Introduction 11
Alternatively, the right end may be insulated. We will see that, mathemati-
cally, this means that
ux(L, t)=0,t> 0. (1.7)
Equations (1.6) and (1.7) are calledboundary conditions,andasystem
like the one consisting of PDE (1.4), subject to conditions (1.5), (1.6) and
(1.7), is called aninitial-boundary-value problem.
As for simpler equations, in Exercises 18 and 19 in the previous section
we were asked to solve a first-order PDE subject to only one side condition.
In practice, one of the variables often will represent time, so the side con-
dition will be an initial condition, and the problem will be aninitial-value
problem. (In fact, when treating first-order PDEs in Chapter 5, we willal-
waysrefer to the side condition as an initial condition and the system as an
initial-value problem.)
Now, it turns out that the initial-boundary-value problem (1.4), (1.5), (1.6),
(1.7) has a unique solution. We call such a problem awell-posedproblem.
Similarly, the problems in Exercises 6c of 1.1, and Exercises 18a, 18b, 19a and
19b of 1.2 all are well-posed. Those in Exercises 6b and 6d of 1.1 and 18c and
19c of 1.2 arenotwell-posed.
To be precise, an initial-value or initial-boundary-value problem is well-
posed if
1) A solution to itexists.2) There is onlyonesuch solution (i.e., the solution isunique).3) The problem isstable.†Property (3), thestability condition, need not concern us. (Most, but not all,
of the problems considered in this book will be stable.)
By the way, remember from ODEs that, if an equation is of ordern,we
generally needninitial conditions to determine a unique solution. For PDEs,
the situation is much more complicated. However, notice that our heat equa-
tion example has one time derivative and one initial condition, while it has
twox-derivatives and twox-boundary conditions. This often is the case. So,
for example, in order that thefinitevibrating string problem be well-posed,
we will require two initial and two boundary conditions.
Exercises 1.3
The idea of well-posedness applies to ODEs, as well. Again, remember that
annth-order linear ODE, withnconditions assigned at the samex-value—
initial conditions—“usually” has a unique solution. This maynotbe the case,
†Basically a problem is stable if, whenever we change the initial or boundary conditions
by a “little bit,” the solution also changes by only a little bit—where we can, of course,
quantify what we mean by alittle bit.