| dc.description.abstract | Partial differential equations (PDE) first appeared over 300 years ago, and the vast
scope of the theory and applications that have since developed makes it challenging
to give a reasonable introduction in a single semester. The modern mathematical
approach to the subject requires considerable background in analysis, including
topics such as metric space topology, measure theory, and functional analysis.
This book is intended for an introductory course for students who do not necessarily
have this analysis background. Courses taught at this level traditionally
focus on some of the more elementary topics, such as Fourier series and simple
boundary value problems. This approach risks giving students a somewhat narrow
and outdated view of the subject.
My goal here is to give a balanced presentation that includes modern methods,
without requiring prerequisites beyond vector calculus and linear algebra. To allow
for some of the more advanced methods to be reached within a single semester, the
treatment is necessarily streamlined in certain ways. Concepts and definitions from
analysis are introduced only as they will be needed in the text, and the reader is
asked to accept certain fundamental results without justification. The emphasis is
not on the rigorous development of analysis in its own right, but rather on the role
that tools from analysis play in PDE applications.
The text generally focuses on the most important classical PDE, which are the
wave, heat, and Laplace equations. Nonlinear equations are discussed to some
extent, but this coverage is limited. (Even at a very introductory level, the nonlinear
theory merits a full course to itself.)
I have tried to stress the interplay between modeling and mathematical analysis
wherever possible. These connections are vital to the subject, both as a source of
problems and as an inspiration for the development of methods. | en_US |
