Elementary PDEs and Applications

This book grew out of a two-quarter sequence of undergraduate courses offered at the University of California (UCSB), for science majors, engineers and mathematicians. These courses along with a two-quarter sequence on ordinary differential equations (ODEs) and dynamical systems constitute the applied mathematics courses for the Program in Scientific Computations, a joint program between the mathematics department and the College of Engineering at UCSB. More information about this program can be obtained at the URL address: www.math.ucsb.edu/~birnir.

One could call the approach taken in this book a qualitative theory of linear PDEs and that is what the more theoretically inclined students can gain from it. A similar approach to nonlinear PDEs is the basis for a rapidly emerging qualitiative theory of nonlinear PDEs that is currently having a great impact on many fields of modern science. Fourier series (seperation of variables) are absent in this volume and this was done on purpose. The philosophy is that Fourier series give different insight into boundary value problems and should be taught seperately. Fourier series and numerical methods of PDEs are also treated in the courses mentioned above, but appear in the sequel to this volume.

This book is designed for students who will eventually be solving partial differential equations (PDEs) numerically. The aim is to teach them enough appreciation for the properties of the solutions to be able to judge with confidence, whether their numerical solutions make sense and how to interpret them. The examples, applications and exercises included in this book should give students a good intuition for the distinct qualities of waves, heat diffusion and fields.

The point of view presented in the book was not developed by the author. It originated with David Hilbert in Gottingen, at the turn of the last century, and the author learned it from his fellow graduate students at the Courant Institute in New York, in the late seventies. They introduced him to a wonderful book: Techniques in Partial Differential Equations, by C. R. Chester, that has unfortunately been out of print for many years. The other book the author owes much to is: Fourier Series and Integrals, by H. P. McKean and H. Dym. The contribution that the author wants to make to the material presented in these books, is to show how simple and elegant linear PDE theory is, if one just approaches it in the right way.

Copy of Book: Elementary PDEs and Applications

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