The partial differential equations that govern scalar and vector fields are the very language used in physics and engineering to model a variety of phenomena in solid mechanics, fluid flow, acoustics, heat transfer, electromagnetism, and many others. Within the limits of linear models, a handful of fundamental equations (principally the Laplace, Poisson, diffusion and wave equations) are sufficient to describe a huge variety of phenomena, and methods for analyzing and solving these equations are essential parts of the toolkit of every working physical scientist and engineer.
Here, Andrea Prosperetti draws on many years' research and teaching experience to produce a guide to many of these methods. Topics range from essential ideas of real and complex analysis and matrices; through Fourier methods, special functions and Green's function theory; up to the theory of distributions and basic functional analysis. A unified view of the various methods is provided by the systematic use of eigenfunction expansions.
Theorems are stated precisely and their meaning explained, though proofs are mostly only sketched, with comments and examples being given more prominence. By avoiding proofs, the author presents powerful versions of theorems and provides a more textured and nuanced presentation of the various results. This approach leads to a deeper and more organic picture of each topic than is usually found in books of a similar nature.
Each chapter is essentially self-contained, and users can fashion their own path through the material with the aid of extensive cross referencing and a detailed index. A unique feature of the presentation is that the main methods are first introduced in the context of applications. The book is designed not only for sequential reading, but also as a resource that practitioners can dip into to learn about the whys, hows, and whens of the methods.
Industry Reviews
'This carefully written book by a well-known expert in the area is also an excellent guide to the present literature, recommended as well to graduate students as to experts in the area. This volume will help the reader in getting acquainted with some mathematical aspects of the modern theory of linear and non-linear phenomena arising in relevant applications to mathematical physics.' Zentralblatt MATH
'A truly wonderful book ... The author succeeded in creating a new type of book, that many will put on their desks, and they should: beginners, physicists, advanced learners, instructors, users of maths in the sciences ... A modern work, showing new ways, unusually multi-layered, applicable in many contexts and at many levels, an exciting book.' Siegfried Grossmann, Philipps-Universitaet Marburg
'This book admirably lays down physical and mathematical groundwork, provides motivating examples, gives access to the relevant deep mathematics, and unifies components of many mathematical areas. This sophisticated topics text, which interweaves and connects subjects in a meaningful way, gives readers the satisfaction and the pleasure of putting two and two together.' Laura K. Gross, SIAM Review
