Computer Algebra Recipes

Chapter 5
Linear PDE Models. Part 1
Because linear partial di®erential equations play such an important role in the
mathematical description of electromagnetic waves, heat °ow, elastic vibrations,
and many other scienti¯c phenomena, there is an abundance of wonderful examples
that can be solved using computer algebra. For this reason, this topic
is split over two chapters. We begin with examples of checking PDE solutions,
either obtained by intelligent guessing or quoted, without derivation, in some
scienti¯c reference. Di®usion and Laplace's equation models are then presented.
5.1 Checking Solutions
5.1.1 The Palace of the Governors
The knowledge of the world is only acquired in the world,
and not in a closet.
Phillip Dormer Stanhope, Earl of Chester¯eld (1694{1773)
While driving back to Phoenix from Los Alamos, where he attended an engineering
conference at the Los Alamos National Laboratory, Russell stops in
Sante Fe to have something to eat and to tour the historic Plaza area of town.
After treating himself to a gourmet lunch, consisting of a tasty rattlesnake
burger washed down with a Corona beer, he strolls around the Plaza. As an
engineer, Russell is particularly impressed by the massively thick walls of the
Palace of the Governors. This long, low adobe structure, which was built in
1610 by the Spanish, is the oldest continuous seat of government in the United
States. The thick walls, which is a design feature of many historic buildings in
the American Southwest, not only o®ered protection from attackers but helped
to keep the heat of the summer sun at bay as well as excluding the cold breath
of winter.
On completing his tour of the Plaza, Russell is intrigued by the question
of how e®ective a very thick adobe wall is in cutting down the incident solar
radiation from the summer sun. Pulling out his laptop computer, and ¯nding
a shady spot, he formulates the following relevant model. For simplicity, he
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