Prelude to Chapter 1
We have seen how physical problems often give rise to ordinary differential
equations (henceforth, ODEs). Thesesame and similar physical problems,
when involving more than one independent variable, lead us to, instead,par-
tialdifferential equations (PDEs). A PDE, therefore, will look very much like
an ODE, except that the unknown function will be a function ofseveralvari-
ables - and, of course, any derivatives that appear must bepartialderivatives.
Although we shall find a number of PDEs which are solved in the same way
that we solved ODEs, this happy state of affairs will be short-lived. Indeed,
two- and higher-dimensional mathematical objects exhibit a wealth of behav-
ior which we do not see in one-dimensional objects. Similarly, PDEs, as a
rule, will exhibit much more complicated behavior and, therefore, be much
harder to solve than ODEs.
In this first chapter, we introduce PDEs, and we point out those which
can be solved like ODEs. Historically, many of these simpler PDEs were
overlooked by earlier mathematicians, simply because they weren’t interesting
(they already knew how to solve ODEs) or important (the really interesting
physical problems led to PDEs which could not be solved in this manner).
So, in the 18th century, we see famous mathematicians jumping right into the
more difficult equations, those which we will begin to discuss in Chapter 2.
We also treat in this chapter the PDE analogs of other ideas that were
studied in ODEs: initial and boundary conditions, and the important concept
of a linear PDE. We then introduce one of the most important tools for solving
linear PDEs, the method of separation of variables. The so-called product
solutions which are derived via separation of variables were studied as early
as the first half of the 18th century, first by Daniel Bernoulli (1700–1782,
son of John Bernoulli) in his study of the vibrating chain, then by the great
Leonhard Euler (1707–1783) and, most notably, by Jean Le Rond d’Alembert
(1717–1783), in their work on the wave equation (which models the vibrations
of a string).
Finally, we turn back to ODEs and look at the so-called eigenvalue problems
which arise when we apply separation of variables to PDEs. At this point, we
consider only simpler special cases of this type of problem, reserving a more
complete study for Chapter 8.