Fractional calculus:
basic theory and applications(Part I)
Diego del-Castillo-Negrete
Oak Ridge National LaboratoryFusion Energy Division
P.O. Box 2008, MS 6169
Oak Ridge, TN 37831-6169
phone: (865) 574-1127
FAX: (865) 576-7926
e-mail: [email protected]
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 references
1. 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: Abels 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
LHopital (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
Fractional integrals
!
aDx
"n # = dx1
a
x
$ dx2a
x1
$ L dxna
xn"1
$ # xn( )
!
a Dx"n # =
1
(n "1)!x " y( )
n"1# y( ) dy
a
x
$
!
a Dx"# $ =
1
%(#)x " y( )
# "1$ y( ) dy
a
x
&
Riemann-Liouville fractional integral
!
0Dx
"#x
=$( +1)
$( + # +1)x
+#Example:
Integer order integration
Interchanging the order
of integration this can be
rewritten as
Extending this expression for non-integer n=! we get
Fractional derivatives
!
a Dx" # =
1
$(m %")&xm # y( )
x % y( )"+1%m
dya
x
'
Riemann-Liouville fractional derivative
!
0Dx
x" =
#(" +1)
#(" $ +1)x"$
Examples:
!
aDx
" =dN
dxN a
Dx
#$ "[ ] N = smallest integer > ! = N "
!
aDx
m " =dN
dxN a
Dx
#(N#m )"[ ] =dm
dxm"
!
m "1
An example: Abels equation
g
x
y
v
(x,h) Let T(h) be the time it takes a particle
to go down sliding on the curve
What is the relation between the function
T(h) and "(y) ?
Energy
conservation
!
1
2v2 = g h " y( )
!
v =ds
dt=1+ "'2dy
dt
!
1+ "'2
2g(h # y)0
h
$ dy = dt = T0
T
$
!
f (y) =1
2g1+ " '2
!
T(h) =f (y)
h " y0
h
# dy
!
T = "0Dh
#1/ 2f
!
f =1
"0Dy1/ 2T
!
x = "(y)
Heat diffusion
!
"(t) = 2#0Dt
$1/ 2S(t)
x Consider a bar that is heated on one
end by a time dependent source S(t).
What is the relation between S(t) and
the temperature T(x,t) of the bar?
T(x,t)
S(t)
!
"tT =# "
x
2T
Heat diffusion equation
!
T(x = 0,t) = S(t)
!
T(x, t = 0) = 0
Boundary conditions Initial conditions
Laplace transform
!
T (x,s) = e" s t
T(x, t) dt0
#
$
!
s T =" #x
2 T
!
T =x
2"#e$
x2
4# ( t$% ) S(%)
t $ %( )3 / 2
0
t
& d%
!
"(t) = T(x,t)dx0
#
$
2. Fractional calculus
Here we present the precise definitions of the left and right Riemann-
Liouville (RL) fractional integrals and derivatives, and briefly discuss
some of the main properties of these operators. Of particular interest is
the singular behavior of the RL operators at the boundaries, and the
definition of the fractional derivatives in the Caputo sense. Further details
on the material discussed here can be found in [Samko-etal-1993],
[Podlubny-1999] and references therein.
Riemann-Liouville integrals
!
a Dx"# $ =
1
%(#)x " y( )
#"1$ y( ) dy
a
x
&
!
x Db"# $ =
1
%(#)y " x( )
#"1$ y( ) dy
x
b
&
Left integral
Right integral
a bx
!
x > a
!
x < b
!
" > 0
!
Q " (x) = " (a + b # x)
!
Q aDx"# $[ ] = xDb"# Q $[ ]
!
Q xDb"# $[ ] = aDx"# Q $[ ]
Given the reflection operator
!
Q aDx"# $[ ] =
1
%(#)
$ u( )
a + b " x " u( )1"#
dua
a+b"x
&
!
=1
"(#)
$ a + b % z( )
z % x( )1%#
dz=xDb%#Q $[ ]
x
b
&!
z = "u + a + b
Semi-group property
!
aDx
"#
aDx
"$ % =aDx
"#"$ %
!
" > 0
!
" > 0
The fractional integrals satisfy the following important semigroup
property
!
xDb
"#
xDb
"$ % =xDb
"#"$ %
!
aDx
"#aDx
"$ % =1
&(#)
1
&($)
dw
(x " w)1"#a
x
'%(u)
(w " u)1"$a
w
' du
w
u
u=w
x
!
dw
a
x
" duL =a
w
" dua
x
" dwLu
x
"
!
=1
"(#)
1
"($)du
a
x
% &(u)dw
(x ' w)1'# (w ' u)1'$u
x
%
!
B ",#( )
x $ u( )1$"$#
!
B ",#( ) =$(") $(#)
$(" + #)
!
=1
"(# + $)
%(u)
(x & u)1&$a
x
' du = aDx&(#+$ )%
Beta function
Fubinis theorem
Riemann-Liouville derivatives
!
aDx
" # =1
$(n %")
& n
& xn# u( )
x % u( )"%n+1
du
a
x
'Left derivative
Right derivative
!
n = ["]+1
a bx
!
xDb
" # =$1( )
n
%(n $")
& n
& xn# u( )
u $ x( )"$n+1
du
x
b
'
[#]=integral part of #
!
aDx
"aDx
#"$ =1
%(")%(n #")
& n
& xndz du
a
z
'$ u( )
x # z( )"#n+1
z # u( )1#"
a
x
'
Exchanging the order of the integrals and using the definition of the
Beta function we get:
!
=1
"(#)"(n $#)
% n
% xndu & u( )
B #,n $#( )(x $ u)1$n
'
( )
*
+ ,
a
x
-
!
=1
"(n)
# n
# xndu
$ u( )(x % u)1%n
a
x
&
!
1
(n "1)!du
# u( )(x " u)1"n
a
x
$ =aDx"nusing
!
=" n
" xnaDx
#n$ = $
!
aDx
"
aDx
#"$ = $
Reciprocity
However
!
a Dx"#
aDx#$ = $(x) " a Dx
#" j$[ ]j=1
k
%x= a
x " a( )#" j
& # " j +1( )
!
j "1#$ < j
!
aDx
"
aDx
#$% =aDx
"#$%
The previous formulae can be generalized as
!
a Dx"#
aDx$% = aDx
$ "#%(x) " a Dx$ " j%[ ]
j=1
k
&x= a
x " a( )#" j
' # " j +1( )!
" # $ # 0
Composition of fractional derivatives
!
a Dx" #
n$
# xn%
& '
(
) * = aDx
"+n$(x) +$ ( j )(a) x + a( )
j+"+n
, 1+ j +" + n( )j= 0
n+1
-!
" n
" xnaDx
# $[ ]=aDxn+# $
!
" # 0
!
a Dx"
a Dx#$[ ] = aDx"+#$(x) % a Dx# % j$[ ]x= a
x % a( )% j%"
& 1%" % j( )j=1
m
'
!
a Dx" # j$[ ]
x= a= 0
!
a Dx"# j$[ ]
x= a= 0
!
aDx
"
aDx
#$[ ] = aDx# a Dx"$[ ]
!
m "1#$ < m!
n "1# $ < n
In particular, fractional
derivatives commute if and only if
!
j =1,Lm
!
j =1,Ln
Behavior near the lower terminal
!
"(x) =" (k )(a)
k!k= 0
#
$ x % a( )kLet
!
aDx
" # =# (k )(a)
k!k= 0
$
% aDx" x & a( )k&"
!
aDx
"x # a( )
k
=$ k +1( )
$ k +1#"( )x # a( )
k#"
using
!
aDx
" # = # (k )(a)x $ a( )
k$"
% k +1$"( )k= 0
&
'
!
limx"a aDx
# $ = limx"a
$ (k )(a)
% k +1( )k= 0
m&1
'1
x & a( )#&k
!
limx"a aDx
# $ =% unless
!
" (k )(a) = 0
!
k =1,Lm "1!
m "1#$ < m
Fourier-Laplace transforms
!
" (s) = e# s t "(t)dt0
$
%
!
" (k) = eikx "(x)dx#$
$
%
!
"(x) =1
2#e$ikx " (k) dx
$%
%
&
!
F "#Dx$ %[ ] = "i k( )
$ % (k)
!
FxD+"
# $[ ] = i k( )# $ (k)
!
"(t) = est " (s) dsc# i$
c+ i$
%
!
L0Dt
" #[
