This book is designed as an advanced undergraduate or a first-year graduate
course for students from various disciplines like applied mathematics,
physics, engineering.

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 problems\emph{. }%
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.