Fractional calculus basic theory and applications Outline

Short Description
random walks. Here we discuss the close connection between fractional calculus (in. particular fractional diffusion equations) and the theory of continuous …

Website: www.red-mat.unam.mx | Filesize: 9105kb

Content
Fractional calculus:
basic theory and applications
(Part I)
Diego del-Castillo-Negrete
Oak Ridge National Laboratory
Fusion Energy Division
P.O. Box 2008, MS 6169
Oak Ridge, TN 37831-6169
phone: (865) 574-1127
FAX: (865) 576-7926
e-mail: delcastillod@ornl.gov
Lectures presented at the
Institute of Mathematics
UNAM. August 2005
Mexico, City. Mexico
Outline
1. Introduction
2. Fractional calculus
3. Fractional diffusion and random walks
4. Numerical methods
5. Applications:
a) Turbulent transport
b) Transport in fusion plasmas
c) Reaction-diffusion systems
6. Some references1. Introduction
Here we introduce the notion of fractional integral as a straightforward
generalization of the standard, integer-order integral, and define the
fractional derivative as the inverse operation. To motivate the concept we
discuss two examples: Abel’s equation and heat diffusion. The concepts
discussed here are further elaborated in the next section.
What is a fractional derivative?
Leibniz (1695):
“This is an apparent paradox from which, one day, useful
consequences will be drawn”
L’Hopital (1695):
“What if n=1/2?”
dn f
dxn
It is a usual practice to extend mathematical operations,
originally defined for a set of objects, to a wider set of objects
r r !R+ ” s s !R
n! n !N+ ” #(s) s !R
! x
n” n #N
+ $ ! x
%” % #C
This mathematical “games” have, eventually, important
physical applications…