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Partial Differential Equations: An Unhurried Introduction

Vladimir A. Tolstykh
De Gruyter
Publication Date: 
Number of Pages: 
De Gruyter Textbook
[Reviewed by
Jason Graham
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Partial Differential Equations An Unhurried Introduction by Tolstykh treats the solvability of first- and second-order partial differential equations (PDEs) in at most two independent variables at a level appropriate for a beginning course in the subject for undergraduate students. The heart of the book is chapters 5 – 7 which develop the theory of first-order quasilinear equations in two independent variables, that is, equations of the form \(a(x,y,u)u_{x}+b(x,y,u)u_{y}=c(x,y,u) \). The focus of the treatment, even for second-order equations, is on initial-value problems and there is no coverage of boundary-value problems, the method of separation of variables, or Fourier series and Fourier transforms. A reoccurring theme of Partial Differential Equations An Unhurried Introduction is the use of a change of variables in order to simplify a problem to a more manageable form, typically an ordinary differential equation or system of ordinary differential equations (ODEs). Even though there are few figures in the book, Tolstykh’s approach throughout is highly geometric. Almost all of the methods used in Partial Differential Equations An Unhurried Introduction are elementary and the only significant prerequisite for reading the book is multi-variable calculus, although some familiarity with ODEs would undoubtedly be beneficial. In fact, the book contains an appendix that provides a nice and thorough coverage of the inverse function theorem in a form that is well-suited for its application to the PDEs studied in the text. One minor exception regarding the previously mentioned required background for the book is the minimal use of holomorphic functions in the final section of the text which treats second-order elliptic equations.
Admittedly when I first skimmed Partial Differential Equations An Unhurried Introduction I was a little skeptical about the book. I have become accustomed to viewing undergraduate introduction to PDE courses as application oriented and geared towards examples of separation of variables and transform methods for physics and engineering students. My position has been that a study of all other aspects of PDEs besides separation of variables and transform methods is best postponed until after students have studied enough analysis to become receptive to the use of function spaces and inequalities to analyze solutions of PDEs.
While reviewing Partial Differential Equations An Unhurried Introduction, I have come to rethink my bias regarding the undergraduate PDE experience. I am immersed in a system where undergraduate students of mathematics, science, or engineering take some semesters of calculus followed by courses in linear algebra and/or ordinary differential equations.  The ODE experience, in my mind, can typically be viewed as both an extension and reinforcement of calculus II (integration techniques, power series, and perhaps parametric curves). On the other hand, the PDE experience that I have come to see as standard is less an extension or reinforcement of calculus III (multi-variable calculus and vector fields) and more an extension and reinforcement of linear algebra because the fundamental ideas behind separation of variables and transform methods is linearity and orthogonality. What I have discerned from reviewing Partial Differential Equations An Unhurried Introduction is that one can pretty easily teach a nice course in PDEs to undergraduate students that serves to extend and reinforce calculus III, especially the geometric aspects of calculus III.
Of course, extending and reinforcing linear algebra has great value and I would not argue against teaching Fourier series/transform, boundary-value problem oriented PDE courses. There are plenty of really nice textbooks for such a course, for example, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems by Haberman, Applied Partial Differential Equations by Logan, Introduction to Applied Partial Differential Equations by Davis, and at a slightly more advanced level Partial Differential Equations: An Introduction by Strauss. What I am suggesting is that with the publication of Partial Differential Equations An Unhurried Introduction, there is cause to consider an alternative offering for a first undergraduate level PDE course, one that extends and reinforces concepts from calculus III in a way that parallels the relationship between calculus II and the undergraduate ODE experience, and one for which there seems to be far fewer accessible texts available.
With my editorializing complete, I should end with a few more specific comments about Partial Differential Equations An Unhurried Introduction.  The text is clear and easy to read. There are a few places in the book where Tolstykh employs Maple or Wolfram Alpha to solve an equation or carry out manipulations. I feel like the book could be improved upon if the use of symbolic algebra systems was fleshed out a little more. Another point in which I feel the book is lacking regards the number of end-of-chapter exercises, there really aren’t very many at all just a handful per chapter. For me, a high point of reading Partial Differential Equations An Unhurried Introduction was the geometric approach that the author takes, I also really like that there is a significant treatment of not-necessarily-linear problems. Overall, Partial Differential Equations An Unhurried Introduction is an interesting textbook option and I believe it is worthy of adoption for a course.
Jason Graham is an Associate Professor in the Department of Mathematics at the University of Scranton. He received his PhD from the program in Applied Mathematical and Computational Sciences at the University of Iowa. His professional interests are in applied mathematics and mathematical biology.