high order numerical methods for the riesz derivatives and ......

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 Applications June 3-5, 2013, Salve Regina University

2

Outline Fractional calculus: partial but important Motivation Numerical methods for Riesz fractional derivatives Numerical methods for space Riesz fractional differential equation Conclusions Acknowledgements

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


Definitions of RL and Caputo derivatives: ( )

0, 0,( ) ( ), 1 .m

mRL t tm

dD x t D x t m m Zdt

α α α− − += − < < ∈

( )0, 0,

d( ) ( ), 1 .d

mm

C t t mD x t D x t m m Zt

α α α− − += − < < ∈

The involved fractional integral is in the sense of RL.

.,)()()(

1)()(0

1,0 ∫ +−− ∈−

Γ=∗=

t

t RdxttxYtxD ατττα

αα

α

1( )

0, 0,0

( ) [ ( ) (0)], 1 .!

kmk

C t RL tk

tD x t D x t x m m Zk

α α α−

+

=

= − − < < ∈∑

5


Once mentioned it, fractional calculus is taken for granted to be the mathematical generalization of calculus.

But this is not true!!!

).()(lim),0()()(limfix ,derivativeCaputo )1(th -For

)(,0

)1()1(,0

)1(txtxDxtxtxD

tZmmm

tCm

mmtC

m=−=

∈<<−

−+ →

−−

−→

+

α

α

α

α

αα

Classical derivative is not the special case of the Caputo derivative.

6


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

α

ε

ε

ε

α

ε

ε

+

− ≤ ≤= − < ≤ + >

+ ∈

∃ ∪ +

∃ （

The existence intervals of two kind of derivatives for the same funtion is not the same, neither relation of inclusion.

7


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


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

α

α

α

α−

−=

= < ≤

= =

=

= 0ome .x

It is here.

9


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 eqnAttention fractional is very possibly a powerful tool for nonsmoot

(

h functions

*).:

.

RL t

RL t x

D x t

D x t

x

α

α

α−

=

= ∈∗ =

=

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 xRL j jf x h f x h f x O hα α α αϖ ϖ− −

−= =

= + +∑ ∑

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

αα α

αα

α

αα

αα

πα α

ξξ

α ξ

ξξ

α ξ

+

− +

− +

∂= −Ψ +

∂

Ψ = − < < ∈

∂=Γ − ∂ −

∂=Γ − ∂ −

∫

∫

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

α

α

α

α

α

φ

ϕ

ψ

−=

−=

∂ ∂= + < < < < < ≤ ∂ ∂

= ≤ ≤ = ≤ ≤ = ≤ ≤

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 Mt n n N h b a M T Nτ τ

= + =

= = = − =

( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )

0 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

αα αα

α α

α

ϖ ϖ

α αϖ

α

−

− += =

∂ Ψ = − + + ∂

− Γ + = − = Γ + Γ + −

∑ ∑

Already existed (typical) numerical schemes:

1) The first order scheme

14


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

αα αα

α α

α

ϖ ϖ

α αϖ

α

+ − +

− + + −= =

∂ Ψ = − + + ∂

− Γ + = − = Γ + Γ + −

∑ ∑

15


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

αα

α α α α

α

α α

α

α

αα αα

αα α

−

−

− + − − −=

−−

+ − +

<

− − ∂ − −Ψ = − +Γ −∂

+ − +

− − − − + +−

+ − +

∑

( ) ( )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 mα α α

− −

+ +=

− −

+

= + − = − − −

∑

16


( )( )

( ) ( ) ( )( )

( )

( )

( ) ( )

( ) ( )

( ) ( )

( )

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

ααα

α α

α

α

α α

α α α

α α

α

α =

− −

+ +

∂ −Ψ= +Γ −∂

< − + = −= + =

+ = +

> +

∑

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 mc c k m

zc k m

k m

j j j k

c j k j k j k k jk j

α

α α

α α α

α

− +

+

+

− −

− − −

− + ≤ −− + ==

= + > +

− − − + == − + − − + − − ≤ ≤ − =

17


( )

( ) ( ) ( )( )( ) ( )

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

α

α α α

α αα

− −

−

− +

− − −

− −

< − = −= − + = − + + ≤ ≤

== − + − − + − − + ≤ ≤ −

− − + − + − − == − +

18



( ) ( ) ( )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

α αα α

∞

=−∞

− Γ +∆ = −

Γ − + Τ + +

∑

5) The second order scheme based on the fractional centered difference method

( ) ( ) ( )2, 1 , .mh m

u x tu x t O h

hx

αα

α α

∂= − ∆ +

∂

19


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

αα αα

α α

α α

ν ϖ ν ϖ

ν ϖ ν ϖ

α αν ν

+ +

− + − += =

− + − +

+ − + −= =

∂ Ψ= − +

∂

+ +

+

− −= =

− −

∑ ∑

∑ ∑

egers.


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

αα αα

α α

α α

α α

κ ϖ κ ϖ

κ ϖ κ ϖ

κ ϖ κ ϖ

κ

+ +

− + − += =

+ − +

− + + −= =

− +− +

+ − + −= =

∂ Ψ= +

∂

+ +

+ +

+

− + +=

∑ ∑

∑ ∑

∑ ∑

( )( )

( )( )

( )

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.

α α

α α α ακ κ

+

− − +

− + + + − + + += =

− − + − − +


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 sufficientlysmooth. For arbitrarily different numbers , and , we have

, ,,

.

Lemma 1

s q p p s qs

p q

q s p s

u x t xp q s

x x u x t x x u x tu x t

x x

O x x x x

− + −=

−

+ − −


23

( )

( ) ( ) ( ) ( )( ) ( )

0

. For any positive number , we have

=0 ,

1 1where 1 .

1

Lemma 2

1

kk

kk

k k k k

α

α

α

ϖ

α αϖ

α

∞

=

− Γ + = − = Γ + Γ + −

∑

( )( ) ( ) ( )Let

, 2 ,, .

2h

u x t f x jh tu x jh tα

αµ+ −

− =


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

α α α

αα

µ

αϖ∞

=

∆ = ∆

∆ = − − ∑

( ) ( ) ( )( )0

Then one has1, 2 , .2AC h k

ku x t u x k h tαα

α ϖ α∞

=

∆ = − −∑


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

α

αα

α

ααα ϖ α

+−∞

−∞

∞

−∞=

∆= +

= − − +∑


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

ααα

α α

α α

α

α ϖ

α αϖ ϖ

α ϖ

−

−

−=

− − += =

+ −=

∂ Ψ = − − ∂

+ + − + +

∑

∑ ∑

∑


( )( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )( )

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 thx

ααα

α α

α α

α

αα

α α

α ϖ

α αϖ ϖ

α ϖ

−

− − −

− −

−=

− + += =

+ +=

∂ Ψ = − + ∂

− + + − +

∂ Ψ= −

∂

∑

∑ ∑

∑

( ) ( )

( ) ( )

( ) ( )

( )

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

α

α

α

α

α ϖ

α ϖ

α ϖ

α ϖ

−

−

− −

− −=

− +=

+ −=

+

+

+ −

+ +

+ −

∑

∑

∑

( )1

22

1)0

, ( ).M m

kk

t O h− −

+=

+

∑

27


New kinds of third order numerical schemes. Skipped… Now directly derive a fourth order scheme.

28


This image cannot currently be displayed.

( ) ( )

( ) ( ) ( ) ( )( )

( ) ( ) ( )

7

1

. Let , lie in whose partial derivativesup 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 Rx

L R L u x t g u x k h t

k

k k

αθ

α

θ θ

αα α

∞

=−∞

= − + = −

− Γ +=

Γ − + Γ + −

∑

( ) ( ) ( ) ( ) ( )41 1 0

one has, 1 , , (1 ) , .

24 24 12u x t

L u x t L u x t L u x t O hhx

α

α α

α α α−

∂ = + − + + ∂

29


( ) ( )( )( )

( )( )( )

( ) ( ) ( )

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

αα

α α

αα

αα

α

α

α

−

− +=− + +

−

− −=− + +

−

−=− + +

∂=

∂

+

− + +

∑

∑

∑

( ) ( ) ( )1 1.

1 12 2

k

k k

α αα α

− Γ +=

Γ − + Γ + −

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

α

α

α

α

α

φ

ϕ

ψ

−=

−=

∂ ∂= + < < < < < ≤ ∂ ∂

= ≤ ≤ = ≤ ≤ = ≤ ≤


( )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

α α α

α α α

α

µ µ µ

τ τµ µ µ

ατµ µ

− − −

− − − + −=− + + =− + + =− + +

− − −− − − − −

− − − + −=− + + =− + + =− + +

− − +

=

+ + − + +

= =

∑ ∑ ∑

∑ ∑ ∑

1 . 2 12

Khα

τ α +


( )

( ) ( )

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 MH G G G

τ τ

µ µ µ

− −

− −

+ −

= =

+ = − + +

=

e expressions of , , are

omitted here.

G G G+ −

33


34

( )( ) ( )2 4

. ( ) Difference scheme (3) (or (2)) is uncoTheorem 3 Stability

Theorem 4 Convergence

nditionally stable.

.

, .km k mu x t u C hτ− ≤ +

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

ααα

α

αα

αα

π αα

α

α

−−

−−

−−

= − ∈

∂ = − + − Γ − ∂

− + − Γ −

+ + − Γ −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

( ) ( )

( ) ( )

( ) ( )

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

αα α

α

α

αα

α

πα αα

α

−+ −

−−

= + − + + − Γ −

− + − + Γ

∂ ∂= <

−

+ < < ≤∂ ∂

( ) ( )

( ) ( ) ( ) ( )

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

αα

α αα α

α

α

α α

−−

− −− −

+

+ − Γ −

− + − + + − Γ − Γ

=

=

−

4(1 ) .x−

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 0.2, 1.9 0.8at t at tα α= = = =

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Numerical Examples

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Numerical Examples Figs. 6.3-6.6 display the numerical and analytical solution surface with 1.1and 1.8α α= =

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Numerical Examples

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Numerical Examples

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Numerical Examples

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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……

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Conclusions

Conclusions？

Not yet, but Go Fractional with some lists.

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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.

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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.

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high order numerical methods for the riesz derivatives and ......

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