My Favourite Applied Math Books

Students often ask me ``What is the best book on . . . <SUBJECT X>?" I don't always have an answer, but in certain areas of applied mathematics (especially related to fluid mechanics or scientific computing) I do have my favourites. When reading this list, please keep in mind that all choices are biased toward my own personal mathematical interests, and that the focus here is more on books suitable for teaching rather than research monographs.

That said, please enjoy perusing this list. Comments, criticisms and suggestions are welcome!

What is the best book on . . . fluid mechanics?

For teaching, I like Acheson's Elementary Fluid Dynamics. Chorin and Marsden's A Mathematical Introduction to Fluid Mechanics is a more advanced treatise that is timeless. An honourable mention also goes to Ockendon and Ockendon's Viscous Flow, which is an absolute pleasure to read.

What is the best book on . . . computational fluid dynamics (CFD)?

This is a tough call. Most texts have a strong engineering bent, which I am not keen on. The best I've come across is Pozrikidis' Introduction to Theoretical and Computational Fluid Dynamics, which as the title indicates covers both the theory and computation. There are lots of practical examples and algorithms for incompressible flow and moving boundary problems. It's missing a discussion of hyperbolic problems (see below).

What is the best book on . . . special topics in fluid dynamics?

My hands-down favourite is John Crank's Free and Moving Boundary Problems which has everything you need to know about the Stefan problem and related models. A beautiful little book on suspension flows is A Physical Introduction to Suspension Dynamics. The clarity and insight of this text is complemented beautifully by the stunning hand-drawn figures!

What is the best book on . . . (applied) PDE?

I really like to teach undergraduates from a book like Gockenbach's Partial Differential Equations: Analytical and Numerical Methods. This book stresses the mind-numbingly beautiful connection with linear algebra by developing everything in terms of linear operators on vector spaces and essentially ignoring the cookbook-style focus on "separation of variables" of most other books. He doesn't go as deeply into the theory as some authors, but I think he strikes just the right balance for an undergraduate level course. And there is the added bonus that the author provides Matlab and Maple code to experiment with.

What is the best book on . . . numerical PDE?

Without a doubt, Strikwerda's Finite Difference Schemes and Partial Differential Equations. I am constantly referring to this book since it's so full of information. I am also a big fan of Morton and Mayer's Numerical Solution of Partial Differential Equations. It's short, sweet and has a collection of very nice exercises.

What is the best book on . . . hyperbolic PDE?

My favourite book is LeVeque's Numerical Methods for Conservation Laws. It's short and very sweet, and its only drawback is the lack of an index. But I've read it cover to cover so many times I don't need the index anyways. I prefer the style of this little book to LeVeque's more recent and expansive text on the subject . . . although that is also a very nice text if you're interested in the details of the algorithms behind his excellent CLAWPACK code. I cannot neglect to mention Tartar's From Hyperbolic Systems to Kinetic Theory: A Personalized Quest, which is a simply beautiful book to read. If only more math texts were written in this style . . .

What is the best book on . . . introductory numerical analysis?

The simple answer here is that there isn't one, at least from a teaching standpoint. I have yet to find a text that I am happy teaching an introductory numerical analysis class from. At SFU, we teach primarily to computer scientists and engineers using Matlab, and a good compromise is Recktenwald's Numerical Methods with MATLAB: Implementations and Applications.

What is the best book on . . . mathematical modelling?

I am a big fan of Fowler's Mathematical Models in the Applied Sciences, which betrays the time I have spent attending Study Groups.

What is the best book on . . . porous media flow?

Here again, there isn't one, but the best I've found is Kaviany's Principles of Heat Transfer in Porous Media.

What is the best . . . mathematics reference book?

Abramowitz and Stegun's Handbook of Mathematical Functions wins in this category, hands down. I don't think this book will ever lose its place on my shelf. Even the indexed electronic version doesn't appeal to me as much as paging through my dusty, musty, yellowed old copy of A&S.

What is the best book on . . . professional development for academics?

There are three books that I recommend to my students, as well as anyone else who asks:

• Handbook of Writing for the Mathematical Sciences by N. Higham, has great advice for writing and publishing.
• Tomorrow's Professor by R. Reis, has some great career advice from an engineer's perspective (cultural differences aside, mathematicians could benefit a lot from freading this gem of a book).
• Survival Skills for Scientists by F. Rosei and T. Johnston, gives a thought-provoking account of the "game of science" and many useful ideas not much discussed in mathematical circles.