Preface
THIS
book aims
to
present a broad
but
reasonably detailed account of the
mathematical solution of free and moving
bound~
problems.
The
mov-
ing boundaries occur mostly in heat-flow problems with phase changes
and in certain diffusion processes. A free boundary does not move but its
position has
to
be
determined as
part
of
the
solution of a steady-state
problem. Seepage through porous media
is
perhaps the most common but
by
no means
the
only source of problems
of
this kind.
The
broad spectrum of active research workers includes three groups:
engineers and others with practical problems; numerical analysts produc-
ing suitable numerical' algorithms;
and
pure mathematicians who decide
that
certain problems and their solutions exist, are properly posed, and
may even
be
unique. They also examine
the
convergence and stability
properties of numerical schemes. A few people fit into all three groups.
It
is
hoped
that
this book will help
to
alleviate
the
usual difficulties of
communication_ between
the
various interested parties.
.
Both
free and moving
boundari~
have been popular subjects for
research in recent years, leading
to
an almost bewildering collection of
new mathematical methods. This seems
to
be
an opportune time to
attempt a systematic presentation of them. Authors have tended to test
their methods by solving a small number of model problems and so
studies of melting ice, simple dams, oxygen diffusion with sorption, and
electrochemical machining are referred to frequently in this book. The
earliest mathematical work concentrated on one-dimensional problems.
Modern computer developments, however, make it possible to handle
free and moving boundaries in two and three space dimensions and to
model practical systems more realistically. Such methods feature largely
in this volume, though it makes no pretence
to
be
a compendium of
industrial problems.
Parallel studies of
the
mathematical properties of
the
differeatial equa-
tions and their solutions, and of
the
numerical algorithms, have been
prolific.
It
has not been possible here
to
do more than indicate some of
the
established results and
to'
include key references in the extensive
bibliography. A separate volume
is
needed
to
do justice to this aspect of
the
subjec~.