PDE. Now,
as we would like to predict the temperature of a particular piece of material,
we would like to find theonesolution of the PDE that does so. Certainly,
there must be some additional requirements at our disposal to narrow down
the general solution to one, unique solution.

x−axis

x= 0 x=L

FIGURE 1.1
Metal rod;uuu(((x, tx, tx, t)=)=)=temperature at pointxxx,attimettt.

First, it seems fairly clear that we cannot know the temperature at later
times if we don’t know the temperaturenowor,atleast,atsomedefinite
point in time. So we should hope that we are given, or can measure, the so-
calledinitialtemperature of the material at each pointx, at some specified
timet=t 0. That is, we would like to be given the functionffor which

u(x, t 0 )=f(x), 0 ≤x≤L. (1.5)

We call this aninitial condition. In practice, the initial time generally is
taken, or arranged, to bet 0 =0.
What additional requirements will we need? Well, it will turn out that
PDE (1.4) is derived under the assumption that the whole piece of material
is insulated except, possibly, at its ends, and that the heat “flows” only in
thex-direction. Therefore,it seems that we will need to know what is going
on at the endpoints. In fact, the endpoints generally are under the control of
the experimenters—so, for example, the left end may be held at a constant
temperature ofu 0 degrees, that is,

u(0,t)=u 0 ,t> 0. (1.6)