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Fractional calculus : models and numerical methods
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Description
Rating
Title
Fractional
calculus
:
models
and
numerical
methods
Contributors
Baleanu, D. (Dumitru)
World Scientific (Firm)
DescriptionAbstract
The
subject
of
fractional
calculus
and its
applications
(that
is
,
convolutiontype
pseudodifferential
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
semiMarkov
continuoustime
random
walks
and the
spacetime
fractional
diffusion
equation
,
which
generalizes
the
usual
theory
relating
random
walks
to the
diffusion
equation
. These
methods
can
be
applied
in
finance
, to
model
tickbytick
(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
tradeoff
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
.
DescriptionTable Of Contents
1
.
Preliminaries
.
1.1
.
Fourier
and
Laplace
transforms
.
1.2
.
Special
functions
and their
properties
.
1.3
.
Fractional
operators

2
. A
survey
of
numerical
methods
for the
solution
of
ordinary
and
partial
fractional
differential
equations
.
2.1
.
Approximation
of
fractional
operators
.
2.2
.
Direct
methods
for
fractional
ODEs
.
2.3
.
Indirect
methods
for
fractional
ODEs
.
2.4
.
Linear
multistep
methods
.
2.5
.
Other
methods
.
2.6
.
Methods
for
terminal
value
problems
.
2.7
.
Methods
for
multiterm
FDE
and
multiorder
FDS
.
2.8
.
Extension
to
fractional
PDEs

3
.
Efficient
numerical
methods
.
3.1
.
Methods
for
ordinary
differential
equations
.
3.2
.
Methods
for
partial
differential
equations

4
.
Generalized
Stirling
numbers
and
applications
.
4.1
.
Introduction
.
4.2
.
Stirling
functions
s([symbol],k)
,
[symbol]
.
4.3
.
General
Stirling
functions
s([symbol]
,
[symbol])
with
complex
arguments
.
4.4
.
Stirling
functions
of the
second
kind
S([symbol],k)
.
4.5
.
Generalized
Stirling
functions
S(n,[symbol])
,
[symbol]
.
4.6
.
Generalized
Stirling
functions
S([symbol],[symbol])
,
[symbol]
.
4.7
.
Connections
between
s([symbol],[symbol])
and
S([symbol],k)

5
.
Fractional
variational
principles
.
5.1
.
Fractional
EulerLagrange
equations
.
5.2
.
Fractional
Hamiltonian
dynamics

6
.
CTRW
and
fractional
diffusion
models
.
6.1
.
Introduction
.
6.2
. The
definition
of
continuoustime
random
walks
.
6.3
.
Fractional
diffusion
and
limit
theorems

7
.
Applications
of
CTRW
to
finance
and
economics
.
7.1
.
Introduction
.
7.2
.
Models
of
price
fluctuations
in
financial
markets
.
7.3
.
Simulation
.
7.4
.
Option
pricing
.
7.5
.
Other
applications
.
Publisher
World Scientific Pub. Co.
Subject
Fractional calculus.
Identifier
9789814355216
(electronic
bk.)
;
9814355208
;
9789814355209
;
http://www.worldscientific.com/worldscibooks/10.1142/8180#t=toc
Language
eng
Type
Electronic books.
FormatExtent
xxiv, 400 p. : ill.
Date
c2012
.
OCLC number
874497682
CONTENTdm number
360
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