Chapter 4
Separation of Variables
There are three paradigmatic linear second order partial differential equations that
have collectively driven the development of the entire subject.
The first two we have
already encountered: The
wave equation
describes vibrations and waves in continuous me
dia, including sound waves, water waves, elastic waves, electromagnetic waves, and so on.
The
heat equation
models diffusion processes, including thermal energy in solids, solutes
in liquids, and biological populations.
Third, and most important of all, is the
Laplace
equation
and its inhomogeneous counterpart, the
Poisson equation
, that govern equilib
rium mechanics. These two equations arise in an astonishing variety of mathematical and
physical contexts, ranging through elasticity and solid mechanics, fluid mechanics, electro
magnetism, potential theory, thermomechanics, geometry, probability, number theory, and
many more. The solutions to the Laplace equation are known as harmonic functions, and
the discovery of their many remarkable properties forms one of the most celebrated chap
ters in the history of mathematics. All three equations, along with their multidimensional
kin, appear repeatedly throughout this text.
The aim of the current chapter is to develop the method of separation of variables
for solving these key partial differential equations in their two independent variable incar
nations.
For the wave and heat equations, the variables are time,
t
, and a single space
coordinate,
x
, leading to initialboundary value problems modeling the dynamical behav
ior of a onedimensional medium. For the Laplace and Poisson equations, both variables
represent space coordinates,
x
and
y
, and the associated boundary value problems model
the equilibrium configuration of a planar body, e.g., the deformations of a membrane. Sep
aration of variables seeks special solutions that can be written as the product of functions
of the individual variables, thereby reducing the partial differential equation to a pair of
ordinary differential equations. More general solutions can then be expressed as infinite
series in the appropriate separable solutions.
For the twovariable equations considered
here, this results in a Fourier series representation of the solution. In the case of the wave
equation, separation of variables serves to focus attention on the vibrational character of
the solution, whereas the earlier d’Alembert approach emphasizes its particlelike aspects.
Unfortunately, for the Laplace equation, separation of variables only applies to boundary
value problems in very special geometries, e.g., rectangles and disks. Further development
of the separation of variables method for solving partial differential equations in three or
more variables can be found in Chapters 12 and 13.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Boundary value problem, Partial differential equation, Boundary conditions, Peter J. Olver

Click to edit the document details