# Fourth-Order Compact Difference Schemes for the Riemann-Liouville and Riesz Derivatives

Authors: Yuxin Zhang; Hengfei Ding; Jincai Luo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We propose two new compact difference schemes for numerical approximation of the Riemann-Liouville and Riesz derivatives, respectively. It is shown that these formulas have fourth-order convergence order by means of the Fourier transform method. Finally, some numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence orders.


---

## Body

## 1. Introduction

Nowadays, fractional derivatives are used to model various different phenomena in science and engineering, such as physics, materials, control theory, biology, and finance [1–3]. Therefore, to obtain highly accurate numerical methods for the fractional derivatives is of great importance, but, due to the nonlocal property of the fractional derivatives, a lot of improvements remain in the present numerical approaches.As we all know, there are several different ways to define the fractional derivatives, and the most commonly used fractional derivatives are Grünwald-Letnikov derivative, Riemann-Liouville derivative, Riesz derivative, and Caputo derivative. Accordingly, there are a lot of different numerical methods for the above fractional derivatives [4–14]. In this paper, we propose two high-order numerical formulas for the Riemann-Liouville and Riesz derivatives.The plan of the remainder of this paper is as follows. In Section2, we introduce the preliminary knowledge for the rest of the paper. Two novel numerical schemes for the Riemann-Liouville and Riesz derivatives are proposed in Section 3. In Section 4, some numerical experiments are carried out to support the theoretical analysis. Finally, the paper concludes with a summary drawn in Section 5.


## 2. The Preliminary Knowledge

In this section, we will introduce the definitions of the Riemann-Liouville and Riesz derivatives and a lemma used throughout the remaining sections of the paper.Definition 1 (see [15]).
Theα order left and right Riemann-Liouville derivatives of the function f
(
x
) are defined as
(1)
RL
D
-
∞
,
x
α
f
(
x
)
=
1
Γ
(
n
-
α
)
(
d
d
x
)
n
∫
-
∞
x
f
(
t
)
(
x
-
t
)
α
-
n
+
1
d
t
,
hhhhhhhhhhhhhhhhlhhhhhhhhh
n
-
1
≤
α
<
n
∈
ℤ
+
,
RL
D
x
,
+
∞
α
f
(
x
)
=
1
Γ
(
n
-
α
)
(
-
d
d
x
)
n
∫
x
+
∞
‍
f
(
t
)
(
t
-
x
)
α
-
n
+
1
d
t
,
hhhhhhhhhhhhhhhhhlhhhhhhhhh
n
-
1
≤
α
<
n
∈
ℤ
+
.Definition 2 (see [15]).
Theα order Riesz derivative of the function f
(
x
) is defined as
(2)
∂
α
f
(
x
)
∂
|
x
|
α
=
-
1
2
cos
⁡
(
π
α
/
2
)
(
RL
D
-
∞
,
x
α
+
RL
D
x
,
+
∞
α
)
f
(
x
)
.Lemma 3 (see [16]).
Letα
>
0, f
∈
C
0
∞
(
ℝ
); the Fourier transform of the left Riemann-Liouville-fractional derivative is
(3)
ℱ
(
R
L
D
-
∞
,
x
α
f
(
x
)
,
ω
)
=
(
i
ω
)
α
f
^
(
ω
)
,
where f
^
(
ω
) denotes the Fourier transform of f
(
x
); that is,
(4)
f
^
(
ω
)
=
∫
ℝ
‍
exp
⁡
(
-
i
ω
x
)
f
(
x
)
d
x
.


## 3. New Numerical Formulas for the Riemann-Liouville and Riesz Derivatives

Firstly, we divide the given interval[
a
,
b
] into
(5)
Λ
:
-
∞
≤
a
=
x
0
<
x
1
<
⋯
<
x
m
=
b
≤
∞
and x
l
=
a
+
l
h, in which h
=
(
b
-
a
)
/
m, l
=
0,1
,
…
,
m.In [17], Tuan and Gorenflo introduced the following fractional center difference operator:
(6)
C
Δ
h
α
f
(
x
)
=
∑
j
=
0
∞
‍
(
-
1
)
j
(
α
j
)
f
(
x
-
(
j
-
α
2
)
h
)
.
It is proved that it has second-order accuracy for the Riemann-Liouville derivative. Inspired by above fractional center difference operator, now, we give a fourth-order approximation formula for the Riemann-Liouville derivative by the following theorem.Theorem 4.
Letf
(
x
) and the Fourier transform of the R
L
D
-
∞
,
x
α
+
4
f
(
x
) all belong to L
1
(
ℝ
). Then, one has
(7)
R
L
D
-
∞
,
x
α
f
(
x
)
=
1
h
α
C
Δ
h
α
f
(
x
)
1
+
(
α
/
24
)
δ
x
2
+
𝒪
(
h
4
)
,
uniformly for x
∈
ℝ, where δ
x
2 is the second-order central difference operator and defied as δ
x
2
f
j
=
f
j
+
1
-
2
f
j
+
f
j
-
1.Proof.
Taking the Fourier transform to fractional center difference operatorC
Δ
h
α and getting
(8)
ℱ
{
C
Δ
h
α
f
(
x
)
h
α
,
ω
}
=
1
h
α
∑
j
=
0
∞
‍
(
-
1
)
j
(
α
j
)
exp
⁡
(
-
i
ω
(
j
-
α
2
)
h
)
f
^
(
ω
)
=
1
h
α
(
∑
j
=
0
∞
(
-
1
)
j
(
α
j
)
exp
⁡
(
-
i
j
ω
h
)
)
exp
⁡
⁡
(
1
2
i
ω
α
h
)
f
^
(
ω
)
=
(
i
ω
)
α
f
^
(
ω
)
[
(
α
2
1152
-
α
2880
)
(
i
ω
h
)
4
1
+
α
24
(
i
ω
h
)
2
hhhhhhhhhhlhh
+
(
α
2
1152
-
α
2880
)
(
i
ω
h
)
4
+
𝒪
(
|
i
ω
h
|
)
6
]
,
similarly, we also have
(9)
ℱ
{
(
1
+
α
24
δ
x
2
)
RL
D
-
∞
,
x
α
f
(
x
)
,
ω
}
=
ℱ
{
[
1
+
α
24
(
h
2
d
2
d
x
2
+
𝒪
(
h
4
)
)
]

RL
D
-
∞
,
x
α
f
(
x
)
,
ω
}
=
(
i
ω
)
α
f
^
(
ω
)
[
1
+
α
24
(
i
ω
h
)
2
+
𝒪
(
h
4
)
]
.
If we denote(10)
ϕ
^
(
ω
,
h
)
=
ℱ
{
C
Δ
h
α
f
(
x
)
h
α
,
ω
}
-
ℱ
{
(
1
+
α
24
δ
x
2
)
RL
D
-
∞
,
x
α
f
(
x
)
,
ω
}
,
then, from the above equation and Lemma 3, we have
(11)
|
ϕ
^
(
ω
,
h
)
|
≤
C
1
h
4
|
(
i
ω
)
α
+
4
f
^
(
ω
)
|
.
Note that the condition ℱ
(
RL
D
-
∞
,
x
α
+
4
f
(
x
)
,
ω
)
∈
L
1
(
ℝ
); then we obtain
(12)
|
C
Δ
h
α
f
(
x
)
h
α
-
(
1
+
α
24
δ
x
2
)
RL
D
-
∞
,
x
α
f
(
x
)
|
=
|
ϕ
(
ω
,
h
)
|
=
1
2
π
|
∫
ℝ
‍
exp
⁡
(
i
ω
h
)
ϕ
^
(
ω
,
h
)
d
ω
|
≤
1
2
π
∫
ℝ
‍
|
ϕ
^
(
ω
,
h
)
|
d
ω
≤
C
1
2
π
(
∫
ℝ
‍
|
(
i
ω
)
α
+
4
f
^
(
ω
)
|
d
ω
)
h
4
=
C
1
2
π
(
∫
ℝ
‍
|
ℱ
(
RL
D
-
∞
,
x
α
+
4
f
(
x
)
)
|
d
ω
)
h
4
≤
C
h
4
=
𝒪
(
h
4
)
,
where C
=
C
1
C
2
/
2
π. This finishes the proof.Remark 5.
(1) Whenα
=
1, (7) becomes the following average-central difference scheme for the first-order derivative:
(13)
d
f
(
x
)
d
x
=
1
h
f
(
x
+
(
1
/
2
)
h
)
-
f
(
x
-
(
1
/
2
)
h
)
1
+
(
1
/
24
)
δ
x
2
+
𝒪
(
h
4
)
.(2) Whenα
=
2, (7) becomes the following average-central difference scheme for the second-order derivative:
(14)
d
2
u
(
x
)
d
x
2
=
1
h
2
f
(
x
+
h
)
-
2
f
(
x
)
+
f
(
x
-
h
)
1
+
(
1
/
12
)
δ
x
2
+
𝒪
(
h
4
)
.For the right Riemann-Liouville derivative, we can also obtain the following fourth-order difference scheme by using the same method:(15)
RL
D
x
,
+
∞
α
f
(
x
)
=
1
h
α
C
Δ
~
h
α
f
(
x
)
1
+
(
α
/
24
)
δ
x
2
+
𝒪
(
h
4
)
,
where
(16)
C
Δ
~
h
α
f
(
x
)
=
∑
j
=
0
∞
‍
(
-
1
)
j
(
α
j
)
f
(
x
+
(
j
-
α
2
)
h
)
.Combing (7) and (15), one can easily get a fourth-order compact difference scheme for Riesz derivative as follows:
(17)
∂
α
f
(
x
)
∂
|
x
|
α
=
-
1
2
cos
⁡
(
π
α
/
2
)
h
α
C
Δ
h
α
f
(
x
)
+
C
Δ
~
h
α
f
(
x
)
1
+
(
α
/
24
)
δ
x
2
+
𝒪
(
h
4
)
.


## 4. Numerical Examples

In order to verify the proposed numerical schemes for the Riemann-Liouville and Riesz derivatives, we give the following two numerical examples. And the numerical results show that the numerical schemes are efficient.Example 6.
Consider the functionf
(
x
)
=
x
4, x
∈
[
0,1
]. The left Riemann-Liouville derivative of the function f
(
x
) is
(18)
RL
D
0
,
x
α
f
(
x
)
=
24
Γ
(
5
-
α
)
x
4
-
α
.
The absolute error and convergence order by the numerical scheme (7) are shown in Table 1.Table 1
The numerical results of Example6 by difference Scheme (7).
α
h
The absolute error
The convergence order
0.2
1
/
10
1.507793
e
-
006
—
1
/
20
9.410104
e
-
008
4.0021
1
/
40
5.876619
e
-
009
4.0012
1
/
80
3.671357
e
-
010
4.0006
0.4
1
/
10
2.250905
e
-
006
—
1
/
20
1.402929
e
-
007
4.0040
1
/
40
8.755965
e
-
009
4.0020
1
/
80
5.468621
e
-
010
4.0010
0.6
1
/
10
2.154827
e
-
006
—
1
/
20
1.342424
e
-
007
4.0047
1
/
40
8.377252
e
-
009
4.0022
1
/
80
5.231917
e
-
010
4.0011
0.8
1
/
10
1.313317
e
-
006
—
1
/
20
8.185957
e
-
008
4.0039
1
/
40
5.110355
e
-
009
4.0017
1
/
80
3.192508
e
-
010
4.0007
1.2
1
/
10
1.371602
e
-
006
—
1
/
20
8.576212
e
-
008
3.9994
1
/
40
5.363498
e
-
009
3.9991
1
/
80
3.354046
e
-
010
3.9992
1.4
1
/
10
2.337915
e
-
006
—
1
/
20
1.464272
e
-
007
3.9970
1
/
40
9.165078
e
-
009
3.9979
1
/
80
5.734311
e
-
010
3.9985
1.6
1
/
10
2.507873
e
-
006
—
1
/
20
1.572017
e
-
007
3.9958
1
/
40
9.842569
e
-
009
3.9974
1
/
80
6.154188
e
-
010
3.9994
1.8
1
/
10
1.690729
e
-
006
—
1
/
20
1.059074
e
-
007
3.9968
1
/
40
6.627397
e
-
009
3.9982
1
/
80
4.135963
e
-
010
4.0021Example 7.
Consider the functionf
(
x
)
=
x
4
(
1
-
x
)
4, x
∈
[
0,1
]. The Riesz derivative of the function f
(
x
) is
(19)
∂
α
f
(
x
)
∂
|
x
|
α
=
-
sec
(
π
2
α
)
×
{
12
Γ
(
5
-
α
)
[
x
4
-
α
+
(
1
-
x
)
4
-
α
]
hhhh
-
240
Γ
(
6
-
α
)
[
x
5
-
α
+
(
1
-
x
)
5
-
α
]
hhhh
+
2160
Γ
(
7
-
α
)
[
x
6
-
α
+
(
1
-
x
)
6
-
α
]
hhhh
-
10080
Γ
(
8
-
α
)
[
x
7
-
α
+
(
1
-
x
)
7
-
α
]
hhhh
+
20160
Γ
(
9
-
α
)
[
x
8
-
α
+
(
1
-
x
)
8
-
α
]
}
.
The absolute error and convergence order by the numerical scheme (17) are shown in Table 2.Table 2
The numerical results of Example7 by difference Scheme (17).
α
h
The absolute error
The convergence order
0.2
1
/
10
1.009205
e
-
006
—
1
/
20
6.374128
e
-
008
3.9848
1
/
40
3.988170
e
-
009
3.9984
1
/
80
2.491356
e
-
010
4.0
07
0.4
1
/
10
2.982868
e
-
006
—
1
/
20
1.886698
e
-
007
3.9828
1
/
40
1.180691
e
-
008
3.9982
1
/
80
7.375345
e
-
010
4.0008
0.6
1
/
10
6.568541
e
-
006
—
1
/
20
4.168414
e
-
007
3.9780
1
/
40
2.612079
e
-
008
3.9962
1
/
80
1.632646
e
-
009
3.9999
0.8
1
/
10
1.281483
e
-
005
—
1
/
20
8.158970
e
-
007
3.9733
1
/
40
5.120258
e
-
008
3.9941
1
/
80
3.202608
e
-
009
3.9989
1.2
1
/
10
4.091054
e
-
005
—
1
/
20
2.613444
e
-
006
3.9684
1
/
40
1.642689
e
-
007
3.9918
1
/
80
1.028262
e
-
008
3.9978
1.4
1
/
10
6.933668
e
-
005
—
1
/
20
4.429158
e
-
006
3.9685
1
/
40
2.783972
e
-
007
3.9918
1
/
80
1.742681
e
-
008
3.9978
1.6
1
/
10
1.145559
e
-
004
—
1
/
20
7.311853
e
-
006
3.9697
1
/
40
4.594486
e
-
007
3.9923
1
/
80
2.875631
e
-
008
3.9980
1.8
1
/
10
1.850464
e
-
004
—
1
/
20
1.179730
e
-
005
3.9714
1
/
40
7.409938
e
-
007
3.9929
1
/
80
4.637042
e
-
008
3.9982


## 5. Conclusion

In this paper, we build two new finite difference schemes for the Riemann-Liouville and Riesz derivatives, respectively. The convergence orders of the difference schemes are proved by Fourier transform method. Finally, numerical experiments have been carried out to support the theoretical claims.



---

*Source: 540692-2014-04-30.xml*