Applied Partial Differential Equations, revised edition
The importance of partial differential equations (PDEs) for both pure analysis and applied mathematics is beyond any doubt and a number of books written on the topic stand testament to this widely accepted fact. In view of this, it is a hard task to write a book that would be recommended as one of the first to read on the topic. The authors seem to have fulfilled the task quite well. Almost all the basic notions of PDEs are covered and studied in the book.
In the context of first order quasilinear systems, we learn about characteristics and a question of blow-up of the smooth solution due to the nonlinear structure of characteristic fields, we are led to solutions with discontinuities and the concept of a weak solution. In the system case, the quasilinear phenomena are studied with mention to the classical Cauchy-Kowalevskaya theorem and proceeding to the concept of hyperbolicity and more non-trivial concepts such as Riemann invariants, domain of dependence, the concept of entropy and the discussion of its connection to the notion of viscosity. For the second order equations, the book confronts the classical classification into elliptic, parabolic and hyperbolic equations, dealing with all classical concepts such as the maximum principle and energy methods and the methods of the Green function. Moreover, there is an extensive and instructive employment of various integral transforms (e.g. Mellin, Hankel, Fourier, and Radon transforms). The reader can also learn some non-standard (or not very frequently used) techniques such as the hodograph method, methods of conformal mappings and the Wiener-Hopf method in 2D setting. The problem of free boundaries is also touched upon. Physicists will be pleased to find not only the notion of the Laplace and Poisson equations, Helmholtz’ equation, heat and mass transfer and convection-diffusion problems but also the Maxwell equations, gravitation, heat transfer, acoustics, etc. The book is very well written, equipped with numerous exercises and applications and will serve as a very good textbook both for masters and PhD students.