Partial Differential Equations in Action (Universitext)

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Product Description

The main purpose is on the one hand to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences; on the other hand to give them a solid theoretical background for numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first one has a rather elementary character with the goal of developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. Ideas and connections with concrete aspects are emphasized whenever possible, in order to provide intuition and feeling for the subject. For this part, a knowledge of advanced calculus and ordinary differential equations is required. Also, the repeated use of the method of separation of variables assumes some basic results from the theory of Fourier series, which are summarized in an appendix. The main topic of the second part is the development of Hilbert space methods for the variational formulation and analysis of linear boundary and initial-boundary value problemsemph{. }%. Given the abstract nature of these chapters, an effort has been made to provide intuition and motivation for the various concepts and results. The understanding of these topics requires some basic knowledge of Lebesgue measure and integration, summarized in another appendix. At the end of each chapter, a number of exercises at different level of complexity is included. The most demanding problems are supplied with answers or hints. The exposition if flexible enough to allow substantial changes without compromising the comprehension and to facilitate a selection of topics for a one or two semester course.

Partial Differential Equations in Action (Universitext) Review

This is a GRADUATE LEVEL general book on modern PDE theory. If you are a Math or Applied Math graduate student preparing for your preliminary examination or qualifying exam I strongly recommend you to buy it. If you are someone looking for a book on cooking recipes on how to solve PDEs with analytic methods this is not the book for you.

You might already know that the facto standard book in PDE is Evans, which is a really well written book, however it is sometimes hard to read. How does the book of Salsa compare to Evans' book? Well, the contents of Salsa's book is the same as the content of the first seven chapters of Evans. However, the writing style is totally different, certainly Salsa's book constitutes a major pedagogical breakthrough. The chapters on classical (strong) solutions of prototypical PDEs (Laplace, Heat, Wave and First order) has most of the theory that you can find in Evans but in addition it also has many examples of these PDEs and how to analyze them and solve them or at least how to find a representation formula for their solution. Examples constitute a major component of the book.

The part of modern theory of second order PDEs (i.e. weak solutions in the context of Sobolev spaces) is at the level (or even better) of Evans. The chapter of Sobolev spaces and funcional analysis is really "user friendly" but still rigorous. It also has a section on distribution theory and Fourier transforms that Evans does not have at all. Then the rest of the book is devoted to Elliptic and Evolution equations in the context of Sobolev spaces.

Five stars for Sandro Salsa, he did a really good job in writing this book, the contents might be similar to those of Evans, but this book has a totally different flavor. The writing style of this book very direct and concrete but with many illustrations and examples that help to understand the message.

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