Preface
Many problems in the physical world can be modeled by partial differential
equations, from applications as diverse as the flow of heat, the vibration of a
ball, the propagation of sound waves, the diffusion of ink in a glass of water,
electric and magnetic fields, the spread of algae along the ocean’s surface,
the fluctuation in the price of a stock option, and the quantum mechanical
behavior of a hydrogen atom. However, as with any area of applied mathe-
matics, the field of PDEs is interesting not only because of its applications,
but because it has taken on a mathematical life of its own. The author has
written this book with both ideas in mind, in the hope that the student will
appreciate the usefulness of the subject and, at the same time, get a glimpse
into the beauty of some of the underlying mathematics.
This text is suitable for a two-semester introduction to partial differen-
tial equations and Fourier series for students who have had basic courses in
multivariable calculus (through Stokes’s and the Divergence Theorems) and
ordinary differential equations. Over the years, the author has taught much of
the material to undergraduate mathematics, physics and engineering students
at Penn State and Fairfield Universities, as well as to engineering graduate
students at Penn State and mathematics and engineering graduate students
at Fairfield. It is assumed that the student hasnot had a course in real
analysis. Thus, we treatpointwiseconvergence of Fourier series and do not
talk aboutmean-squareconvergence until Chapter 8 (and, there, in terms
of the Riemann, and not the Lebesgue, integral). Further, we feel that it is
not appropriate to introduce so subtle an idea asuniformconvergence in this
setting, so we discuss it only in the Appendices.
Approach and Suggestions for Instructor
One may approach the teaching of PDEs in one of two ways: either based
on type of equation, or based on method of solution. While appreciating the
importance of the former idea, we have chosen the latter approach, as it
- allows us to treat problems in one spatial dimension before dealing with
those in higher dimensions, and
xi