Partial differential equations in action

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\emph{. }%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.