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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.
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R398
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