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The subject of fractional calculus and its applications (that is,
convolution-type pseudo-differential operators including integrals
and derivatives of any arbitrary real or complex order) has gained
considerable popularity and importance during the past three
decades or so, mainly due to its applications in diverse fields of
science and engineering. These operators have been used to model
problems with anomalous dynamics, however, they also are an
effective tool as filters and controllers, and they can be applied
to write complicated functions in terms of fractional integrals or
derivatives of elementary functions, and so on.This book will give
readers the possibility of finding very important mathematical
tools for working with fractional models and solving fractional
differential equations, such as a generalization of Stirling
numbers in the framework of fractional calculus and a set of
efficient numerical methods. Moreover, we will introduce some
applied topics, in particular fractional variational methods which
are used in physics, engineering or economics. We will also discuss
the relationship between semi-Markov continuous-time random walks
and the space-time fractional diffusion equation, which generalizes
the usual theory relating random walks to the diffusion equation.
These methods can be applied in finance, to model tick-by-tick
(log)-price fluctuations, in insurance theory, to study ruin, as
well as in macroeconomics as prototypical growth models.All these
topics are complementary to what is dealt with in existing books on
fractional calculus and its applications. This book was written
with a trade-off in mind between full mathematical rigor and the
needs of readers coming from different applied areas of science and
engineering. In particular, the numerical methods listed in the
book are presented in a readily accessible way that immediately
allows the readers to implement them on a computer in a programming
language of their choice. Numerical code is also provided.
This book will give readers the possibility of finding very
important mathematical tools for working with fractional models and
solving fractional differential equations, such as a generalization
of Stirling numbers in the framework of fractional calculus and a
set of efficient numerical methods. Moreover, we will introduce
some applied topics, in particular fractional variational methods
which are used in physics, engineering or economics. We will also
discuss the relationship between semi-Markov continuous-time random
walks and the space-time fractional diffusion equation, which
generalizes the usual theory relating random walks to the diffusion
equation. These methods can be applied in finance, to model
tick-by-tick (log)-price fluctuations, in insurance theory, to
study ruin, as well as in macroeconomics as prototypical growth
models.All these topics are complementary to what is dealt with in
existing books on fractional calculus and its applications. This
book will keep in mind the trade-off between full mathematical
rigor and the needs of readers coming from different applied areas
of science and engineering. In particular, the numerical methods
listed in the book are presented in a readily accessible way that
immediately allows the readers to implement them on a computer in a
programming language of their choice.The second edition of the book
has been expanded and now includes a discussion of additional,
newly developed numerical methods for fractional calculus and a
chapter on the application of fractional calculus for modeling
processes in the life sciences.
Fractional calculus was first developed by pure mathematicians in
the middle of the 19th century. Some 100 years later, engineers and
physicists have found applications for these concepts in their
areas. However there has traditionally been little interaction
between these two communities. In particular, typical mathematical
works provide extensive findings on aspects with comparatively
little significance in applications, and the engineering literature
often lacks mathematical detail and precision. This book bridges
the gap between the two communities. It concentrates on the class
of fractional derivatives most important in applications, the
Caputo operators, and provides a self-contained, thorough and
mathematically rigorous study of their properties and of the
corresponding differential equations. The text is a useful tool for
mathematicians and researchers from the applied sciences alike. It
can also be used as a basis for teaching graduate courses on
fractional differential equations.
Eine Gruppe, deren Mitglieder sich zwischen mehreren zur Wahl
stehenden Alternativen entscheiden mussen, hat eine grosse Anzahl
von Moeglichkeiten, aus den Praferenzen der Einzelnen eine von der
Gemeinschaft getragene Entscheidung zu ermitteln. Wie lasst sich
sicherstellen, dass diese gemeinschaftliche Entscheidung den Willen
der Gruppe sinnvoll widerspiegelt? Die Untersuchung von Methoden,
die die vielen individuellen Meinungen zu einer einzigen
Entscheidung fur die gesamte Gruppe zusammenfassen, und die
Darstellung der wichtigsten Eigenschaften dieser Verfahren sind
Inhalt dieses Buches. Neben theoretischen UEberlegungen steht dabei
gleichberechtigt die Betrachtung zahlreicher Beispiele, die oft
unerwartete Eigenschaften erkennen lassen.
The NASA Technical Reports Server (NTRS) houses half a million
publications that are a valuable means of information to
researchers, teachers, students, and the general public. These
documents are all aerospace related with much scientific and
technical information created or funded by NASA. Some types of
documents include conference papers, research reports, meeting
papers, journal articles and more. This is one of those documents.
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