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International Symposium on Fractional PDEs: Theory, Numerics and Applications June 3-5, 2013, Salve Regina University. High Order Numerical Methods for the Riesz Derivatives and the Space Riesz Fractional Differential Equation. Hengfei Ding * , Changpin Li * , YangQuan Chen ** - PowerPoint PPT Presentation
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1
High Order Numerical Methods for the Riesz Derivatives and the Space Riesz Fractional
Differential Equation
Hengfei Ding*, Changpin Li*, YangQuan Chen**
* Shanghai University ** University of California, Merced
International Symposium on Fractional PDEs: Theory, Numerics and ApplicationsJune 3-5, 2013, Salve Regina University
2
Outline
Fractional calculus: partial but importantMotivationNumerical methods for Riesz fractional derivatives Numerical methods for space Riesz fractional differential equationConclusionsAcknowledgements
3
Fractional calculus: partial but important
Calculus= integration + differentiation
Fractional calculus= fractional integration + fractional differentiation
Fractional integration (or fractional integral) Mainly one: Riemann-Liouville (RL) integral
Fractional derivatives More than 6: not mutually equivalent. RL and Caputo derivatives are mostly used.
4
Fractional calculus: partial but important
Definitions of RL and Caputo derivatives:( )
0, 0,( ) ( ), 1 .m
mRL t tm
dD x t D x t m m Zdt
( )0, 0,
d( ) ( ), 1 .d
mm
C t t mD x t D x t m m Zt
The involved fractional integral is in the sense of RL.
.,)()()(
1)()(0
1,0
t
t RdxttxYtxD
1( )
0, 0,0
( ) [ ( ) (0)], 1 .!
kmk
C t RL tk
tD x t D x t x m m Zk
5
Fractional calculus: partial but important
Once mentioned it, fractional calculus is taken for granted to be the mathematical generalization of calculus.But this is not true!!!
).()(lim),0()()(lim
fix ,derivativeCaputo )1(th -For )(
,0)1()1(
,0)1(
txtxDxtxtxD
tZmmm
tCm
mmtC
m
Classical derivative is not the special case of the Caputo derivative.
6
Fractional calculus: partial but important
On the other hand, see an example below:
RL derivative can not be regarded as the mathematical generalization of the classical derivative.
0,
Investigate a function in [0,1 ],1 , 0 1,
( )1, 1 1
( ) in 0,1 ], (0,1).
'( ) in [0,1) (1,1 ]
, 0.
.RL tD x t
x
t tx t
t t
t
(
The existence intervals of two kind of derivatives for the same funtion is not the same, neither relation of inclusion.
7
Fractional calculus: partial but important
Therefore, fractional calculus, closely related to classical calculus, is not the direct generalization of classical calculus in the sense of rigorous mathematics. For details, see the following review article:[1] Changpin Li, Zhengang Zhao, Introduction to fractional integrability and differentiability, The European Physical Journal-Special Topics, 193(1), 5-26, 2011.
8
Fractional calculus: partial but important
Where is fractional calculus?
0
0,
10, 0 0
( ) ,0 1/ 2, is not a solution to
( , ), arbitrarily given,
(0) , arbitrarily given.
But it is a solution to
( ) ( , ), for some ( , ),
( ) |
Exampl
, for s
e
RL t
RL t t
x t tdx f x tdtx x
D x t f x t f x t
D x t x
0ome .x
It is here.
9
Fractional calculus: partial but important
0
1, (0,1], ( , ) 1,( )
0, others in [0,1],is not a solution to
( , ), arbitrarily given,
(0) ,
A
arbitrarily give
nother Examp
n.
But it is
le
a solut
qt p qpx t
dx f x tdtx x
0,
10, 0
ion to
( ) 0, (0,1),( )
( ) | 0.
Actually, 0 a.e. solves eqn
Attention fractional is very possibly a powerful tool for nonsmoot
(
h functions
*).
:.
RL t
RL t x
D x t
D x t
x
Motivation
For Riemann-Liouvile derivative, high order methods were very possibly proposed by Lubich (SIAM J. Math. Anal., 1986)
For Caputo derivative, high order schemes were constructed by Li, Chen, Ye (J. Comput. Phys., 2011)
In this talk, we focus on high order algorithms for Riesz fractional derivatives and space Riesz fractional differential equation.
10
,,0 0
( ) ( ) ( ) ( ).D n
n sp
n n j j n j ja xRLj j
f x h f x h f x O h
11
Motivation
The Riesz fractional derivative is defined on (a, b) as follows,
.
, ,
, 1
, 1
( , ) , ,
1 sec , 1 , 2 2
,1( , )
in whi
,
,1( , ) .
ch,
RL a x RL x b
xn
RL a x nna
bn
RL x b nnx
u x t D D u x tx
n n Z
u tD u x t d
n x x
u tD u x t d
n x x
12
Motivation
The space Riesz fractional differential equation reads as
2, ;
2, ;
, ,, ,1 2, ,0
( , ) | , 0 , (1)
( , ) | ( ), 0 ,
( ,0) , .
The above boundary value conditions are of type.DirichletNeuma
RL a x t x a
RL x b t x b
u x t u x tK f x t a x b t T
t x
D u x t t t T
D u x t t t T
u x x a x b
or boundary value conditions can be similarlyproposed.Unsuitable initial or boundary value conditions il
nn
l-
Rubi
po
n
: sed.
13
Numerical methods for Riesz fractional derivatives
Let be the step size with , 0,1, , , and
, 0,1, , , where / , / .m
n
h x a mh m M
t n n N h b a M T N
0 0
( )
( , ), , ( ),
1 1in which 1 .
1 1
m M mm
k m k k m kk k
kk
k
u x tu x t u x t O h
hx
k k k
Already existed (typical) numerical schemes:
1) The first order scheme
14
Numerical methods for Riesz fractional derivatives
2) The shifted first order scheme
1 1
1 10 0
( )
( , ) , ( , ) ( ),
1 1in which 1 .
1 1
m M mm
k m k k m kk k
kk
k
u x t u x t u x t O hhx
k k k
15
Numerical methods for Riesz fractional derivatives
3) The L2 numerical scheme
1 001
1
1 10
11
1
If 1< 2, then one has
2 , ,, 1 2 ,3
( , ) 2 ( , ) ,
2 , ,1 2 ,
( , ) 2 ( , )
m
m
k m k m k m kk
M MM
k m k m k
u x t u x tu x t u x th m mx
d u x t u x t u x t
u x t u x tu x tmM m
d u x t u x t u x
1
10
( ) 2 2
, ,
in which ( 1) , 0,1, , max( 1, 1).
M m
m kk
k
t O h
d k k k m M m
16
Numerical methods for Riesz fractional derivatives
2,
0
,
,
, 1 , 1
, , ,
, 1 , 1
,
,, ,
4
in which is defined below,
, 1,
, 1,
, ,
, 1,
, 1.
Mm
m k kk
m k
m k
m m m m
m k m m m m
m m m m
m k
u x tz u x t O h
hx
z
z k m
z z k m
z z z k m
z z k m
z k m
4) The second order numerical scheme based on the spline interpolation method
1, , 1,
, 1,,
1,
3 2
3 3 3,
2 , 1,
2 , ,
, 1,
0 , 1,in which
1 3 , 0,
1 2 1 , 1 1,
1 , ;
m k m k m k
m k m km k
m k
j k
c c c k m
c c k mz
c k m
k m
j j j k
c j k j k j k k j
k j
17
Numerical methods for Riesz fractional derivatives
1, 1,
, 1,
1, , 1,
3 3 3,
2 3
0 , 1,, 1,
2 , ,
2 , 1 ,
in which
1,
1 2 1 , 1 1
3 1 ,
with 1, , 1.
m mm k
m m m m
m k m k m k
j k
k mc k m
zc c k m
c c c m k M
k j
c k j k j k j j k M
M j M j M j k M
j m m m
18
Numerical methods for Riesz fractional derivatives
Numerical methods for Riesz fractional derivatives
The so called is defined fractional centered differen by
1 1, ( , ).
1 1
ce operator
2 2
k
hk
u x t u x kh tk k
5) The second order scheme based on the fractional centered difference method
2, 1 , .mh m
u x tu x t O h
hx
19
Numerical methods for Riesz fractional derivatives
6) The second order scheme based on the weighted and shifted Grünwald-Letnikov (GL) formula
20
1 2
1 2
1 2
1 2
1 20 0
1 20 0
2
2 11 2 1 2
1 2 1 2
,, ,
, ,
,
in which 2 2, , , are two int
2 2
m mm
k m k k m kk k
M m M m
k m k k m kk k
u x tu x t u x t
hx
u x t u x t
O h
egers.
Numerical methods for Riesz fractional derivatives
7) The third order scheme based on the weighted and shifted GL formula
21
1 2
1 2
3 1
3 1
32
2 3
1 20 0
3 10 0
2 30 0
3
2 3 2 31
,, ,
, ,
, ,
,
12 6 6 1in which,
m mm
k m k k m km k
m M m
k m k k m kk m
M mM m
k m k k m km m
u x tu x t u x t
hx
u x t u x t
u x t u x t
O h
2
22 3 1 2 1 3 1
2 21 3 1 3 1 2 1 2
2 32 21 3 1 2 2 3 2 1 2 1 3 2 3 3
1 2 3
3,
12
12 6 6 1 3 12 6 6 1 3, ,
12 12
, , are three integers.
Numerical methods for Riesz fractional derivatives
22
In our work, we construct new three kinds of second order schemes and a fourth order numerical scheme. We first derive second order schemes.
. Assume that , with respect to sufficiently
smooth. For arbitrarily different numbers , and , we have
, ,,
.
Lemma 1
s q p p s qs
p q
q s p s
u x t x
p q s
x x u x t x x u x tu x t
x x
O x x x x
Numerical methods for Riesz fractional derivatives
23
0
. For any positive number , we have
=0 ,
1 1where 1 .
1
Lemma 2
1
kk
kk
k k k k
Let
, 2 ,, .
2h
u x t f x jh tu x jh t
Numerical methods for Riesz fractional derivatives
24
0
Define
, , ,
where
, , .2
AC h h C h
C h kk
u x t u x t
u x t u x k h t
0
Then one has1, 2 , .2AC h k
k
u x t u x k h t
Numerical methods for Riesz fractional derivatives
25
2,
1
2,
2,
0
. Let , and the Fourier transform of the u ,
with respect to both be in , then
,, , i
Theorem 1
.e.,
12 )
, 2 , .(
RL x
AC hRL x
RL x kk
u x t D x t
x L R
u x tD u x t O h
h
D u x t u x k h t O hh
Numerical methods for Riesz fractional derivatives
26
neBy w cho threices of e
kinds o
, and , o
f second o
ne
rde
has
r sch :emss p qx x x
2
2 2
2
20
2( 1) 20 0
22( 1)
0
,1 ,
22
, 1 , 2 2
, ( )
(I)
,2
m
m M m
M m
m nk m k
k
k m k k m kk k
k m kk
u x tu x t
hx
u x t u x t
u x t O h
Numerical methods for Riesz fractional derivatives
12
1 12 2
12
20
2( 1) 20 0
22( 1)
0
,1 ,
22
, 1 , 2 2
, ( ),2
, 1
(II)
22
m
m M m
M m
m nk m k
k
k m k k m kk k
k m kk
m n
u x tu x t
hx
u x t u x t
u x t O h
u x t
hx
12
12
12
2( 1)0
2( 1)0
2( 1)0
2(
,4
1( ) ,2 4
1 , ( 2 4
1( )
I
2 4
II )
m
m
M m
k m kk
k m kk
k m kk
k m
u x t
u x t
u x t
u x
1
22
1)0
, ( ).M m
kk
t O h
27
Numerical methods for Riesz fractional derivatives
New kinds of third order numerical schemes. Skipped…
Now directly derive a fourth order scheme.
28
Numerical methods for Riesz fractional derivatives
7
1
. Let , lie in whose partial derivatives
up to order seven with respect to belong to
. Set , , , 1,0,1,
1 1in which g ,
1 1
Theor
2 2t
em 2
hen
kk
k
u x t C R
x
L R L u x t g u x k h t
k
k k
41 1 0
one has
, 1 , , (1 ) , .24 24 12
u x tL u x t L u x t L u x t O h
hx
29
Numerical methods for Riesz fractional derivatives
1
11
1
11
14
1
Based on Therom 2, can be established as
,,
24
a fourth order schem
,24
11 , ,12
where
g
e m
mk m k
k M m
m
k m kk M m
m
k m kk M m
k
u x tg u x t
hx
g u x th
g u x t O hh
1 1.
1 12 2
k
k k
30
Numerical method for space Riesz fractional differential equation
Recall Equ (1).
31
2, ;
2, ;
, ,, ,1 2, ,0
( , ) | , 0 , (1)
( , ) | ( ), 0 ,
( ,0) , 0 .
RL a x t x a
RL x b t x b
u x t u x tK f x t a x b t T
t x
D u x t t t T
D u x t t t T
u x x x L
Numerical method for space Riesz fractional differential equation
Let be the approximation solution of , , one
h finite differas th ence e followin scheme forg Equ (1 ).
nm m nu u x t
32
1 1 1
1 1 1 1 21 1 1
1 1 11 1 1 1 1
1 1 1 1 21 1 1
1 2
, (2) 2 2
where ,48
m m mn n n nm k m k k m k k m k
k M m k M m k M m
m m mn n n n n nm k m k k m k k m k m m
k M m k M m k M m
u g u g u g u
u g u g u g u f f
Kh
1 . 2 12Kh
Numerical method for space Riesz fractional differential equation
1 2 1 1 2 1
1 1
2 1 1
Set , , , , ( , , , ).
Then system (2) can by written in a compact form:
, (3)2 2
where is an identity matrix with order -1,
- - .Th
Tn n n n n n n nM M
n n n n
U u u u F f f f
I H U I H U F F
I M
H G G G
e expressions of , , are
omitted here.
G G G
33
Numerical method for space Riesz fractional differential equation
34
2 4
. ( ) Difference scheme (3) (or (2)) is uncoTheorem 3 Stability
Theorem 4 Convergence
nditionally stable.
.
, .km k mu x t u C h
Numerical Examples
22
22
33
44
Example 1
Consider the function ( ) 1 , 0,1 .
Its exact Riesz fractional derivative is
1sec 12 3
6 14
12 1 .5
}
u x x x x
u xx x
x
x x
x x
35
Numerical ExamplesWe use the second order numerical fomulas (I),(II)and (III) to compute the given function, the absolute errors and convergence orders at x=0.5 are shown in Tables 1-3.
36
Numerical Examples
37
Numerical Examples
38
Numerical Examples
39
Numerical Examples
40
Numerical Examples
41
Numerical Examples
42 4 4 2 1 4
55
Example 2Consider the following Riesz fractional differential equation
, ( , ) , , 0 1
where
12( ) (2 1) (1 ) sec( ) 12 5
240 211
, 0 1
6
f t t x x
u x t u x tK f x t xx
x
tt
t x
x x
66
7 87 8
2 1 4
together with the initial condition ( ,0) 0 and homoge
60 17
10080 201601 18 9
(
.
The exact solution i
nerous boundaryvalue conditio
s
n
)
s
,
x x
x x x x
u x t t
u x
x
4(1 ) .x
42
Numerical ExamplesThe following table shows the maximum error, time and space convergence orders which confirms with our theoretical analysis.
43
Numerical Examples
44
Numerical ExamplesFigs.6.1 and 6.2 show the comparison of the analytical and numerical solutions with 1.1 0.2, 1.9 0.8at t at t
45
Numerical Examples
46
Numerical ExamplesFigs. 6.3-6.6 display the numerical and analytical solution surface with 1.1and 1.8
47
Numerical Examples
48
Numerical Examples
49
Numerical Examples
50
51
Comments
1) Proper and exact applications of fractional calculus
2) Long-term integrations at each step, induced by historical dependencies and/or long-range interactions, require more computational time and storage capacity.
Therefore the effective and economical numerical methods
are needed.
3) Fractional calculus + Stochastic process Stochastics and nonlocality may better characterize our
complex world.
So numerical fractional stochastic differential equations have been placed on the agenda.
And more……
52
Conclusions
Conclusions ?Not yet, but Go Fractional with
some lists.
53
Conclusions
Numerical calculations:
[1] Changpin Li, An Chen, Junjie Ye, J Comput Phys, 230(9), 3352- 3368, 2011.[2] Changpin Li, Zhengang Zhao, Yangquan Chen, Comput Math Appl 62(3), 855-875, 2011.[3] Changpin Li, Fanhai Zeng, Finite difference methods for fractional differential equations, Int J Bifurcation Chaos 22(4), 1230014 (28 pages), 2012. Review Article[4] Changpin Li, Fanhai Zeng, Fawang Liu, Fractional Calculus and Applied Analysis 15 (3), 383-406, 2012.[5] Changpin Li, Fanhai Zeng, Numer Funct Anal Optimiz 34(2), 149- 179, 2013.[6] Hengfei Ding, Changpin Li, J Comput Phys 242(9), 103-123, 2013.
54
Conclusions
Fractional dynamics:
[1] Changpin Li, Ziqing Gong, Deliang Qian, Yangquan Chen, Chaos 20(1), 013127, 2010.[2] Changpin Li, Yutian Ma, Nonlinear Dynamics 71(4), 621-633, 2013. [3] Fengrong Zhang, Guanrong Chen, Changpin Li, Jürgen Kurths, Chaos synchronization in fractional differential systems, Phil Trans R Soc A 371 (1990), 20120155 (26 pages), 2013. Review article
Acknowledgements
Q & A !Thank Professor George Em Karniadakis for cordial invitation.
Thank you all for coming.
55