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Communications on Applied Mathematics and Computation (2020) 2:1–29https://doi.org/10.1007/s42967-019-00023-y

1 3

ORIGINAL PAPER

A High Order Formula to Approximate the Caputo Fractional Derivative

R. Mokhtari1 · F. Mostajeran1

Received: 13 August 2018 / Revised: 23 February 2019 / Accepted: 24 February 2019 / Published online: 4 July 2019 © Shanghai University 2019

AbstractWe present here a high-order numerical formula for approximating the Caputo fractional deriv-ative of order � for 0 < 𝛼 < 1. This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of frac-tional ordinary/partial differential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is 4 − �, while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The effectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical exam-ples. Moreover, an application of the new formula in solving some fractional partial differential equations is presented by constructing a finite difference scheme. A PDE-based image denois-ing approach is proposed to demonstrate the performance of the proposed scheme.

Keywords Caputo fractional derivative · Fractional partial differential equation · Finite difference scheme

Mathematics Subject Classification 65M15 · 65M06

1 Introduction

Nowadays, to provide a better approach for describing complex phenomenon in nature, the fractional calculus has been developed in both theory and application. By entering the fractional derivative into the engineering and science problems, dozens of fractional prob-lems such as the fractional diffusion equation, the fractional advection–diffusion equation, the fractional Fokker–Planck equation, the fractional cable equation and many other equa-tions have appeared in recent years. Among various definitions of the fractional deriva-tive, the Caputo fractional derivative is one of the most important and applicable tools in

* R. Mokhtari [email protected]

F. Mostajeran [email protected]

1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

http://crossmark.crossref.org/dialog/?doi=10.1007/s42967-019-00023-y&domain=pdf

2 Communications on Applied Mathematics and Computation (2020) 2:1–29

1 3

time-dependent problems. In fact in almost all of the time-fractional partial differential equations, the time-fractional derivative is usually considered in the sense of Caputo. A concise and somewhat complete literature survey can be found in [18].

Due to the complexity of computing of some special functions, such as the Fox H func-tion and the hyperbolic geometry function [23], and the difficulties of finding exact solutions for most fractional problems, many researchers are working on extracting useful numerical algorithms. Some successful methods are the finite difference method (FDM) [2, 6, 21], the finite element method [24], the method of discontinuous Galerkin [8, 9, 26, 27], the meshless method [28], and the spectral approximation [19]; among them, the FDM is especially favored thanks to its simplicity in both calculation and analysis.

Obviously, a suitable numerical differentiation formula is necessary for numerically solving some fractional differential equations involving fractional derivatives. To the best of our knowl-edge, the Grünwald–Letnikov (GL) formula and the L1 formula are two more important numer-ical differentiation formulae for discretizing fractional derivatives. For the fractional derivative of order � , the accuracy of the GL formula is of order one [7, 23], while the L1 formula can achieve an improved accuracy of order 2 − � [16]. In fact, the L1 formula is established by a piecewise linear interpolation approximation for the integrand function on each small interval. By applying a higher-order interpolant instead of the linear interpolant to improve the numeri-cal accuracy, Cao et al. [5], Gao et al. [12], and Lv and Xu [20] proposed some (3 − �)-order formulae for the �-order Caputo fractional derivative. A three-point L1 approximation of the Caputo derivative with order 3 − � has been also presented in [11]. Recently, Cao et al. [4] have presented a high-order algorithm with convergence order 4 − � for the Caputo derivative, in which the coefficients and truncation errors of the proposed scheme are not discussed in detail. Moreover, the proposed method was used to present a higher-order numerical scheme to solve a time fractional advection–diffusion equation and its stability analysis was discussed by using the Fourier method. We also independently introduce a 4 − � order algorithm for the Caputo derivative with a slightly different approach, where the implementation of the proposed scheme is distinctive, the weight coefficients are exactly examined, and the truncation error in our work is precisely inspected. For this purpose, three distinct numerical examples for testing the trunca-tion error of the proposed formula and two different numerical examples of the time-fractional partial differential equations with the Caputo fractional derivative are carried out.

The objective of this paper is to establish an explicit high-order numerical differentiation formula for the Caputo fractional derivative and apply it to some schemes for numerically solving some fractional differential equations. Here after presenting a brief discussion about previous fractional numerical differentiation formulae for Caputo fractional of order � , i.e., L1 and L1-2 [12], we introduce L1-2-3 formula which has a better numerical accuracy compared to the L1 and L1-2 formulae. After that, we use the new L1-2-3 formula in various examples and make a comparison between the new and older formulae. Then, we construct a finite dif-ference scheme for solving a time-fractional diffusion equation involving the Caputo deriva-tive and provide some examples. Moreover, this new scheme is applied to the image denois-ing. Finally, we conclude the paper with a brief conclusion.

2 New Formula, Construction and Properties

In this section, to fully understand the proposed process, after representing both the L1 for-mula and the L1-2 formula, we give a new process of deriving the fractional numerical dif-ferentiation formula called L1-2-3. For this purpose, we denote Δt as the temporal step length and for the integer k ≥ 0 , we set tk = kΔt , tk+1∕2 = (tk+1 + tk)∕2 , and

3Communications on Applied Mathematics and Computation (2020) 2:1–29

1 3

Denoting linear Lagrange interpolant of f on each small interval [tj−1, tj] (1 ≤ j ≤ k) as Π1,j f , we get

which follows from the linear interpolation theory that

In a similar way, by constructing the second-degree Lagrange interpolating polynomial Π2,j f of f using three points (tj−2, f (tj−2)), (tj−1, f (tj−1)) , and (tj, f (tj)) and taking a constraint of the result onto small interval [tj−1, tj] (2 ≤ j ≤ k) , we have

and

For j ≥ 3 , constructing the third degree Lagrange interpolating polynomial Π3,j f of f using four points (tj−n, f (tj−n)), n = 3, 2, 1, 0 and taking a constraint of the result onto the small interval [tj−1, tj] (2 ≤ j ≤ k) , we get

and immediately

�t fk− 1

2

=

f (tk) − f (tk−1)

Δt, �2

tfk =

1

Δt(�t fk+ 1

2

− �t fk− 1

2

),

�3tfk−

1

2

=

1

Δt2(�t fk+ 1

2

− 2�tfk− 1

2

+ �t fk− 3

2

).

Π1,j f (t) = f (tj−1)tj − t

Δt+ f (tj)

t − tj−1

Δt,

f (t) − Π1,j f (t) =f��

(�j)

2(t − tj−1)(t − tj),

t ∈ [tj−1, tj], �j ∈ (tj−1, tj), 1 ≤ j ≤ k.

Π2,j f (t) = f (tj−2)(t − tj−1)(t − tj)

2Δt2

+ f (tj−1)(t − tj−2)(tj − t)

Δt2+ f (tj)

(t − tj−1)(t − tj−2)

2Δt2,

f (t) − Π2,j f (t) =f��

(�j)

6(t − tj−2)(t − tj−1)(t − tj),

t ∈ [tj−1, tj], �j ∈ (tj−2, tj), 2 ≤ j ≤ k.

Π3,j f (t) = f (tj−3)(tj − t)(t − tj−1)(t − tj−2)

6Δt3+ f (tj−2)

(t − tj)(t − tj−1)(t − tj−3)

2Δt3

+ f (tj−1)(tj − t)(t − tj−3)(t − tj−2)

2Δt3+ f (tj)

(t − tj−3)(t − tj−1)(t − tj−2)

6Δt3

= f (tj−3)(tj − t)(t − tj−1)(t − tj−2)

6Δt3+ f (tj−2)

(t − tj)(t − tj−1)

2Δt2

(1 +

t − tj−2

Δt

)

+ f (tj−1)(tj − t)(t − tj−2)

Δt2

(1 +

t − tj−1

2Δt

)+ f (tj)

(t − tj−1)(t − tj−2)

2Δt2

(1 +

t − tj

3Δt

)

= Π2,j f (t) +1

6(t − tj)(t − tj−1)(t − tj−2) �

3tfj−

3

2

,

(Π3,j f (t))�

= �t fj− 1

2

+ (�2tfj−1) (t − t

j−1

2

) +

1

6Φj(t) �

3tfj−

3

2

,


1 3

where Φj(t) = 3t2 − 2t(tj−2 + tj−1 + tj) + tjtj−1 + tjtj−2 + tj−2tj−1 . Moreover, we know that

For further details about Lagrange interpolants, we refer to [3]. Letting f ∈ C1[0, tk] ,

the Caputo fractional derivative of order � is defined as [19]

In this paper, we assume that 0 < 𝛼 < 1 . In (1), we use Π1,j f to approximate f on the first small interval [t0, t1] , Π2,j f to approximate f on the second small interval [t1, t2] and for j ≥ 3 , Π3,j f to approximate f on the interval [tj−1, tj] . Noticing

where

we can obtain an improved numerical approximation of the Caputo fractional derivative of order � for function f in the following form:

f (t) − Π3,j f (t) =f (4)(�j)

4!

3∏i=0

(t − tj−i),

t ∈ [tj−1, tj], �j ∈ (tj−3, tj), 3 ≤ j ≤ k.

(1)

C0�

tf (t)

||||t=tk=

1

Γ(1 − �) �tk

0

f�

(s)

(tk − s)�ds

=

1

Γ(1 − �)

k∑j=1

�tj

tj−1

f�

(s)

(tk − s)�ds.

∫tj

tj−1

Φj(s)(tk − s)−� ds =6Δt3−�

1 − �

�(�)

k−j,

�(�)

j= −

(1

6((j + 1)1−� + 2 j1−�) +

1

2 − �

j2−� −1

(2 − �)(3 − �)

((j + 1)3−� − j3−�)

), j ≥ 0,

C0�

tf (t)

||||t=tk=

1

Γ(1 − �)

k∑j=1

�tj

tj−1

f�

(s)

(tk − s)�ds

≈

1

Γ(1 − �)

(�

t1

t0

(tk − s)−� (Π1,1f (t))�

ds + �t2

t1

(tk − s)−� (Π2,2f (t))�

ds

+

k∑j=3

�tj

tj−1

(tk − s)−� (Π3,jf (t))�

ds

)

=

1

Γ(1 − �)

(�tf 1

2 �t1

t0

(tk − s)−� ds + �t2

t1

(tk − s)−� (�tf 32

+ �2tf1(s − t 3

2

)) ds

+

k∑j=3

�tj

tj−1

(tk − s)−� (�tfj− 1

2

+ �2tfj−1(s − t

j−1

2

) +

1

6Φj(s) �

3tfj−

3

2

) ds

)

=

1

Γ(1 − �)

( k∑j=1

(�tfj− 1

2

)�tj

tj−1

(tk − s)−�ds +

k∑j=2

(�2tfj−1)�

tj

tj−1

(tk − s)−�(s − tj−

1

2

)ds

+

k∑j=3

1

6(�

3tfj−

3

2

)�tj

tj−1

(tk − s)−� Φj(s) ds

).


1 3

According to [12], D�

tf is the classical L1 operator which is derived from a piecewise lin-

ear interpolation approximation of f on each small interval [tj−1, tj] (1 ≤ j ≤ k) defined by

with a(�)j

= (j + 1)1−� − j1−� , 0 ≤ j ≤ k − 1 , and ��

tf is the L1-2 operator which is

derived from a second Lagrange interpolation approximation of f on each small interval [tj−1, tj] (1 ≤ j ≤ k) defined by

with b(�)j

= [( j + 1)2−� − j2−�]∕(2 − �) − [( j + 1)1−� + j1−�]∕2, j ≥ 0 . Now, we define

The operator �̃�

t is the improved fractional numerical differentiation operator for the Caputo

fractional derivative C0�

tf and we call it L1-2-3 operator. Looking at (3) and (4), we under-

stand that at t = tk the new L1-2-3 operator �̃�

tf is actually a modification of the L1-2 operator

��

tf by adding a correction term Δt3−�

Γ(2−�)

k∑j=3

�(�)

k−j�3tfj−

3

2

for k ≥ 3 . The following lemma remarks

the properties of coefficient {�(�)j} in the correction term.

Lemma 2.1 For any � , we have

(i) 𝛽(𝛼)

j> 0, j ≥ 0;

(ii) �(�)

0≥ �

(�)

1≥ �

(�)

2≥ ⋯ , i.e., {�(�)

j} is strictly monotonically decreasing with respect

to j.

Proof The only root of Φj in [tj−1, tj] is t∗j= tj−1 + Δt∕

√3 . Φj is an increasing function on

[tj−1, tj] . Moreover, ∫ t∗j

tj−1−Φj(s) ds = ∫ tj

t∗j

Φj(s) ds = 2√3Δt3∕9 = A , and (tk − s)−� is a con-

tinuous, positive and increasing function of s on [tj−1, tj] . Using the weighted mean value theorem for integrals [3], we have

(a) there is c1 ∈ [tj−1, t∗

j] such that

(b) there is c2 ∈ [t∗j, tj] such that

(2)

D�

tf (t)

||||t=tk=

Δt1−�

Γ(2 − �)

k∑j=1

a(�)

k−j�t fj− 1

2

=

Δt−�

Γ(2 − �)

(f (tk) −

k−1∑j=1

(a(�)

k−j−1− a

(�)

k−j)f (tj) − a

(�)

k−1f (t0)

),

(3)��

tf (t)

||||t=tk= D�

tf (t)

||||t=tk+

Δt2−�

Γ(2 − �)

k∑j=2

b(�)

k−j�2tfj−1,

(4)�̃�

tf (t)

||||t=tk= �

�

tf (t)

||||t=tk+

Δt3−�

Γ(2 − �)

k∑j=3

�(�)

k−j�3tfj−

3

2

.

0 < ∫t∗j

tj−1

−Φj(s) (tk − s)−𝛼 ds = (tk − c1)−𝛼 ∫

t∗j

tj−1

−Φj(s) ds = A (tk − c1)−𝛼

< A (tk − t∗j)

−𝛼 ,


1 3

Therefore,

and it means that 𝛽(𝛼)j

> 0 . To prove that {�(�)j} is strictly monotonically decreasing with

respect to j, we must show

Again, using the weighted mean value theorem for integrals, we can write

(a) there is c1 ∈ [tj−1, t∗

j] such that

(b) there is c2 ∈ [t∗j, tj] such that

where g(s) = (tk − s − Δt)−� − (tk − s)−� which is an increasing function of s on [tj−1, tj] since g�

(s) = 𝛼 ((tk − s − Δt)−(𝛼+1) − (tk − s)−(𝛼+1)) > 0 . Therefore, Ag(c1) < Ag(c2) , and it means that

Then,

Knowing Φj(s) = Φj(s + Δt), s ∈ [tj−1, tj] , we have

∫tj

t∗j

Φj(s) (tk − s)−𝛼 ds = (tk − c2)−𝛼 ∫

tj

t∗j

Φj(s) ds = A (tk − c2)−𝛼

> A (tk − t∗j)

−𝛼 .

0 < ∫t∗j

tj−1

−Φj(s) (tk − s)−𝛼 ds < ∫tj

t∗j

Φj(s) (tk − s)−𝛼 ds,

(5)0 < ∫tj

tj−1

Φj(s) (tk − s)−𝛼 ds < ∫tj+1

tj

Φj+1(s) (tk − s)−𝛼 ds.

0 < ∫t∗j

tj−1

−Φj(s) g(s) ds = g(c1) ∫t∗j

tj−1

−Φj(s) ds = Ag(c1),

0 < ∫tj

t∗j

Φj(s) g(s) ds = g(c2) ∫tj

t∗j

Φj(s) ds = A g(c2),

0 < ∫t∗j

tj−1

−Φj(s) g(s) ds < ∫tj

t∗j

Φj(s) g(s) ds.

0 < ∫tj

tj−1

Φj(s) g(s) ds = ∫tj

tj−1

Φj(s) (tk − s − Δt)−𝛼 − Φj(s) (tk − s)−𝛼 ds.

0 < ∫tj+1

tj

Φj+1(s) (tk − s)−𝛼 − ∫tj

tj−1

Φj(s) (tk − s)−𝛼 ds.


1 3

Therefore, (5) holds, and so the proof is complete.The new, improved fractional numerical differentiation formula (4) can be rewritten as

where for k = 1 , � (�)0

= 1 ; for k = 2 , � (�)0

= a(�)

0+ b

(�)

0, �

(�)

1= a

(�)

1− b

(�)

0 ; for k = 3,

and for k ≥ 4,

For k = 1, (4) is the same as (2); for k = 2 , (4) is the same as (3); for k = 3,

(6)

�̃�

tf (t)

||||t=tk=

Δt1−�

Γ(2 − �)

( k∑j=1

a(�)

k−j(�t fj− 1

2

) +

k∑j=2

b(�)

k−j(�t fj− 1

2

− �t fj− 3

2

)

+

k∑j=3

�(�)

k−j(�t fj− 1

2

− 2 �t fj− 3

2

+ �t fj− 5

2

)

)

=

Δt1−�

Γ(2 − �)

( k∑j=1

a(�)

k−j(�t fj− 1

2

) +

k∑j=2

b(�)

k−j�t fj− 1

2

−

k−1∑j=1

b(�)

k−j−1�t fj− 1

2

+

k∑j=3

�(�)

k−j�t fj− 1

2

− 2

k−1∑j=2

�(�)

k−j−1�t fj− 1

2

+

k−2∑j=1

�(�)

k−j−2�t fj− 1

2

)

=

Δt1−�

Γ(2 − �)

k∑j=1

�(�)

k−j�t fj− 1

2

=

Δt−�

Γ(2 − �)

(�(�)

0fk −

k−1∑j=1

(�(�)

k−j−1− �

(�)

k−j) fj − �

(�)

k−1f0

),

�(�)

l=

⎧⎪⎨⎪⎩

a(�)

l+ b

(�)

l+ �

(�)

l, l = 0,

a(�)

l+ b

(�)

l− b

(�)

l−1− 2�

(�)

l−1, l = 1,

a(�)

l− b

(�)

l−1+ �

(�)

l−2, l = 2,

(7)�(�)

l=

⎧⎪⎪⎨⎪⎪⎩

a(�)

l+ b

(�)

l+ �

(�)

l, l = 0,

a(�)

l+ b

(�)

l− b

(�)

l−1+ �

(�)

l− 2�

(�)

l−1, l = 1,

a(�)

l+ b

(�)

l− b

(�)

l−1+ �

(�)

l− 2�

(�)

l−1+ �

(�)

l−2, 2 ≤ l ≤ k − 3,

a(�)

l+ b

(�)

l− b

(�)

l−1− 2�

(�)

l−1+ �

(�)

l−2, l = k − 2,

a(�)

l− b

(�)

l−1+ �

(�)

l−2, l = k − 1.

�(�)

0=

1

3+

1

2 − �

+

1

(2 − �) (3 − �)

∈

(1,

11

6

),

�(�)

1= 2−� +

22−� − 2

2 − �

−

2

3−

2

(2 − �) (3 − �)

∈

(−

7

6, 1),

�(�)

2= 31−� − 2−� −

22−� − 1

2 − �

+

1

3+

1

(2 − �) (3 − �)

∈ (0, 1).


1 3

Lemma 2.2 For any � and � (�)j(0 ≤ j ≤ k − 1, k ≥ 4) defined in (7), we have

(i) 𝛾(𝛼)

0> |𝛾 (𝛼)

1|;

(ii) 𝛾(𝛼)

j> 0, j ≠ 1;

(iii) � (�)2

≥ �(�)

3≥ ⋯ ≥ �

(�)

k−3 ≥ �

(�)

k−2≥ �

(�)

k−1;

(iv) 𝛾 (𝛼)0

> 𝛾(𝛼)

2;

(v) k−1∑j=0

�(�)

j= k1−�.

Proof For any � and k ≥ 4 , noticing � (�)k

in (7), we have

Let g(x) = 21−x

3+

22−x−3

2−x+

23−x−3

(2−x)(3−x)− 1 . Then, g�

(x) < 0 for x ∈ (0, 1) . Therefore, g is monotonically decreasing on (0, 1), and then − 7

6= 𝛾

(1)

1< 𝛾

(𝛼)

1< 𝛾

(0)

1= 1 ; thus, 𝛾 (𝛼)

0> |𝛾 (𝛼)

1| ,

i.e., (i) holds. Moreover, the root of g on (0, 1) is about x0 = 0.532 49 , thus, 𝛾 (𝛼)1

> 0 for 𝛼 < x0 , and 𝛾 (𝛼)

1< 0 for 𝛼 > x0 . For 2 ≤ j ≤ k − 3,

where

�(�)

0=

1

3+

1

2−�+

1

(2−�) (3−�)∈

(1,

11

6

), �

(�)

1=

21−�

3+

22−� − 3

2 − �

+

23−� − 3

(2 − �)(3 − �)

− 1.

�(�)

j= a

(�)

l+ b

(�)

l− b

(�)

l−1+ �

(�)

l− 2�

(�)

l−1+ �

(�)

l−2

=

(1

3((j + 1)1−� − 2j1−� + (j − 1)1−�) +

1

2 − �

((j + 1)2−� − 2j2−� + (j − 1)2−�)

+

1

(2 − �)(3 − �)

((j + 1)3−� − 2j3−� + (j − 1)3−�)

)

−

(1

3(j1−� − 2(j − 1)1−� + (j − 2)1−�) +

1

2 − �

(j1−� − 2(j − 1)1−� + (j − 2)1−�)

+

1

(2 − �)(3 − �)

(j1−� − 2(j − 1)1−� + (j − 2)1−�)

)

= Ij − Ij−1,

Ij =1

3((j + 1)1−� − 2j1−� + (j − 1)1−�) +

1

2 − �

((j + 1)2−� − 2j2−� + (j − 1)2−�)

+

1

(2 − �)(3 − �)

((j + 1)3−� − 2j3−� + (j − 1)3−�)

=

(1

3(j + 1)1−� +

1

2 − �

(j + 1)2−� +1

(2 − �)(3 − �)

(j + 1)3−�)

− 2

(1

3j1−� +

1

2 − �

j2−� +1

(2 − �)(3 − �)

j3−�)

+

(1

3(j − 1)1−� +

1

2 − �

(j − 1)2−� +1

(2 − �)(3 − �)

(j − 1)3−�), j ≥ 1.


1 3

Let h(x) = 1

3x1−� +

1

2−�x2−� +

1

(2−�)(3−�)x3−� , x ≥ 1 . Then Ij = h(j + 1) − 2h(j) + h(j − 1) .

For x ≥ 1 , we have h�

(x) > 0, h��

(x) > 0, h��

(x) > 0 , and h(4)(x) < 0 . Therefore,

where j − 2 < 𝜁j < j + 1 and j − 2 < 𝜎j < j + 2 . That is to say that � (�)2

≥ �(�)

3≥ ⋯ ≥ �

(�)

k−3 .

In addition,

is continuous for all � ∈ [0, 1] , then it attains its maximum and minimum values on [0, 1]. Because the minimum of this function on [0, 1] is zero, � (�)

k−3− �

(�)

k−2≥ 0 , so � (�)

k−3≥ �

(�)

k−2 .

Similarly,

is continuous for all � ∈ [0, 1] , then due to attaining its minimum values on [0, 1], which is zero, � (�)

k−2− �

(�)

k−1≥ 0 , then � (�)

k−2≥ �

(�)

k−1 . Moreover,

is continuous for all � ∈ [0, 1] . Because p�

(𝛼) < 0 on [0, 1], this function is decreasing on [0, 1], and then 0 = p(1) < p(𝛼) for all � ∈ (0, 1) ; thus, 𝛾 (𝛼)

k−2> 0 . In a similar way,

is continuous, thus for all � ∈ [0, 1] . Because q�

(𝛼) < 0 on [0, 1], 0 = q(1) < q(𝛼) for all � ∈ (0, 1) , 𝛾 (𝛼)

k−1> 0 . That is to say, (ii) and (iii) hold. Due to the fact that

�(�)

0− �

(�)

2= 3 h(2) − 2 h(1) − h(3) is positive, continuous, and increasing on (0, 1), we

have 𝛾 (𝛼)0

> 𝛾(𝛼)

2 , so (iv) holds. The validity of (v) can be directly derived from the definition

(7) of � (�)j

. The proof is completed.

The truncation errors of the new L1-2-3 formula (6) are illustrated in the following theorem.

Theorem 2.1 Suppose f ∈ C4[0, tk]. For any � , �̃�

tf (t)|t=tk is defined in (6). Denote

|R( f (tk))| =C0�

tf (t)|t=tk − �̃

�

tf (t)|t=tk. Then, we have

𝛾(𝛼)

j= Ij − Ij−1 = h(j + 1) − 3 h(j) + 3 h(j − 1) − h(j − 2) = h

��

(𝜁j) > 0,

𝛾(𝛼)

j− 𝛾

(𝛼)

j+1= −h(j + 2) + 4 h(j + 1) − 6 h(j) + 4 h(j − 1) − h(j − 2) = −h(4)(𝜎j) > 0,

�(�)

k−3− �

(�)

k−2= −h(k − 1) + 3 h(k − 2) − 6 h(k − 3) + 4 h(k − 4) − h(k − 5) +

(k − 1)3−�

(2 − �)(3 − �)

�(�)

k−2− �

(�)

k−1= −3 h(k − 2) + 4 h(k − 3) − h(k − 4) − k1−� + (k − 1)1−� +

2 (k − 1)2−�

2 − �

�(�)

k−2= −2 h(k − 2) + 3 h(k − 3) − h(k − 4) +

1

2(k − 1)1−� +

(k−1)2−�

2−�∶= p(�)

�(�)

k−1= h(k − 2) − h(k − 3) + k1−� −

1

2(k − 1)1−� −

(k−1)2−�

2−�∶= q(�)


1 3

Proof For k = 1, (6) is just the L1 formula, and we have

where �1 ∈ (t0, t1) . For k = 2, (6) is just the L1-2 formula, and we have

where

and

where �2 ∈ (t1, t2) . For k ≥ 3 , we get

(8)

�R( f (t1))� ≤ �

2Γ(3−�)Δt2−� max

t0≤t≤t1f��

(t),

�R( f (t2))� ≤ �

Γ(1−�)

�1

12maxt0≤t≤t1� f

��

(t)� (t2 − t1)−�−1

Δt3

+

1

3(1−�)(2−�)

�1

2+

1

3−�

�maxt0≤t≤t2� f

��

(t)�Δt3−��,

�R( f (tk))� ≤ �

Γ(1−�)

�12 (tk − t1)

−�−1 maxt0≤t≤t1� f

��

(t)�Δt3 + 1

8(tk − t2)

−�−1 maxt0≤t≤t2� f

��

(t)�Δt4

+

�1

2+

1

12

27−10�+�2∏4

i=1(�−i)

�maxt0≤t≤tk� f

(4)(t)�Δt4−�

�, k ≥ 3.

R( f (t1)) =1

Γ(1 − �)∫t1

t0

[ f (s) − Π1,1( f (s))]�

(t1 − s)−� ds =�

2Γ(3 − �)

f��

(�1)Δt2−� ,

R( f (t2)) =1

Γ(1 − �)

{∫

t1

t0

[ f (s) − Π1,1(f (s))]�

(t2 − s)−� ds + ∫t2

t1

[ f (s) − Π2,2( f (s))]�

(t2 − s)−� ds

}

=

−�

Γ(1 − �)

{∫

t1

t0

[ f (s) − Π1,1( f (s))](t2 − s)−�−1 ds + ∫t2

t1

[ f (s) − Π2,2( f (s))](t2 − s)−�−1 ds

},

||||�t1

t0

[ f (s) − Π1,1( f (s))](t2 − s)−�−1 ds|||| =

||||�t1

t0

f��

(�1)

2(s − t0)(s − t1)(t2 − s)−�−1 ds

||||=

||||1

2f��

(�1)�t1

t0

(s − t0)(s − t1)(t2 − s)−�−1 ds||||

≤ 1

12(t2 − t1)

−�−1Δt3 max

t0≤t≤t1| f��

(t)|,

∫t2

t1

[ f (s) − Π2,2( f (s))](t2 − s)−�−1 ds = ∫t2

t1

f��

(�2)

6(s − t0)(s − t1)(t2 − s)−� ds

=

1

6f��

(�2)∫t2

t1

(s − t0)(s − t1)(t2 − s)−� ds

= −

1

6| f �� (�2)|Δt3−� 2

(1−�)(2−�)

(1

2+

1

3−�

),


1 3

where

is used. Moreover,

where �1 ∈ (t0, t1),

R( f (tk)) =1

Γ(1 − �)

{∫

t1

t0

[ f (s) − Π1,1( f (s))]�

(tk − s)−� ds + ∫t2

t1

[ f (s) − Π2,2( f (s))]�

(tk − s)−� ds

+

k∑j=3

∫tj

tj−1

[ f (s) − Π3,j( f (s))]�

(tk − s)−� ds

}

=

1

Γ(1 − �)

{[ f (s) − Π1,1( f (s))](tk − s)−�

||||t1

t0

− � ∫t1

t0

[ f (s) − Π1,1( f (s))](tk − s)−�−1 ds

+ [ f (s) − Π2,2( f (s))](tk − s)−�||||t2

t1

− � ∫t2

t1

[ f (s) − Π2,2( f (s))](tk − s)−�−1 ds

+

k∑j=3

{[ f (s) − Π3,j( f (s))](tk − s)−�

||||tj

tj−1

− � ∫tj

tj−1

[ f (s) − Π3,j( f (s))](tk − s)−�−1 ds

}}

=

−�

Γ(1 − �)

{∫

t1

t0

[ f (s) − Π1,1( f (s))](tk − s)−�−1 ds

+ ∫t2

t1

[ f (s) − Π2,2( f (s))](tk − s)−�−1 ds

+

k∑j=3

{∫

t2

t1

[ f (s) − Π2,2( f (s))](tk − s)−�−1 ds

}},

[f (s) − Π3,k( f (s))

](tk − s)−�

||||tk

tk−1

= −

1

4!f (4)(�k) (s − tk−3)(s − tk−2)(s − tk−1)(tk − s)1−�

||||tk

tk−1

= 0

|||||�t1

t0

[ f (s) − Π1,1( f (s))](tk − s)−�−1 ds|||||=

|||||�t1

t0

1

2f��

(�1) (s − t0)(s − t1)(tk − s)−�−1 ds|||||

=

|||||1

2f��

(�1)�t1

t0

(s − t0)(s − t1)(tk − s)−�−1 ds|||||

≤ 1

8| f �� (�1)|Δt2 �

t1

t0

(tk − s)−�−1 ds

≤ 12| f �� (�1)|(tk − t1)−�−1

Δt3,


1 3

where �2 ∈ (t1, t2) , and

where �j ∈ (tj−3, tj), 3 ≤ j ≤ k − 1 , � ∈ (t0, tk−1) . In addition,

where �k ∈ (tk−3, tk) . Hence, (8) holds.

3 Numerical Examples

To verify our constructed numerical formula for the Caputo fractional derivative and show that L1-2-3 formula (6) can act better than the L1 and L1-2 formulae, we test the follow-ing three numerical examples. Although the first two examples ( f (t) = t4+r and f (t) = t3 ) are polynomial-based examples with zero initial values, the convergence order of them is different. In addition, the third one ( f (t) = exp(2t) ) is a non-polynomial example with non-zero initial value and its exact solution does not have a closed form. Actually, we aim to present these three examples to show that the convergence order of the formula depends on the function f. In fact, the convergence order of the following three examples are, respec-tively, 4 − � , 4 and 3.

||||�t2

t1

[ f (s) − Π2,2( f (s))](tk − s)−�−1 ds|||| =

||||�t2

t1

1

6f��

(�2) (s − t0)(s − t1)(s − t2)(tk − s)−�−1 ds||||

=

||||1

6f��

(�2)�t2

t1

(s − t0)(s − t1)(s − t2)(tk − s)−�−1 ds||||

≤ 1

12| f �� (�2)|Δt3 �

t2

t1

(tk − s)−�−1 ds

≤ 1

8| f �� (�2)| (tk − t2)

−�−1Δt4,

||||k−1∑j=3

�tj

tj−1

[ f (s) − Π3,j( f (s))](tk − s)−�−1 ds|||| =

||||k−1∑j=3

�tj

tj−1

1

4!f (4)(�j)

3∏i=0

(s − tj−i)(tk − s)−�−1 ds||||

=

1

4!| f (4)(�)|

k−1∑j=3

�tj

tj−1

3∏i=0

(s − tj−i)(tk − s)−�−1 ds

≤ 1

4!| f (4)(�)|Δt4 �

tk−1

t2

(tk − s)−�−1 ds

≤ 1

2| f (4)(�)|Δt4−� ,

∫tk

tk−1

[ f (s) − Π3,k( f (s))](tk − s)−�−1 ds = ∫tk

tk−1

1

4!f (4)(�k)

3�i=0

(s − tk−i)(tk − s)−�−1 ds

= −

1

4!f (4)(�4)∫

tk

tk−1

3�i=1

(s − tk−i)(tk − s)−� ds

= −

1

4!f (4)(�4) Δt

4−�

�2(27−10�+�2)∏4

i=1(�−i)

�,


1 3

For 0 ≤ k ≤ N, denote

and assume that tN = T = 1.

Example 1 Let f (t) = t4+r , where 0 ≤ r < 1 . We compute the �-order Caputo fractional derivative of f(T) numerically. The exact solution is given by

We use three methods presented in Sect. 2, and list the computational errors and numeri-cal convergence orders for different parameters � = 0.9, 0.5, 0.1 and different temporal step sizes in Table 1. Noticing Table 1, we find that the computed errors by the L1-2-3 formula (6) are obviously much smaller than that by the L1-2 and L1 formulae (3) and (2), and the attain-able convergence order of the L1-2-3 formula (6) is 4 − � , which is also higher than the con-vergence order of 3 − � and 2 − � for the L1-2 and L1 formulae (3) and (2). From the above test example, we can draw the conclusion that the computational efficiency and numerical accuracy of the new formula (6) are superior to that of the L1-2 formula (3) and L1 formula (2), and the convergence order of (6) reaches 4 − � if N ≠ 1 and N is relatively large.

Example 2 Consider f (t) = t3 . We compute the �-order Caputo fractional derivative of f(T) numerically. The exact solution is given by

Taking the same varying temporal step sizes and parameters � as in Example 1, we use formulae (2), (3), and (6), and list the computational errors and numerical convergence orders in Table 2. According to (8), for f (t) = t3 , we have

which shows that the convergence order of (6) for this example reaches 4 since (tk − t1)

−�−1= O(1) and (tk − t2)

−�−1= O(1) . The results of Table 2 confirm this fact.

Example 3 Suppose f (t) = exp(2t) . We compute the �-order Caputo fractional derivative of f(T) numerically. The exact solution is given [10] by

where E�,�(z) is the Mittag-Leffler function with two parameters defined by

Fk=

C0D

�

tf (t)

||||t=tk, f k

1= D�

tf (t)

||||t=tk, f k

2= �

�

tf (t)

||||t=tk, f k

3= �̃

�

tf (t)

||||t=tk,

ENj= |FN

− f Nj|, orderj = log2

ENj(Δt)

ENj(Δt∕2)

, j = 1, 2, 3

C0�

tf (t)

||||t=1 =

t4−�+r Γ(5 + r)

Γ(5 − � + r)

||||t=1 =

Γ(5 + r)

Γ(5 − � + r).

C0�

tf (t)

||||t=1 =

6

Γ(4 − �)

t3−�||||t=1 =

6

Γ(4 − �)

.

|R( f (tk))| ≤ �

Γ(1 − �)

( 72 (tk − t1)−�−1

+

6

8(tk − t2)

−�−1) Δt4, k ≥ 3,

C0�

tf (t)

||||t=1 = 2t1−� E1,2−�(2t)||||t=1 = 2E1,2−�(2),

E�,�(z) =

∞∑n=0

zn

Γ(�n + �)

.


1 3

Table 1 (Example 1) Computational errors and convergence orders

� r Δt EN1(Δt) order1 EN

2(Δt) order2 EN

3(Δt) order3

≃ 2 − � ≃ 3 − � ≃ 4 − �

0.9 0 1/10 3.859 7E−1 0.00 4.980 0E−2 0.00 3.995 3E−3 0.001/20 1.871 4E−1 1.04 1.211 9E−2 2.04 4.663 0E−4 3.101/40 8.905 4E−2 1.07 2.886 5E−3 2.07 5.440 6E−5 3.101/80 4.197 3E−2 1.08 6.803 7E−4 2.08 6.346 7E−6 3.101/160 1.968 6E−2 1.09 1.595 4E−4 2.09 7.403 0E−7 3.101/320 9.209 3E−3 1.10 3.731 4E−5 2.10 8.634 6E−8 3.10

0.9 1/10 5.949 4E−1 0.00 1.070 5E−1 0.00 1.556 4E−2 0.001/20 2.930 2E−1 1.02 2.699 7E−2 2.00 1.930 6E−3 3.011/40 1.405 8E−1 1.06 6.545 5E−3 2.04 2.319 3E−4 3.061/80 6.653 8E−2 1.08 1.556 7E−3 2.07 2.744 7E−5 3.081/160 3.127 5E−2 1.09 3.666 9E−4 2.08 3.224 5E−6 3.091/320 1.464 8E−2 1.09 8.596 0E−5 2.09 3.774 5E−7 3.10






1 3

Due to the lack of closed form for the exact solution of this problem, the Mittag-Leffler function E

�,�(z) which depends on the precise values of the parameters � and � and the argu-ment z has been numerically computed. Table 3 is reported to make a comparison with [4]. According to the results of Table 3, numerical convergence order of formula (6) reaches 3. Also, being determined by (8), for f (t) = exp(2t) , we have

��R( f (tk))�� ≤ �

Γ(1 − �)

⎧⎪⎪⎨⎪⎪⎩

48(tk − t1)−�−1

Δt3

+ 16

⎛⎜⎜⎜⎜⎝

1

2+

1

12

27 − 10� + �2

4∏i=1

(� − i)

⎞⎟⎟⎟⎟⎠Δt4−� + (96(tk − t1)

−�−1+ (tk − t2)

−�−1)Δt4

+32k

⎛⎜⎜⎜⎜⎝

1

2+

1

12

27 − 10� + �2

4∏i=1

(� − i)

⎞⎟⎟⎟⎟⎠Δt5−� + 4(tk − t2)

−�−1Δt5

⎫⎪⎪⎬⎪⎪⎭

, k ≥ 3,


� Δt EN1(Δt) order1 EN

2(Δt) order2 EN

3(Δt) order3

≃ 2 − � ≃ 3 − � ≃ 4





1 3

which shows that the convergence order of (6) for this example must be 3, since (tk − t1)

−�−1= O(1) and (tk − t2)

−�−1= O(1) . Since truncation error of the new L1-2-3

formula has not been obtained correctly in [4], the authors claimed and showed that the convergence order of (6) for this example is 4 − �, which is not correct. Specifically, for � = 0.8 , when the temporal step size has decreased to 1/5 120, the convergence order has increased to 3.196 2, which seems to reach 4 − � . Nevertheless, for this value of � , the convergence order will steadily decrease and reach 3 when Δt decreases to less than 1/40 960. For these three numerical tests, Fig. 1 shows that the computational error curves with � = 0.5 and N = 20 using three different formulae, the L1 formula (2), L1-2 formula (3), and L1-2-3 formula (6). According to Fig. 1, it is clearly obvious that the computational results by (6) are better than the ones by (2) and (3).


� Δt EN1(Δt) order1 EN

2(Δt) order2 EN

3(Δt) order3

≃ 2 − � ≃ 3 − � ≃ 3

0.6 1/10 2.820 3E−1 0.00 3.165 6E−2 0.00 4.230 8E−3 0.001/20 1.131 8E−1 1.31 6.572 5E−3 2.26 4.581 9E−4 3.201/40 4.440 0E−2 1.35 1.310 4E−3 2.32 4.683 9E−5 3.291/80 1.718 6E−2 1.36 2.555 4E−4 2.35 4.647 9E−6 3.331/160 6.599 5E−3 1.38 4.922 6E−5 2.37 4.546 7E−7 3.351/320 2.521 8E−3 1.38 9.418 0E−6 2.38 4.422 6E−8 3.361/ 640 9.607 6E−4 1.39 1.794 8E−6 2.39 4.299 5E−9 3.361/1 280 3.653 2E−4 1.39 3.412 6E−7 2.39 4.191 9E−10 3.351/2 560 1.387 4E−4 1.39 6.479 9E−8 2.39 4.109 5E−11 3.351/5 120 5.265 0E−5 1.39 1.229 4E−8 2.39 4.060 4E−12 3.34

0.4 1/10 1.095 0E−1 0.00 1.186 6E−2 0.00 1.615 6E−3 0.001/20 3.917 7E−2 1.48 2.187 1E−3 2.43 1.568 1E−4 3.361/40 1.365 5E−2 1.52 3.862 5E−4 2.50 1.447 7E−5 3.431/80 4.681 6E−3 1.54 6.655 0E−5 2.53 1.310 3E−6 3.461/160 1.587 5E−3 1.56 1.129 9E−5 2.55 1.185 5E−7 3.461/320 5.342 9E−4 1.57 1.901 0E−6 2.57 1.087 5E−8 3.441/640 1.788 6E−4 1.57 3.179 6E−7 2.57 1.022 6E−9 3.411/1 280 5.965 4E−5 1.58 5.297 4E−8 2.58 9.941 2E−11 3.361/ 2 560 1.984 0E−5 1.58 8.801 8E−9 2.58 1.004 6E−11 3.301/5 120 6.585 0E−6 1.58 1.459 6E−9 2.59 1.057 5E−12 3.24

0.2 1/10 3.218 2E−2 0.00 3.398 7E−3 0.00 4.902 4E−4 0.001/20 1.039 4E−2 1.63 5.629 0E−4 2.59 4.409 3E−5 3.471/40 3.264 0E−3 1.67 8.924 9E−5 2.65 3.863 7E−6 3.511/80 1.005 6E−3 1.69 1.378 0E−5 2.69 3.423 1E−7 3.491/160 3.056 7E−4 1.71 2.091 9E−6 2.71 3.149 7E−8 3.341/320 9.196 3E−5 1.73 3.139 5E−7 2.73 3.061 3E−9 3.361/640 2.745 0E−5 1.74 4.673 1E−8 2.74 3.160 9E−10 3.271/1 280 8.142 5E−6 1.75 6.913 1E−9 2.75 3.451 7E−11 3.191/2 560 2.403 1E−6 1.76 1.017 7E−9 2.76 3.943 3E−12 3.121/5 120 7.062 8E−7 1.76 1.492 5E−10 2.76 4.654 6E−13 3.08


1 3

3.1 Time‑Fractional Partial Differential Equations

We carry out in this section a series of numerical examples of time-fractional partial differ-ential equations with the Caputo fractional derivatives and present some results to confirm our theoretical statements. The main purpose is to check the convergence behavior of the numerical solution with respect to the time step Δt . Consider the time-fractional diffusion equation of the form

subject to the following boundary and initial conditions:

where � is the order of the Caputo fractional derivative, T > 0 , and f ,�,� , and u0 are given functions. Introducing a finite difference approximation to discretize the time-fractional derivative, let Ω

Δt = {tk|tk = kΔt , k = 0, 1,… ,N}, where Δt = T∕N is the time step size. By using (6), Eq. (9) is converted to

(9)C0�

tu(x, t) =

�2u(x, t)

�x2+ f (x, t), x ∈ (0, 1), t ∈ [0, T],

(10)u(0, t) = �(t), u(1, t) = �(t), t ∈ (0, T],

u(x, 0) = u0(x), x ∈ [0, 1],

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

t

The

abso

lute

err

ors

L1L1−2L1−2−3

(a) Example 1 : f(t) = t4

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

t

The

abso

lute

err

ors

L1L1−2L1−2−3

(b) Example 1 : f(t) = t4+α

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

t

The

abso

lute

err

ors

L1L1−2L1−2−3

(c) Example 3 : f(t) = (2t)

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

t

The

abso

lute

err

ors

L1L1−2L1−2−3

(d) Example 2 : f(t) = t3exp

Fig. 1 The error curves of absolute errors with � = 0.5,N = 20


1 3

where � = (Δ t)�Γ(2 − �) , and according to Theorem 2.1 for k ≥ 3,

Let uk = uk(x) be the numerical approximation to u(x, tk) , and f k = f (x, tk) . Then, Eq. (11) can be discretized as the following scheme:

To discretize (11) in space, let ΩΔx = {xi|xi = iΔx, i = 0, 1,… ,M} , where Δx = 1∕M is

the space step size. Suppose u = {uki|0 ≤ i ≤ M, 0 ≤ k ≤ N} is a grid function on

ΩΔx × Ω

Δt , and introduce �xuki−1∕2 = (uki− uk

i−1)∕Δx , where �xuki−1∕2 is the first-order differ-

ence quotient of u on the points (xi, tk) and (xi−1, tk) , and �2xuki= (�xu

ki+1∕2

− �xuki−1∕2

)∕Δx , where �2

xuki is the second-order difference quotient of u on the points (xi−1, tk) , (xi, tk) and

(xi+1, tk) . The finite difference scheme which we consider for (9) and (10) is as follows:

Remark 3.1 The truncation error of (13) is

(11)

�(�)

0u(x, tk) − �

�2u(x, tk)

�x2= �

(�)

k−1u(x, t0) +

k−1∑i=1

(�(�)

i−1− �

(�)

i)u(x, tk−i) + �f (x, tk) + Rk,

�Rk� = �R(u(x, tk))� ≤ �

Γ(1 − �)

�12 (tk − t1)

−�−1 maxt0≤t≤t1

��2u

�t2(x, t)

��Δt3

+

1

8(tk − t2)

−�−1 maxt0≤t≤t2

��3u

�t3(x, t)

��Δt4

+

⎛⎜⎜⎜⎜⎝

1

2+

1

12

27 − 10� + �2

4∏i=1

(� − i)

⎞⎟⎟⎟⎟⎠maxt0≤t≤tk

��4u

�t4(x, t)

��Δt4−�

⎫⎪⎪⎬⎪⎪⎭

.

(12)�(�)

0uk − �

�2uk

�x2= �

(�)

k−1u0 +

k−1∑i=1

(�(�)

i−1− �

(�)

i)uk−i + �f k.

(13)

⎧⎪⎨⎪⎩

�̃�

tuki= �

2xuki+ f k, 1 ≤ i ≤ M − 1, 1 ≤ k ≤ N,

uk0= �(tk), u

kM= �(tk), 1 ≤ k ≤ N,

u0i= u0(xi), 0 ≤ i ≤ M.

�T(x, tk)� ≤ 𝛼

Γ(1 − 𝛼)

�12 (tk − t1)

−𝛼−1 maxt0≤t≤t1

��𝜕2u

𝜕t2(x, t)

��Δt3

+

1

8(tk − t2)

−𝛼−1 maxt0≤t≤t2

��𝜕3u

𝜕t3(x, t)

��Δt4

+

⎛⎜⎜⎜⎜⎝

1

2+

1

12

27 − 10𝛼 + 𝛼2

4∏i=1

(𝛼 − i)

⎞⎟⎟⎟⎟⎠maxt0≤t≤tk

��𝜕4u

𝜕t4(x, t)

��Δt4−𝛼

⎫⎪⎪⎬⎪⎪⎭+

1

12max0<x<1

��𝜕4u

𝜕x4(x, t)

��Δx2, k ≥ 3.


1 3

The corresponding difference schemes using the L1-2 formula [12] and L1 formula [25] for the Caputo derivative are, respectively,

Letting {Uki} , {(u1)ki } , {(u2)

ki} , and {(u3)ki } for 0 ≤ i ≤ M and 0 ≤ k ≤ N be, respectively,

solutions of problem (9) and (10), finite difference schemes (15), (14), and (13), we denote

where j = 1, 2, 3 . To test the efficiency of the difference scheme (13), two numerical exam-ples, which have different nature (in particular, in the time-dependent part of the solution and therefore in the initial condition which affect the convergence order of the formula) are examined.

Example 4 In (9) and (10), take f (x, t) = ext4[Γ(5+�)

24− t�], �(t) = t4+� , �(t) = et4+� , and

u0(x) = 0 . The exact solution is u(x, t) = ext4+�.

(14)

⎧⎪⎨⎪⎩

��

tuki= �

2xuki+ f k, 1 ≤ i ≤ M − 1, 1 ≤ k ≤ N,

uk0= �(tk), u

kM= �(tk), 1 ≤ k ≤ N,

u0i= u0(xi), 0 ≤ i ≤ M,

(15)

⎧⎪⎨⎪⎩

D�

tuki= �

2xuki+ f k, 1 ≤ i ≤ M − 1, 1 ≤ k ≤ N,

uk0= �(tk), u

kM= �(tk), 1 ≤ k ≤ N,

u0i= u0(xi), 0 ≤ i ≤ M.

EUNj(Δt) = max

0≤i≤M||||U

Ni− (uj)

Ni

||||, and order(uj) = log2

EUNj(Δt)

EUNj(Δt∕2)

,

Table 4 (Example 4) Numerical results of finite difference schemes with Δx = 1∕20 000

� Δt EUN1(Δt) order (u1) EUN

2(Δt) order (u2) EUN

3(Δt) order (u3)

≃ 2 − � ≃ 3 − � ≃ 4 − �

0.9 1/10 9.883 3E−2 0.00 1.860 0E−2 0.00 2.935 5E−3 0.001/20 4.831 6E−2 1.03 4.722 8E−3 1.98 3.666 5E−4 3.001/40 2.310 5E−2 1.06 1.148 8E−3 2.04 4.418 8E−5 3.051/80 1.092 0E−2 1.08 2.736 7E−4 2.07 5.921 8E−6 3.081/160 5.129 9E−3 1.09 6.451 5E−5 2.08 6.190 0E−7 3.08




1 3

In Table 4, the numerical results using difference schemes (15), (14), and (13) for T = 1 are computed. Due to verifying the temporal numerical accuracy, a fixed and sufficient small spatial step size Δx (Δx = 1∕20 000) is chosen. Noticing Table 4, we find that the computa-tional errors of the difference scheme (13) are smaller, and its numerical accuracy is superior to the results of 3 − � order of accuracy of (14) and 2 − � order of accuracy of (15) in time. Moreover, for T = 1 and � = 0.1 with Δx = 1∕1 000 and different temporal step sizes, the cell numbers N and the CPU time (on the same machine equipped with Intel 2.4-GHz Intel Core processor, Matlab software) are recorded when the absolute errors of three schemes are on

80 90 100 110 120 130 140 150 1601.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2x 10−8

N

The

abso

lute

err

ors

L1−2−3

80 90 100 110 120 130 140 150 1600.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

N

CP

U ti

me/

s

L1−2−3

410 420 430 440 450 460 470 480 4901.82

1.84

1.86

1.88

1.9

1.92

1.94

1.96

1.98

2x 10−8

N

The

abso

lute

err

ors

L1−2

410 420 430 440 450 460 470 480 4902

2.5

3

3.5

N

CP

U ti

me/

s

L1−2

6 000 6 100 6 200 6 300 6 400 6 500 6 600 6 700 6 8002.4

2.45

2.5

2.55

2.6

2.65x 10−8

N

The

abso

lute

err

ors

L1

6 000 6 100 6 200 6 300 6 400 6 500 6 600 6 700 6 800340

360

380

400

420

440

460

N

CP

U ti

me/

s

L1

Fig. 2 The absolute errors and CPU time versus cell number N with � = 0.1,Δx = 1∕1 000


1 3

the same orders. According to Fig. 2 showing the relationship between N, absolute errors, and CPU time for three schemes, the cell number N would be much smaller for the scheme (13) than for the schemes (14) and (15). It is clearly obvious that the larger temporal step size can be used for the scheme (13) than for the other two schemes. In addition, Fig. 2 shows that for the same absolute errors, the CPU time is much shorter by using the scheme (13) than that by using the schemes (14) and (15). To reach absolute error less than 5E−8 when T = 1, � = 0.1 , and Δx = 1∕1 000 , we must compute u(80)

3, u

(410)

2 , and u(6 800)

1 which are, respectively, the solu-

tion of scheme (13), (14), and (15). For obtaining u(80)3

, we need three arrays with dimension 80 × 1 , 79 × 1 , and 78 × 1, respectively, to compute a(�)

j, b

(�)

j , and �(�)

j ; for obtaining u(410)

2 , we

need two arrays with dimension 410 × 1 and 409 × 1, respectively, to compute a(�)j

and b(�)j

; and for obtaining u(6 800)

1 , we need one array with dimension 6 800 × 1 to compute a(�)

j . There-

fore, we can draw a conclusion that less memory for computing u(80)3

(obtained by L1-2-3) is required.

Example 5 In (9) and (10), take T = 1, f (x, t) = ex[2t1−�E1,2−�(2t) − e2t], �(t) = e2t,

�(t) = e1+2t , and u0(x) = ex . The exact solution is u(x, t) = ex+2t.

The computational results listed in Table 5 state the advantages of the difference scheme (13) over (14) and (15). Noticing this table, we find that the computational efficiency and numerical accuracy of the difference scheme (13) are better; moreover, its convergence order of (13) is superior.

To investigate the numerical stability of scheme (13), we only follow up some tests and report corresponding results in Tables 6 and 7. Results imply that even by setting large time and space step sizes, satisfactory results may be produced and there is no track of the instabil-ity. So it seems that the method is unconditionaly stable, but proving this fact needs a rigorous and strict theoretical analysis which we leave it to future works.

Table 5 (Example 5) Numerical results of finite difference schemes with Δx = 1∕20 000

� Δt EUN1(Δt) order (u1) EUN

2(Δt) order (u2) EUN

3(Δt) order (u3)

≃ 2 − � ≃ 3 − � ≃ 3





1 3

3.2 PDE‑Based Image Denoising

Image noise removal constitutes a very important process; the most common noise results from the image acquisition system can be modeled as Gaussian random noise in most cases. Gaussian noise represents statistical noise having the probability density function equal to that of the normal distribution. Numerous image denoising models have been introduced in the past few decades. Although the conventional filters such as averaging fil-ter, median filter, and the 2D Gaussian filter are efficient in smoothing the noise, they have the disadvantage of blurring image edges [13, 15]. PDE-based methods for image process-ing (denoising, restorations, inpainting, segmentation, etc.) have been largely studied in the literature [1] due to their remarkable advantages in both theory and computation. To pre-serve the image structures when removing the noise, Perona and Malik (PM) [22] proposed a nonlinear equation which replaced isotropic diffusion expressed through a linear heat equation with an anisotropic diffusion. While the backward diffusion of the PM equation results in enhancing the edges, it is an ill-posed process in the sense that it is very sensi-tive to perturbations in the initial noisy data. Since the work of Perona and Malik, a large number of nonlinear PDE-based anisotropic diffusion models have been proposed [14, 17].

In this section, to confirm the ability of the new L1-2-3 formula to solve fractional dif-ferential equations in two dimension and illustrate the applications of it in image denoising, the fractional-order PDE has been applied to the image processing and computer vision. We proposed a fractional-order equation for image denoising as

(16)C0�

tu(�, t) = Δu(�, t), (�, t) ∈ Ω × [0, T],

Table 6 Numerical stability analysis of finite difference scheme (13) (Part I)

� Δx Δt Example 4 Example 5

EUN1(Δt) EU

N2(Δt) EUN

3(Δt) EUN

1(Δt) EUN

2(Δt) EUN

3(Δt)

0.9 1/10 1/10 9.77E−02 1.85E−02 3.02E−03 1.70E−01 2.12E−02 3.87E−031/20 4.78E−02 4.78E−03 4.81E−04 8.24E−02 6.14E−03 1.45E−031/40 2.29E−02 1.25E−03 1.63E−04 3.96E−02 2.30E−03 1.12E−031/80 1.09E−02 3.89E−04 1.24E−04 1.92E−02 1.37E−03 1.08E−031/160 5.18E−03 1.83E−04 1.20E−04 9.59E−03 1.14E−03 1.08E−03

1/100 1/10 9.88E−02 1.86E−02 2.93E−03 1.71E−01 2.03E−02 2.83E−031/20 4.83E−02 4.72E−03 3.67E−04 8.23E−02 5.12E−03 3.86E−041/40 2.31E−02 1.15E−03 4.53E−05 3.90E−02 1.25E−03 5.87E−051/80 1.09E−02 2.74E−04 6.44E−06 1.83E−02 3.06E−04 1.69E−051/160 5.13E−03 6.57E−05 1.82E−06 8.62E−03 8.04E−05 1.17E−05

1/1 000 1/10 9.88E−02 1.86E−02 2.93E−03 1.71E−01 2.03E−02 2.82E−031/20 4.83E−02 4.72E−03 3.66E−04 8.23E−02 5.11E−03 3.75E−041/40 2.31E−02 1.14E−03 4.42E−05 3.90E−02 1.24E−03 4.79E−051/80 1.09E−02 2.73E−04 5.24E−06 1.83E−02 2.95E−04 6.18E−061/160 5.12E−03 6.45E−05 6.27E−07 8.61E−03 6.96E−05 9.05E−07


1 3

where Δ denotes the Laplacian operator, Ω ⊂ ℝ2 represents the image domain, and

C0�

t stands for the Caputo fractional time derivative of order � . Assume that an

image has (Mx + 2) × (My + 2) pixels, and uki,j≃ u(i, j, tk) for i = 0, 1,… ,Mx + 1 and

j = 0, 1,… ,My + 1 . The following explicit finite difference scheme by using L1-2-3 for-mula (6) to approximate the time-fractional derivative and the second-order central differ-ence quotient has been considered for (16):

(17)

�(�)

0uki,j= �

(�)

k−1u0i,j+

k−1∑r=1

(�(�)

r−1− �

(�)

r)uk−r

i,j+ � (uk−1

i−1,j+ uk−1

i+1,j+ uk−1

i,j−1+ uk−1

i,j+1− 4uk−1

i,j),

Table 7 Numerical stability analysis of finite difference scheme (13) (Part II)

� Δx Δt Example 4 Example 5

EUN1(Δt) EUN

2(Δt) EUN

3(Δt) EUN

1(Δt) EUN

2(Δt) EUN

3(Δt)








1 3

to treat the spatial derivative term. Here � = (Δt)�Γ(2 − �) , u0i,j

is an image affected by a noise, and the boundary conditions are zero. Similarly, the following explicit finite differ-ence scheme by using L1-2 formula to approximate the time-fractional derivative term has been considered for (16)

with c(�)r

introduced in [12]. To assess the performance levels of each image denoising tech-nique, performance evaluation of the proposed methods are measured by the peak signal to noise ratio (PSNR)

where U is an initial image and u is a restored version. Due to the purpose of the paper, developing a fractional numerical differentiation formula called the L1-2-3 formula to approximate the Caputo fractional derivative of order � , the denoising performance of our method has not been compared with performances of other noise removal techniques. The images shown in Fig. 3a represents the original [512 × 512] image of Baboon; (b) repre-sents the degraded image by Gaussian noise characterized by parameters 0.1 (mean) and 0.01 (variance); (c, e) represent the image denoising result produced by the L1-2 model (18) with � = 0.4,N = 25 and � = 0.5,N = 8 ; (d, f) represent the image denoising result produced by the L1-2-3 model (17) with � = 0.4,N = 25 and � = 0.5,N = 8 . Another two image denoising method comparison examples are displayed in Figs. 4 and 5. The same noise removal approaches are applied on the standard Lena and Peppers images cor-rupted by the same amount of Gaussian noise, their results being represented in Figs. 4c–f and 5c–f. According to Table 8, the values of PSNR for noise reduction approaches cor-responding to Figs. 3−5, the L1-2-3 model (17) can smooth the noise better than the L1-2 model (18).

4 Conclusion

In this work, a new fractional numerical differentiation formula called the L1-2-3 for-mula, obtained by constructing a third degree Lagrange interpolation function for the integrand f, is established to approximate the Caputo fractional derivative of order � . After providing some analysis for the coefficient features and truncation errors of the L1-2-3 formula and considering two test examples which are conducted to effectively confirm its computational validity and numerical accuracy, we apply it in solving some time-fractional sub-diffusion equations on a bounded spatial domain. To illustrate the efficiency and advantages of the proposed scheme, we use it for smoothing the noise of some digital images. Some numerical tests imply that the scheme is unconditional stable. Providing a rigorous theoretical stability analysis of the proposed scheme can be consid-ered as a future work.

(18)

c(�)

0uki,j= c

(�)

k−1u0i,j+

k−1∑r=1

(c(�)

r−1− c(�)

r)uk−r

i,j+ � (uk−1

i−1,j+ uk−1

i+1,j+ uk−1

i,j−1+ uk−1

i,j+1− 4uk−1

i,j),

PSNR = 10 log10(2552∕MSE), MSE =

Mx+1∑i=0

My+1∑j=0

(u(i, j) − U(i, j))2

(Mx + 1) (My + 1),


1 3

(a) Original Baboon image (b) Image affected by Gaussian noise

(c) Denoising result of with α = 0.4 (d) Denoising result of (17) with α = 0.4

(e) Denoising result of (18) with α = 0.5 (f) Denoising result of (17) with α = 0.5

(18)

Fig. 3 Gaussian noise removal results produced by the L1-2 model (18) and the L1-2-3 model (17)


1 3

(a) Original Lena image (b) Image affected by Gaussian noise

(c) Denoising result of (18) with α = 0.4 (d) Denoising result of (17) with α = 0.4




1 3

(a) Original Peppers image (b) Image affected by Gaussian noise

(c) Denoising result of (18) with α = 0.4 (d) Denoising result of (17) with α = 0.4




1 3

In summary, in comparison with the previous methods such as the L1-2 and L1 formu-lae, the convergence order of the L1-2-3 formula is higher and its computational efficiency and numerical accuracy are superior. Moreover, this new formula can be used for both numerically solving other various fractional differential equations that involve the Caputo time-fractional derivatives and smoothing the noise. Experiments show improved perfor-mances in these areas.

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Table 8 Values of PSNR for noise reduction approaches

� = 0.4 � = 0.5

L1-2 L1-2-3 L1-2 L1-2-3

Baboon 35.319 6 36.567 0 30.842 5 32.915 4Lena 38.556 1 41.438 4 31.220 1 37.411 2Peppers 38.672 0 41.335 0 31.674 3 35.931 7


1 3

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ahorformula tapproximatcaputfractional dative...tive and provide some examples. moreover, this new...

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