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**
Shanghai University
** University of California, Merced
*
1
Outline
Fractional calculus: partial but important
Motivation
Numerical methods for Riesz fractional derivatives
Numerical methods for space Riesz fractional
differential equation
Conclusions
Acknowledgements
2
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.
3
Fractional calculus: partial but important
Definitions of RL and Caputo derivatives:
m
d

 ( m  )

D
x
(
t
)

D
x
(
t
),
m

1



m

Z
.
RL 0,t
0,t
m
dt

C
 ( m  )
0,t
D0,t x(t )  D

D
C 0,t x(t ) 
dm

x
(
t
),
m

1



m

Z
.
m
dt
m1 k
t (k )
x (0)],
k 0 k !

D
RL 0,t [ x(t )  
m 1    m  Z  .
The involved fractional integral is in the sense of RL.
1 t
 1
D x(t )  Y  x(t ) 
(
t


)
x( )d ,

0
( )

0 ,t
4
  R .
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!!!
For  - th (m  1    m  Z  ) Caputo derivative , fix t
lim
 ( m 1)

( m 1)
( m 1)
D
x
(
t
)

x
(
t
)

x
(0),
0 ,t
C
lim  C D0,t x(t )  x ( m) (t ).
 m
Classical derivative is not the special case of the
Caputo derivative.
5
Fractional calculus: partial but important
On the other hand, see an example below:
Investigate a function in [0,1   ],
0  t  1,
1  t ,
x(t )  
1  t  1   ,   0.
t  1,

RL D0,t x (t )  in（0,1   ],   (0,1).
x '(t )  in [0,1)  (1,1   ].
The existence intervals of two kind of
derivatives for the same funtion is not
the same, neither relation of inclusion.
RL derivative can not be regarded as the mathematical
generalization of the classical derivative.
6
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.
7
Fractional calculus: partial but important
Where is fractional calculus?
It is here.
Example
x(t )  t  , 0    1 / 2, is not a solution to
 dx
 f ( x, t ),
arbitrarily given,

 dt

arbitrarily given.
 x(0)  x0 ,
But it is a solution to


D
 RL 0,t x(t )  f ( x, t ),

 1
D

 RL 0,t x (t ) |t 0  x0 ,
8
for some f ( x, t ),
for some x 0 .
Fractional calculus: partial but important
Another Example

1,
x (t )  
0,

q
 (0,1], ( p, q )  1,
p
others in [0,1],
t 
is not a solution to
 dx
 f ( x, t ),
arbitrarily given,

 dt

arbitrarily given.
 x (0)  x0 ,
But it is a solution to


D
 RL 0,t x (t )  0,   (0,1),
() 
 1
D

RL
0,
t x (t ) | x  0  0.

Actually, x  0 a.e. solves eqn (*).
Attention: fractional is very possibly a powerful tool
for nonsmooth functions.
9
Motivation
For Riemann-Liouvile derivative, high order methods
were very possibly proposed by Lubich (SIAM J. Math.
Anal., 1986)

RL
D
a , xn
f ( xn )  h

n
 n j f ( x j )  h
j 0


s
p

f
(
x
)

O
(
h
).
 n, j j
j 0
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
Motivation
The Riesz fractional derivative is defined on (a, b) as
follows,
  u ( x, t )
x

   RL Da, x  RL Dx,b  u  x, t  ,
in which,
1   
  sec   , n  1    n  Z  ,
2  2 
n x
1

. u  , t 

d ,
RL Da , x u ( x, t ) 
  n 1
n 
  n    x a  x   
1

D
u
(
x
,
t
)

RL
  n    x n
n b

x ,b
11
u  , t 
   x 
x
 n 1
d .
Motivation
The space Riesz fractional differential equation reads as
 u  x, t 
  u  x, t 
K
 f  x, t  , 1    2, a  x  b, 0  t  T


x
 t

 2
(1)
 RL Da , x;t u ( x, t ) |x  a    t  , 0  t  T ,

 2
D
 RL x ,b;t u ( x, t ) |x b   (t ), 0  t  T ,
 u ( x, 0)    x  , a  x  b.

The above boundary value conditions are of Dirichlet type.
Neumann or Rubin boundary value conditions can be similarly
proposed.
Unsuitable initial or boundary value conditions: ill-posed.
12
Numerical methods for Riesz fractional
derivatives
Already existed (typical) numerical schemes:
Let h be the step size with xm  a  mh, m  0,1, , M , and
tn  n , n  0,1, , N , where h   b  a  / M ,  T / N .
1) The first order scheme
 u ( xm , t )
x

M m
   m  

 
     k u  xmk , t     k u  xm k , t    O(h),
h  k 0
k 0

1  1   
 

  1   
.
 k   1  k   1    k 
k
in which 
( )
k
k
13
Numerical methods for Riesz fractional
derivatives
2) The shifted first order scheme
 u ( xm , t )
x

M  m 1
  m1  

     k u  xmk 1 , t     k u ( xm k 1 , t )   O(h),
h  k 0
k 0

1  1   
 

  1   
.
 k   1  k   1    k 
k
in which 
( )
k
k
14
Numerical methods for Riesz fractional
derivatives
3) The L2 numerical scheme
If 1<  2, then one has
 u  xm , t 
x



  3    h

 1    2    u  x0 , t   2    u  x1 , t   u  x0 , t  



 1
m
m


m 1
  d k  u ( xm  k 1 , t )  2u ( xm  k , t )  u  xm  k 1 , t  
k 0
1    2    u  xM , t   2    u  xM , t   u  xM 1 , t  




 1
m
M

m



 
d
u
(
x
,
t
)

2
u
(
x
,
t
)

u
x
,
t


 m k 1    O  h  ,

k
m k
 m  k 1
k 0

 (k  1) 2  k 2 , k  0,1, , max( m  1, M  m  1).

in which d k( )
M  m 1
15
Numerical methods for Riesz fractional
derivatives
4) The second order numerical scheme based on
the spline interpolation method
 u  xm , t 
 x

 

  4    h
 
2
z
u
x
,
t

O
h



,
 m,k k
M

k 0
 
in which zm
, k is defined below,

 
zm
,k

 zm, k , k  m  1,

 zm, m 1  zm, m 1 , k  m  1,

 

  zm , m  zm , m , k  m,
  
 
z

z
 m , m 1
m , m 1 , k  m  1,
  

 zm , k , k  m  1.
16
Numerical methods for Riesz fractional
derivatives
cm 1,k  2cm,k  cm 1,k , k  m  1,

   2cm , k  cm 1,k , k  m,
zm , k  
cm 1,k , k  m  1,
0 , k  m  1,

in which
c j ,k
 j  13  j 2  j  3    , k  0,

3
3
3

  j  k  1  2  j  k    j  k  1 , 1  k  j  1,
1 , k  j;

17
Numerical methods for Riesz fractional
derivatives
0 , k  m  1,
c
, k  m  1,
   m 1, m 1
zm , k  
2cm,m  cm1,m , k  m,
cm1,k  2cm,k  cm1,k , m  1  k  M ,

in which
1 , k  j

3
3
3
c j ,k   k  j  1  2  k  j    k  j  1 , j  1  k  M  1

2 
3
 3    M  j  M  j    M  j  1 , k  M
with j  m  1, m, m  1.
18
Numerical methods for Riesz fractional
derivatives
5) The second order scheme based on the fractional
centered difference method
 u  xm , t 
x

1 
    h u  xm , t   O  h 2  .
h
The so called fractional centered difference operator is defined by
 u  x, t  

h


k 
 1    1
k

 

   k  1    k  1
2
 2

19
u ( x  kh, t ).
Numerical methods for Riesz fractional
derivatives
6) The second order scheme based on the weighted
and shifted Grünwald-Letnikov (GL) formula
 u  xm , t 
x


 
h
 1
m 2
 m  1  
 


u
x
,
t



 1 k
mk  1
2 
k u xm  k  2 , t
k 0
 k 0
M m


1
k 0

 k u  xm k  , t   2
 
1

M m

k 0
2
 k u  xm  k 
 
 O  h2  ,
in which
 2
1 
2 1 
2 1 
, 2 
, 1,
2 1  2 
2
2
20
2
are two integers.


,t 
2


Numerical methods for Riesz fractional
derivatives
7) The third order scheme based on the weighted and shifted
GL formula
m
 u  xm , t     m 
 
 
 x




 1   k u xm  k  1 , t   2
h  m 0
1
m


  3   k u xm  k  3 , t  1
3
 
k 0
 2
M m

2

in which,1 
2 
1
,
12
2
1 3
12 
,
3
3
12 
 6
1 3
2
 k u  xm  k  , t    3
 

1
 6
2
6
1
2
3
3

2

6
1
2
3

1 3
 1   3 2
2
are three integers.
21
3

2
2



,3 
12

1
m0


u xm  k  2 , t
k
 k u  xm  k  , t 
M m
 1   3 2
2
1
1
m0
 O  h3  ,
12
k 0
M m
2
m0

2
3
 k u  xm  k  , t 
3
,
1
2
12 
 6
1
2

1
6
2
1 3

 1   3 2
2
3

2
3

,
Numerical methods for Riesz fractional
derivatives
In our work, we construct new three kinds of second
order schemes and a fourth order numerical scheme.
We first derive second order schemes.
Lemma 1. Assume that u  x, t  with respect to x sufficiently
smooth. For arbitrarily different numbers p, q and s , we
have
u  xs , t 
x


s
O
 xq  u  x p , t    x p  xs  u  xq , t 
 x
q
x p  xq

 xs  x p  xs  .
22
Numerical methods for Riesz fractional
derivatives
Lemma 2. For any positive number  , we have

 

 k =0 ,
k 0
 
where  k
1  1   
 

  1   
.
 k   1  k   1    k 
k
k
Let
u  x, t   f  x  2 jh, t 
h  u  x  jh, t   
.

2

23
Numerical methods for Riesz fractional
derivatives
Define




u
x
,
t



 h  C h u  x, t   ,
AC h 
where
 h u  x, t    k

C

 
k 0

 

u  x   k   h, t .
2 


Then one has
1

AC  h u  x, t   
2

 

 k u  x   2 k    h, t  .
k 0
24
Numerical methods for Riesz fractional
derivatives
Theorem 1. Let u  x, t  and the Fourier transform of the
 2
D
RL  , x u  x, t 
with respect to x both be in L1  R  , then


AC h u  x, t 


h

2
D
u
x
,
t

O
h


  , i.e.,
RL  , x
1

RL D , x u  x, t  
(2h)

 
2

u
x

2
k


h
,
t

O
h




 .
 k
k 0
25
Numerical methods for Riesz fractional
derivatives
By choices of xs , x p and xq , one has new three
kinds of second order schems:
 u  xm , tn 

x



 2h 

m
2

 



1    k u  xm  2 k , t 
 2  k  0

 M m 
 2 


m
2
 

 
 
  k u  xm  2( k 1) , t   1     k u  xm  2 k , t  (I)
2 k 0
 2  k 0


2

 
2


u
x
,
t

O
(
h
),
 m2(k 1) 

k
k 0

 M m 
 2 


26
Numerical methods for Riesz fractional
derivatives
 u  xm , tn 
 x


 x

m 
 2 1






 
 1 
   k u  xm  2 k , t 
2  k 0


 2h 


m 
 2 1


 M m 
 2 1




 
, t   1 
   k u  xm  2 k , t 
2 k 0
2  k 0

 M m 
1

  2   

 k u  xm  2( k 1) , t    O (h 2 ),

2 k 0


m 
 1


1    2   

 

   k u  xm  2( k 1) , t 

2
4
 k 0
 2h  

 u  xm , tn 

(
    u  x
m  2( k 1)
k
(II)
m 
 2 1


1 
 )   k u  xm  2( k 1) , t 
2
4 k 0
 M m 
1

 1    2   
 
   k u  xm  2( k 1) , t 
4  k 0
2
 M m 
1


1   2   
 (  )   k u  xm  2( k 1) , t    O ( h 2 ).
2
4 k 0


27
(III)
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
Theorem 2. Let u  x, t  lie in C 7  R  whose partial derivatives
up to order seven with respect to x belong to
L1  R  . Set L u  x, t  
 
in which g k 


k 
g k u  x   k    h, t  ,   1,0,1,

 1k    1

 

   k  1    k  1
2
 2

,
then one has
  u  x, t 
x

1
 
h




4
L
u
x
,
t

L
u
x
,
t

(1

)
L
u
x
,
t

O
h
.








1
0
 24 1

24
12

29
Numerical methods for Riesz fractional
derivatives
Based on Therom 2, a fourth order scheme can be established as
 u  xm , t 
x



m 1
24h




k  M  m 1
m 1

24h

k  M  m 1
   1
 1   
 12  h




g k u xm  k 1 , t
g k u xm  k 1 , t
m 1

k  M  m 1
g k u  xm  k , t   O  h 4  ,
where
g k  
k   1

1
  


 

   k  1    k  1
2
 2

30
.
Numerical method for space Riesz fractional
differential equation
Recall Equ (1).
 u  x, t 
  u  x, t 
K
 f  x, t  , 1    2, a  x  b, 0  t  T


x
 t

 2
D
(1)
 RL a , x;t u ( x, t ) |x a    t  , 0  t  T ,

 2
D
 RL x ,b;t u ( x, t ) |x b   (t ), 0  t  T ,
 u ( x, 0)    x  , 0  x  L.

31
Numerical method for space Riesz fractional
differential equation
Let umn be the approximation solution of u  xm , tn  , one
has the following finite difference scheme for Equ (1).
u  1
n
m
m 1

k  M  m 1
 
gk u
n
m  k 1
 1
m 1

k  M  m 1
 
gk u
n
m  k 1
 2
m 1

k  M  m1
g k umn k

u  1
n 1
m
m 1

k  M  m 1
where 1 
 
gk u
n 1
m  k 1
 1
m 1

k  M  m 1
 K
K   
,


1 .
2

 
48h
2h  12 
32
 
gk u
n 1
m  k 1
 2
m 1

k  M  m1
 
gk u
n 1
m k
 n  n1
 f m  f m , (2)
2
2
Numerical method for space Riesz fractional
differential equation
Set U   u , u ,
n
n
1
n
2
,u

T
n
M 1
, F n  ( f1n , f 2n ,
, f Mn 1 ).
Then system (2) can by written in a compact form:
 n  n 1
n
n 1
 I  H U   I  H U  F  F ,
2
2
where I is an identity matrix with order M -1,
H  2G - 1G  - 1G  .The expressions of G, G  , G  are
omitted here.
33
(3)
Numerical method for space Riesz fractional
differential equation
Theorem 3. (Stability)
Difference scheme (3) (or (2)) is unconditionally stable.
Theorem 4.  Convergence 
u  xm , tk   umk  C  2  h 4  .
34
Numerical Examples
Example 1
Consider the function u ( x)  x 1  x  , x   0,1.
2
2
Its exact Riesz fractional derivative is
 u  x 
x

1
2 

 
2 

  sec    
x  1  x  



3




 2 

6
3
3


x  1  x  

 4   
}
12
4 
4 


x  1  x   .

 5    
35
Numerical Examples
We 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
Example 2
Consider the following Riesz fractional differential equation
u  x, t 
  u ( x, t )
K
 f  x, t  , 0  x  1, 0  t  1

t
 x
where

 12
 x 4   1  x 4 
f (t )  (2  1)t x (1  x)  t
sec(  ) 


2

  5   
240  5
2160  6
5 
6 

x
 1  x   
x
 1  x  
  7   

 6    
2
4
4
2 1

10080  7 
20160  8
7 
8  



x
 1  x 

x
 1  x   
  9    

 8    

together with the initial condition u ( x, 0)  0 and homogenerous boundary
value conditions.
The exact solution is u ( x, t )  t 2 1 x 4 (1  x) 4 .
42
Numerical Examples
The following table shows the maximum error, time and
space convergence orders which confirms with our
theoretical analysis.
43
Numerical Examples
44
Numerical Examples
Figs.6.1 and 6.2 show the comparison of the analytical and
numerical solutions with   1.1 at t  0.2,   1.9 at t  0.8
45
Numerical Examples
46
Numerical Examples
Figs. 6.3-6.6 display the numerical and analytical solution
surface with   1.1 and   1.8
47
Numerical Examples
48
Numerical Examples
49
Numerical Examples
50
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……
51
Conclusions
Conclusions？
Not yet, but Go Fractional with
some lists.
52
Conclusions
Numerical calculations:
[1] Changpin Li, An Chen, Junjie Ye, J Comput Phys, 230(9), 33523368, 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), 149179, 2013.
[6] Hengfei Ding, Changpin Li, J Comput Phys 242(9), 103-123,
2013.
53
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
54
Acknowledgements
Q & A！
Thank Professor George Em Karniadakis for cordial invitation.
Thank you all for coming.
55