Contemporary Mathematics
Volume 379, 2005
An Introduction to Wave Equations
Ronald E. Mickens
ABSTRACT. This paper provides a concise, but heuristic introduction to vari-
ous wave equations that serve important roles as mathematical models of many
interesting dynamical systems in the natural and engineering sciences. We be-
gin our discussion with a general introduction as to why the physical equations
should be reformulated, by means of scaling of their variables, to dimensionless
equations. This procedure is illustrated by applying it to a number of both lin-
ear and nonlinear ordinary and partial differential equations. The remainder
of the paper is devoted to analyzing certain of the mathematical properties of
some linear and nonlinear wave equations.
This paper is a summary of the topics presented by the author, in two lectures,
given at the Regional Research Conference, "Mathematical Methods in Nonlin-
ear Wave Propagation," held May 15-19, 2002 at the Mathematics Department of
North Carolina A and T State University in Greensboro, NC. My main task was
to provide general background information and physical explanations for many of
the issues discussed by the principal lecturer in his ten lectures. This person was
Kenneth Shaw (Departments of Mathematics, and Computer and Elec-
trical Engineering, Virginia Polytechnic Institute and State University, Blacksburg,
VA). Following an exchange of emails between us, we agreed that whatever each of
us presented, it should not contradict what the other gave! This compact worked
quite well as evidenced by the good reception of both of our presentations.
In the material to follow, I give a quick overview of several important topics
required if one is to understand the nature of wave equations, especially their use
as mathematical models of important physical and engineering systems. The paper
is organized into {our main sections. Section 2 introduces the concept of scaling of
variables such that the original "physical equations" get transformed into dimen-
sionless "mathematical equations." These are the differential equations that can
be studied using standard procedures from the area of differential equations. The
Mathematics Subject Classification.
Primary 35K15, 35K55, 35L05; Secondary 35L15,
35L45, 35L60.
Key words and phmses.
Nonlinear wave equations, scaling and dimensionless variables, soli-
tons, traveling waves.
The work reported here was funded in part by research grants from DOE and the MBRS-
SCORE Program at Clark Atlanta University.
2005 American Mathematical Society
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