Chapter 12
Partial Differential Equations in Space
At last we have reached the ultimate rung of the dimensional ladder (at least for
those of us living in a threedimensional universe): partial differential equations in physical
space. As in the one and twodimensional settings developed in the preceding chapters,
the three key examples are the threedimensional Laplace equation, modeling equilibrium
configurations of solid bodies, the threedimensional wave equation, governing vibrations of
solids, liquids, gasses, and electromagnetic waves, and the threedimensional heat equation,
modeling spatial diffusion processes.
Fortunately, almost everything of importance has already appeared, and appending
a third dimension is, for the most part, simply a matter of appropriately adapting the
constructions.
We have already developed the basic solution techniques:
separation of
variables and Green’s functions or fundamental solutions. (Unfortunately, the most pow
erful of our planar tools, conformal mapping, does
not
carry over to higher dimensions.) In
threedimensional problems, separation of variables is applicable in rectangular, cylindrical
and spherical coordinates (and eight additional exotic coordinate systems that we do not
treat, [
94
]). The first two do not produce anything fundamentally new, and are therefore
relegated to the exercises. Separation in spherical coordinates leads to spherical harmon
ics and spherical Bessel functions, whose properties are investigated in some detail. These
new special functions play important roles in a number of physical systems, including the
quantum theory of atomic structure that underlies the spectral and chemical properties of
atoms.
The Green’s function for the threedimensional Poisson equation in space can be iden
tified as the classic Newtonian (or Coulomb) 1
/r
gravitational (electrostatic) potential.
The fundamental solution for the threedimensional heat equation can be easily guessed
from its one and twodimensional versions.
The threedimensional wave equation, sur
prisingly, has an explicit, although more intricate, solution formula of d’Alembert form,
originally due to Poisson. Counterintuitively, the best way to handle the twodimensional
wave equation is by “descending” from the simpler (!) threedimensional formula. This
strategy highlights a remarkable difference between waves in planar and spatial media.
Huygens’ Principle states that threedimensional waves emanating from a localized initial
disturbance remain localized as they propagate through space.
In contrast, in two di
mensions, initially concentrated disturbances leave a slowly decaying remnant that never
entirely disappears.
In the final section, we analyze the threedimensional Schr¨
odinger equation associated
with the hydrogen atom, that is, the quantum mechanical system governing the motion of
a single electron around a nucleus. As we will see, the spherical harmonic eigensolutions
account for the observed energy levels of atoms that underly the periodic table and hence
molecular chemistry.
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Spherical Harmonics, Boundary value problem, Partial differential equation, Green's function, Peter J. Olver

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