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3. The Heat Equation
90
Here, one important question arises. When can a function f be expanded
in the way indicated in (3.6)? Obviously this is not possible for a completely
general family of functions {u k (x, t)}. Informally speaking, the family must
contain sufficiently many different functions in order to span a wide class
of functions. This is referred to as the problem of completeness, which will
be discussed in Chapters 8 and 9. Here we simply state that the family
{u k (x, t)} has an infinite number of members, and that it spans a very
large class of functions f
The observation that we need infinite series to expand initial functions
as in (3.6), has some serious implications. The arguments outlined above
assume finite linear combinations. When dealing with an infinite series, we
have to verify that this series converges towards a well-defined function u,
and that u is a solution of our problem. These tasks can be summarized as
follows:
Step 4:
(a) Verify that the series in (3.5) converges toward a well-defined
function u = u(x, t).
(b) Verify that the limit u solves the differential equation (3.1).
(c) Verify that the limit u satisfies the boundary condition (3.2).
(d) Verify that the limit u satisfies the initial condition (3.3).
The rest of this section will be devoted to the steps 1, 2, and 3. Here we
will simply leave the questions of convergence open, and just derive formal
solutions of our problems. When we refer to a solution as formal, it means
that not every step in the derivation of the solution is rigorously justified.
Formal solutions are often used in preliminary studies of problems, leaving
the justification to a later stage. This is often is a fruitful way of working.

3.2 Separation of Variables

Now we return to step 1 above, and the task is to find particular solutions
{u k (x, t)} of the form (3.4) satisfying the differential equation

for x ∈ (0, 1), t > 0, (3.7)
(u k ) t =(u k ) xx
subject to the boundary conditions

u k (0,t)= u k (1,t)=0. (3.8)

By inserting the ansatz
u k (x, t)= X k (x) T k (t) (3.9)