Partial Differential Equations

This new textbook in partial differential equations (PDEs) is aimed at three distinct groups: students in “pure” mathematics, those in applied mathematics, and a broad category of others including students in the sciences, engineering, and economics. The author feels that PDEs have been neglected in many respects – that attention to them has been too limited, often restricted to the method of separation of variables for boundary value problems. He is passionate about expanding the scope to match the broad importance of PDEs.

 

The book is designed for a one or two-semester course at the senior undergraduate level, and the goal is to provide students with a clear understanding of the ideas at the heart of the discipline. The seven core topics are clearly identified: first order PDEs and the method of characteristics; the wave equation and wave propagation; the Fourier transform; diffusion and the diffusion equation; harmonic functions and the Laplacian; the fundamental solution of Laplace’s equation and Green’s functions, and; Fourier series and separation of variables techniques. Prerequisites include only proficiency with advanced calculus. A separate index includes short summaries of the primary topics and tools of advanced calculus.

 

The author has carefully structured the book into modules that address individual topics concisely, often in only a few pages. He also has provided a very detailed table of contents with indications of sections he considers basic material. With this there is clear documentation of the dependencies between chapters and identification of prerequisites for each chapter.

 

Perhaps the most unusual aspect of the book is its extensive treatment of distributions (aka generalized functions). The author feels strongly that this is a critical topic – that students need to understand delta functions, how to differentiate a function in terms of distributions, and how to take the limit of functions in the sense of distributions. He devotes two entire chapters to that subject.

 

Most of the applications that the author includes are in physics, and this may trouble students with special interests outside physics, but he argues that physical intuition is an important tool for guiding students through the analysis. Nonetheless, even a few more applications beyond physics would have been very helpful to motivate and encourage other students.

 

The text has some proofs and previous exposure to proofs at the level of an analysis course would be helpful but is not necessary. Some basic experience with ordinary differential equations would also be useful. The exposition throughout the book is clear, with good examples and generally good guidance for readers. Each chapter has a large collection of exercises. The appendix on advanced calculus provides review and reinforcement of its most relevant elements.

 

Computational aspects of PDEs are not considered here. The author believes his book’s material provides the core material needed for students to explore the deeper computational questions later. Some numerical questions are considered as they arise for particular equations (e.g., the finite difference method for the transport and wave equations).

 

Overall this is a clearly written and very thorough text, one that would support either classroom use or individual study for well-prepared students. The author provides a small set of references.

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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. He did his PhD work in dynamical systems and celestial mechanics.