Partial Differential Equations in Action

This work is an updated version of a book evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences and on the other to provide them with a solid theoretical background for numerical methods such as finite elements. Accordingly this textbook is divided into two parts. The first part chapters 2 to 5 is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion propagation and transport waves and vibrations. In the second part chapters 6 to 10 concentrate on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems while Chapter 11 deals with vector-valued conservation laws extending the theory developed in Chapter 4. The main differences with respect to the previous editions are: a new section on reaction diffusion models for population dynamics in a heterogeneous environment; several new exercises in almost all chapters; a general restyling and a reordering of the last chapters. The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines including applied mathematics physics and engineering.