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
levelof
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.
General
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