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Preface

Our understanding of the fundamental processes of the natural world is based to a large
extent on partial differential equations. Examples are the vibrations of solids, the flow
of fluids, the diffusion of chemicals, the spread of heat, the structure of molecules, the
interactions of photons and electrons, and the radiation of electromagnetic waves. Partial
differential equations also play a central role in modem mathematics, especially in geome-
try and analysis. The availability of powerful computers is gradually shifting the emphasis
in partial differential equations away from the analytical computation of solutions and
toward both their numerical analysis and the qualitative theory.
This book provides an introduction to the basic properties of partial differential equa-
tions (PDEs) and to the techniques that have proved useful in analyzing them. Our
purpose is to provide for the student a broad perspective on the subject, to illustrate the
rich variety of phenomena encompassed by it and to impart a working knowledge of the
most important techniques of analysis of the solutions of the equations. One of the most
important techniques is the method of separation of variables (Fourier’s method). Many
textbooks heavily emphasize this technique to the point of excluding other points of view.
The problem with that approach is that only certain kinds of partial differential equations
can be solved by it, whereas others cannot. In this book it plays a very important but
not an overriding role. Other texts, which bring in relatively advanced theoretical ideas,
require too much mathematical knowledge for the typical undergraduate student. We
have tried to minimize the advanced concepts and the mathematical jargon in this book.
However, because partial differential equations is a subject at the forefront of research in
modem science, I have not hesitated to mention advanced ideas as further topics for the
ambitious student to pursue.
This is an undergraduate textbook. It is designed for juniors and seniors who are
science, engineering, or mathematics majors. Graduate students, especially in the sciences,
could surely learn from it, but it is in no way conceived ofas a graduate text.
The main prerequisite is a solid knowledge of calculus, especially multivariate. The
other prerequisites are small amounts of ordinary differential equations and of linear al-
gebra, each much less than a semester’s worth. However, since the subject of partial
differential equations is by its very nature not an easy one, we have recommended to our
own students that they should already have taken full courses in these two subjects.
The presentation is based on the following principles. Motivate with physics but then
do mathematics. Focus on the three classical equations: All the important ideas can be
understood in terms of them. Do one spatial dimension before going on to two and three
dimensions with their more complicated geometries. Do problems without boundaries

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