Invitation to Partial Differential Equations

This text offers an introduction to linear partial differential equations (PDEs) with topics that include classical and modern approaches. It is based on notes for a course the author gave at Moscow State University. He worked for several years to finish a book derived from that course, but he died before it was completed. His colleagues at Northeastern University edited, revised and expanded the work to prepare it for publication.

 

The book is designed for a one-semester course for graduate students in mathematics. It requires a good background in graduate level analysis. Some prior acquaintance with PDEs would also be particularly helpful.

 

The author’s preface suggests that he wanted the book to be both modern and concise, but that he found this requirement to be contradictory. He settled on what he called “pictures at an exhibition”, typical methods and problems selected according to his taste. The distinction between modern and classical approaches to PDEs is nebulous, but “modern” generally refers to the use of Green’s function methods, distributions and Fourier transforms, while “classical” usually means techniques related to separation of variables and Fourier series.

 

It is evident from the beginning that the author regards mathematical physics as his prime motivation. He explores the Laplace, Poisson, heat and wave equations, beginning with the simplest form of the wave equation for a vibrating string. The general wave equation is then discussed in greater detail later in the book. Derivations of these equations are based primarily on physical and mechanical principles. In addition to the treatment of these equations and the general classes that include them, the book also includes discussion of distributions (aka generalized functions), Sobolev spaces, and a more general theory of potentials.

 

Each section of the book takes the reader from introductory material to more advanced topics. All of the usual topics of an introductory course at the graduate level are included. There are relatively few examples, and most of them appear as exercises. Within each chapter the development is clear and easy to follow.

 

The author is indeed concise, and the pace is fast. Readers new to the subject might wish for a clearer idea of where the author is going and why, and a better sense of the connections between topics and chapters. The chapter on distributions begins with only hints about why they are important, and only with a later chapter on Fourier transforms and convolutions does that become more evident. Sobolev spaces are also first introduced without explanation or motivation.

 

A modest collection of exercises is provided in each chapter. They come with solutions and hints in an appendix.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films, material science and the odd bit of high performance computing. He did his PhD work in dynamical systems and celestial mechanics.