Introduction to Partial Differential Equations

Description

Altmetric | student solutions manual and Corrections to second printing appended to end of PDF (TAR file is of the accompanying movies)

1. PDF pp. 199ff. is on Cauchy's method of characteristics
2. PDF p. 40: "method of characteristics is able to solve a first-order linear partial differential equation by reducing it to one or more first-order nonlinear ordinary differential equations."

The author of this undergraduate textbook is a highly-regarded researcher, and the author of several other mathematical texts. His goal in writing this book was to supply "a well-written, systematic, modern introduction to the basic theory, solution techniques, qualitative properties, and numerical approximation schemes for the principal varieties of partial differential equations that one encounters in both mathematics and applications''. It differs from many other textbooks in the target market in that it is intended for a year-long course, and the typical emphasis on separation of variables is supplemented by an extensive treatment of characteristics , Green's functions, and numerical methods. While the abstract theory of linear differential transformations on inner product spaces is discussed, the approach to this material remains at an undergraduate level.
Early on, the author states that "[w]hen entering a new mathematical subject … one should first analyze and fully understand the very simplest examples''. He then proceeds to convince the reader that this approach is a good one by treating simple examples with great care and clarity. The author evidently does not consider elementary material to be drudgery, since he writes with an enthusiasm that the reviewer found to be contagious.
The book is structured in such a way that the heat, wave, and Poisson equations are all first studied in one space dimension before proceeding to two-dimensional problems, and then problems in R3. The advantage to this method is that concepts can be introduced without distraction from complicating geometry. The disadvantage to this method is that there is substantial repetition in the higher-dimensional material, but this will probably be less noticeable over a year-long course than over a two-week review.
From beginning to end, the book is well written. Treatment of the Dirac delta function manages to be simultaneously accessible and honest. There is a particularly nice motivation given for the Fourier integral as the limit of Fourier series. The author chose to deemphasize modeling, and that choice works out well. There are many exercises, and they seem to have been written with great care. Students are not just asked to plug parameters into templates but are given the opportunity to interpret formulas, to apply them in interesting contexts, and to discover new things. The book has a dedicated web site with many helpful animations of solutions to PDEs.
There seem to be very few typographical errors for a book of this size. As appears to be true of almost all books from Springer Verlag nowadays, this book is print-on-demand, and the glossy toner used to print the review copy was not as easy to read as the ink of an offset-printed copy would have been. Figure 11.1, which is intended to show the use of a diffusion equation to denoise a grayscale photograph, is (in this copy) too blurry to be of much use. But at least the text is crisper than earlier print-on-demand volumes tended to be. Reviewed by Christopher P. Grant

"This textbook furnishes the basis for a 1-year introductory course in partial differential equations for advanced undergraduates. … The book is written with great care and great attention to detail throughout. At the end of every chapter there are well-chosen exercises that genuinely add depth to the concepts treated in the text. … this book can be wholeheartedly recommended."

• M. Kunzinger, Monatshefte für Mathematik, Vol. 181, 2016

"This book easily covers all the material one might want in a course aimed at first-time students of PDEs. … I recommend this one highly: It provides the best first-course introduction to a vast and ever-more relevant and active area. Students, and perhaps instructors too, will learn much from it. If they wish to go beyond the material taught in a first course, this text will prepare them better than any other I know."

• SIAM Review, Vol. 56 (3), September, 2014

"Introduction to Partial Differential Equations is a complete, well-written textbook for upper-level undergraduates and graduate students. Olver … thoroughly covers the topic in a readable format and includes plenty of examples and exercises, ranging from the typical to independent projects and computer projects. … Instructors teaching an introduction to partial differential equations course will want to consider this textbook as a viable option for their students. Summing Up: Highly Recommended. Upper-division undergraduates, graduate students, and faculty."

• S. L. Sullivan, Choice, Vol. 51 (11), July, 2014

"This introduction to partial differential equations is addressed to advanced undergraduates or graduate students … . an imposing book that includes plenty of material for two semesters even at the graduate level. … The author succeeds at maintaining a good balance between solution methods, mathematical rigor, and applications. With appropriate selection of topics this could serve for a one semester introductory course for undergraduates or a full year course for graduate students. … the author has clearly taken pains to make it readable and accessible."

• William J. Satzer, MAA Reviews, January, 2014

This textbook is designed for a one year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, engineering, and elsewhere. The exposition carefully balances solution techniques, mathematical rigor, and significant applications, all illustrated by numerous examples. Extensive exercise sets appear at the end of almost every subsection, and include straightforward computational problems to develop and reinforce new techniques and results, details on theoretical developments and proofs, challenging projects both computational and conceptual, and supplementary material that motivates the student to delve further into the subject.

No previous experience with the subject of partial differential equations or Fourier theory is assumed, the main prerequisites being undergraduate calculus, both one- and multi-variable, ordinary differential equations, and basic linear algebra. While the classical topics of separation of variables, Fourier analysis, boundary value problems, Green's functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and similarity, the Maximum Principle, financial models, dispersion and solitons, Huygens'

Principle, quantum mechanical systems, and more make this text well attuned to recent developments and trends in this active field of contemporary research. Numerical approximation schemes are an important component of any introductory course, and the text covers the two most basic approaches: finite differences and finite elements.

Peter J. Olver is professor of mathematics at the University of Minnesota. His wide-ranging research interests are centered on the development of symmetry-based methods for differential equations and their manifold applications. He is the author of over 130 papers published in major scientific research journals as well as 4 other books, including the definitive Springer graduate text, Applications of Lie Groups to Differential Equations , and another undergraduate text, Applied Linear Algebra.

A Solutions Manual for instrucors is available by clicking on "Selected Solutions Manual" under the Additional Information section on the right-hand side of this page.