Partial Differential Equations in Action: From Modelling to Theory (Universitext) 1st 2008. Corr edition by Salsa, Sandro (2010) Paperback

by Sandro Salsa

This book is designed as an advanced undergraduate or a first-year graduatecourse for students from various disciplines like applied mathematics,physics, engineering.The main purpose is on the one hand to train the students to appreciate theinterplay between theory and modelling in problems arising in the appliedsciences; on the other hand to give them a solid theoretical background fornumerical methods, such as finite elements.Accordingly, this textbook is divided into two parts.The first one has a rather elementary character with the goal ofdeveloping and studying basic problems from the macro-areas of diffusion,propagation and transport, waves and vibrations. Ideas and connections withconcrete aspects are emphasized whenever possible, in order to provideintuition and feeling for the subject.For this part, a knowledge of advanced calculus and ordinary differentialequations is required. Also, the repeated use of the method of separation ofvariables assumes some basic results from the theory of Fourier series,which are summarized in an appendix.The main topic of the second part is thedevelopment of Hilbert space methods for the variational formulation andanalysis of linear boundary and initial-boundary value problems mph{. }%Given the abstract nature of these chapters, an effort has been made toprovide intuition and motivation for the various concepts and results.The understanding of these topics requires some basic knowledge of Lebesguemeasure and integration, summarized in another appendix.At the end of each chapter, a number of exercises at different level ofcomplexity is included. The most demanding problems are supplied withanswers or hints.The exposition if flexible enough to allow substantial changes withoutcompromising the comprehension and to facilitate a selection of topics for aone or two semester course.

Genres: Mathematics

Paperback

First published May 5, 2004

Reviews

[Old Man · Rating 2 out of 5]

My review applies to the third edition of this book, focusing specially on my experience reading chapters 6 and 7.

Against:
Too many typos, present in previous editions...
In many sections there's simply too much hand waving, for a first time student of this subject. This book would gain much by having maybe less chapters, and a more profound, and a slower pace.
Many statements( in some definitions, theorems, etc) are too imprecise, which made me recurrently search the internet for an equivalent statement. In chapter 7 later sections, for example 7.7 where we want to prove that the set of restrictions in dense in the associated Sobolev space, the author simply does a lot of hand waving, some proofs cannot even be called sketches.

Examples: 1) The statement for Riesz Representation theorem is not correct if we consider complex functions, instead of just real ones. Even though in that section the author states he's focusing on real functions, he changes focus when later on we tend to deal with Fourier transform, and complex functions
2) the common compact condition for the test functions in the notion of convergence definition for the test function space is very imprecise.

I could give many others.

If you're learning by yourself, you might be more satisfied with Evans' bible on PDE.

For:
Even though there are many flaws, I still feel I'm able to learn something by myself.
Easy to get a free copy on the internet.