Nonstandard finite difference method for solving complex-order fractional Burgers’ equations

Journal of Advanced Research 25 (2020) 19–29

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Journal of Advanced Research
journal homepage: www.elsevier.com/locate/jare

Nonstandard finite difference method for solving complex-order
fractional Burgers’ equations
N.H. Sweilam a,⇑, S.M. AL-Mekhlafi b, D. Baleanu c,d
a

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
Department of Mathematics, Faculty of Education, Sana’a University, Yemen
c
Department of Mathematics, Cankaya University, Turkey
d
Institute of Space Sciences, Magurele-Bucharest, Romania
b

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history:
Received 25 February 2020
Revised 8 April 2020
Accepted 15 April 2020
Available online 15 May 2020

Keywords:
Burgers’ equations
Complex order fractional derivative
Nonstandard weighted average finite
difference method
Stability analysis

a b s t r a c t
The aim of this work is to present numerical treatments to a complex order fractional nonlinear onedimensional problem of Burgers’ equations. A new parameter rt is presented in order to be consistent
with the physical model problem. This parameter characterizes the existence of fractional structures in
the equations. A relation between the parameter rt and the time derivative complex order is derived.
An unconditionally stable numerical scheme using a kind of weighted average nonstandard finitedifference discretization is presented. Stability analysis of this method is studied. Numerical simulations
are given to confirm the reliability of the proposed method.
Ó 2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University. This is an open access article
under the CC BY-NC-ND license ( />
Introduction
It is known that the complex order fractional derivative is a generalization of fractional order derivative and the integer order
derivative when the imaginary part of complex order is equal to

q

Peer review under responsibility of Cairo University.

⇑ Corresponding author.

E-mail addresses: (N.H. Sweilam),
(S.M. AL-Mekhlafi), (D. Baleanu).

zero [1]. In recent years, mathematical systems could be depicted
suitability and more accurately by employing the fractional order

derivative. There are several definitions for derivatives of fractional
order. The most common is Caputo its have several applications
[3]. More recently, Atangana-Baleanu Caputo sense (ABC) defined
a modified Caputo fractional derivative by introducing generalized
Mittag–Leffler function as the nonlocal and non-singular kernel
[18]. These new type of derivatives have been used in modeling
of real life applications in different fields ([4–7]). In order to a
better understanding of some mistakes and limitations of the

/>2090-1232/Ó 2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University.
This is an open access article under the CC BY-NC-ND license ( />

20

N.H. Sweilam et al. / Journal of Advanced Research 25 (2020) 19–29

fractional classical mathematical models can be seen in the comment of Baleanu in [2]. Recently, in [20] Fernandez proposed the
complex analysis approach to Atangana-Baleanu fractional calculus. The integer-order derivatives cannot describe systems with
the effects of history memory and hereditary properties of materials and processes as fractional order derivatives and complex order
fractional derivative [8–10]. In [10], Pinto and Carvalho presented a
new mathematical model for complex order fractional model for
HIV infection with drug resistance. They concluded that, the complex order fractional system has many advantages such as its
dynamics are rich, moreover, the changes of the complex order
derivative value can sheds a new light on the modeling of the intracellular delay. Also, in [22] the complex-order approximation to
the forced van der Pol oscillator is proposed.
Burgers’ equations can describe the communication between
acoustic waves, reaction apparatuses, convection effects, heat conduction, diffusion transports, and modeling of dynamics, for more
details see [11–14,16,17]. Several authors have investigated studied Burgers’ model for various physical flow problem in fluid
dynamics. The structure of Burgers’ equation is roughly similar to
that of Navier–Stokes equations due to the presence of the nonlinear convection term and the occurrence of the diffusion term

with viscosity coefficient. So this equation can be considered as a
simplified form of the Navier–Stokes equations. The one dimensional coupled Burgers’ equation can be taken as a simple model
of sedimentation and evolution of scaled volume of two kinds of
particles in fluid, suspensions and colloids under the effect of gravity [15].
In this work, we present applications for the new definition of
complex fractional order which given in [20], these applications
are Burgers’ equation with proportional delays in onedimensional (1-D) and the coupled Burgers’ equations in 1-D. In
order to characterize the existence of complex fractional
structure in the model, a parameter rt is added to the model problem [2]. A relation between rt and the complex order derivative
ðl þ kiÞ is derived. Moreover, a numerical scheme is constructed
using weighted average nonstandard finite-difference method
(WANFDM) ([24–27]) to solve numerically the proposed equations.
To our knowledge the nonstandard finite difference method for
solving complex-order fractional Burgers’ equations was never
explored before.
This paper is organized as follows: In Section 2, we explain
some of the required mathematical concepts and preliminaries of
complex fractional order derivatives. In Section 3, two complex
order fractional Burgers’ equations models are introduced and
the construction of WANFDM to solve these equations. Moreover,
the stability of this scheme is studied in Section 4. Numerical simulations for the proposed equations are given in Section 5. Finally,
the conclusions are given in Section 6.
Preliminaries and notations

Dtlþki yðtÞ ¼ f ðt; yðtÞÞ; 0 < t 6 T; ðl þ kiÞ 2 C;

ð1Þ

yð0Þ ¼ y0 :
The Atangana-Baleanu fractional order derivative in Caputo

sense (ABC) given is defined as follows [18]:
ABC

MðlÞ
Dt yðtÞ ¼
ð1 À lÞ
l

Z

t
0



ðt À qÞl
_
yðqÞdq;
El Àl
ð1 À lÞ

ABC

DtðlþkiÞ yðtÞ ¼

where,

Mðl þ kiÞ
2pið1 À ðl þ kiÞÞ
!

Z t
ðt À qÞðlþkiÞ
_
EðlþkiÞ Àðl þ kiÞ
Â
yðqÞdq;
ð1 À ðl þ kiÞÞ
0

Mðl þ kiÞ ¼ 1 À ðl þ kiÞ þ CððllþkiÞ
,
þkiÞ

ð3Þ

Reðl þ kiÞ > 0

and

Cðl þ kiÞ is the Stirling asymptotic formula of gamma function
[21].
Numerical discretization for the ABC complex order derivatives
In this section we aim to construct WANFDM with ABC complex
order fractional derivative to obtain the discretization of complex
order fractional derivative numerically. Using (3) let a ¼ ðl þ kiÞ
2C. Then the discretization of complex order fractional derivative
is given numerically as follows:

Z



Àaðt À sÞa duðsÞ
ds;
ds
1Àa

Dat u ¼

MðaÞ
2pið1 À aÞ

ABC

Dat u ¼


 jþ1Àp
jÀ1 Z
À ujÀp
MðaÞÞ X tpþ1
Àaðt À sÞa ui
i
Ea
ds;
2pið1 À aÞ p¼0 tp
1Àa
uð4tÞ

ABC

Dat u ¼


Z tpþ1 
jÀ1 jþ1Àp
À ujÀp
MðaÞ X ui
Àaðt jþ1 À sÞa
i
ds;
Ea
2pið1 À aÞ p¼0
uð4tÞ
1Àa
tp

ABC

Dat u ¼ H

ABC

tj

Ea
0

jÀ1 jþ1Àp
X

u
À ujÀp
i

p¼0

uð4tÞ

i

ð4Þ

Hp;j ;

ð5Þ

where

H¼

MðaÞ
;
2pið1 À aÞ

Hp;j ¼

R tpþ1
tp

Ea



Àaðt jþ1 ÀsÞa
1Àa

¼ ðt jþ1 À t pþ1 ÞEa




ds

Àaðt jþ1 Àt pþ1 Þa
1Àa



À ðt jþ1 À tp ÞEa



Àaðt jþ1 Àt p Þa
1Àa



:

Complex order fractional Burgers’ equations

In the following, two nonlinear complex order fractional Burgers’ models in 1-D are presented as follows:
1-D Burgers’ equation
Consider the Burgers’ equation in 1-D as follows ([12,23]):

Let us consider the complex order fractional differentiation
equation as follows:
ABC

The Atangana-Baleanu complex order fractional derivative in
Caputo sense is defined as follows [20]:

ð2Þ

where, 0 < l < 1, MðlÞ ¼ 1 À l þ CðllÞ is normalization function,
P
Zn
Z 2 C.
El is Mittag–Leffler function, where, El ðZÞ ¼ 1
n¼0 Cðlnþ1Þ,

ut ðt; xÞ þ k1 uðt; xÞux ðt; xÞ þ l1 uðt; xÞ À quxx ðt; xÞ ¼ 0;

ð6Þ

with the initial and the boundary conditions given as follows:

uðt 0 ; xÞ ¼ gðxÞ; L0 6 x 6 L;
uðt; L0 Þ ¼ uðL; tÞ ¼ f ðtÞ;

t > 0;

where, k1 ; q > 0 and l1 are constants, ut ðt; xÞ is the variation term,
uðt; xÞ is the velocity component, q is diffusion coefficient,
uðt; xÞux ðt; xÞ is the nonlinear convective term and uxx is the diffusion
term, gðxÞ and f ðtÞ are known functions. t 0 is the initial time.
In the following, the ordinary time derivative will be replaced
by the complex order derivative.


21

N.H. Sweilam et al. / Journal of Advanced Research 25 (2020) 19–29

lþki

d
d
! lþki :
dt
dt

ð7Þ

It can be seen that (7) is not quite right, from a physical point of
view, because the time derivative operator dtd has dimension of
inverse time T À1 , while the fractional complex time derivative operdlþki
dtlþki

ator

1

has, T

d

lþki

lþkiÞ
lþki
r1Àð
dt
t

ÀðlþkiÞ

. Now we introduce

rt in the following way:

¼ ðTÞÀ1 :

Construction of WANFDM
In the following, we aim to construct WANFDM in order to
obtain the discretization of the model problems.
1-D complex fractional order Burgers’ equation
The discretization of 1-D complex fractional order Burgers’
Eq. (6) and the nonstandard finite differences approximation can
be claimed as follows:

ð8Þ
1

1Àa
t

r

H

p¼0

In the case the expression (7) becomes an ordinary derivative operator dtd in case l ¼ 1; k ¼ 0. In this way (7) is dimensionally consistent if and only if the new parameter rt , has dimension of time
½rt Š ¼ T. Put

ABC

DðtlþkiÞ ¼

dlþki
,
dt ðlþkiÞ

jÀ1
X

Now, we can write a fractional com-

plex differential equation corresponding to the fractional complex
order Burgers’ equation in the following way:

ABC

1
lþkiÞ
r1Àð
t

DtðlþkiÞ uðt; xÞ þ k1 uðt; xÞux ðx; tÞ þ l1 uðt; xÞ À quxx ðt; xÞ ¼ 0;
ð9Þ

ABC

1
lþkiÞ
r1Àð
t

a
r1À
t

H

jÀ1
X

Dt uðt; xÞ þ l1 uðt; xÞ ¼ 0:

lþkiÞ
r1Àð

t

ð11Þ

By using the same steps in [28], the numerical solution of (11) when

l ¼ 1; k ¼ 0, i.e., a ¼ 1 is given as follows:
À1

l1 t

U ¼ U0 e :

ð12Þ

In this case the relation between a and

a¼



jþ1
jþ1
jþ1
jþ1
jþ1
uiþ1 Àui
uiþ1 À2ui þuiÀ1
jþ1
þð1 À hÞ k1 ujþ1

þ
l
u
À
q
2
1
i
i
wð4xÞ
wð4xÞ

1-D complex fractional order coupled Burgers’ equation
The discretization form of 1-D complex fractional order coupled
Burgers’ Eqs. (13) given as follows:
1

H

a
r1À
t



jÀ1 jþ1Àp jÀp
X
u
Àu
i

uð4tÞ

Hp;j þ ð1 À hÞ ðk1 ujþ1
þ b1 v jþ1
i
i Þ

i

p¼0

jþ1

jþ1

uiþ1 Àui

wð4xÞ



h
þ h ðk1 ujÀ1
þ b1 v jÀ1
i
i Þ

jÀ1
jÀ1

ujÀ1
ÀujÀ1
v iþ1
Àv ijÀ1
ujÀ1 À2ujÀ1
þuiÀ1
iþ1
i
i
¼ Rj1;i ;
þ b1 ujÀ1
À q iþ1 wð4xÞ
2
i
wð4xÞ
wð4xÞ

rt is given by [28]:

þb1 ujþ1
i

jþ1
jþ1
v iþ1
Àv i

wð4xÞ

Àq

jþ1

Consider the complex order coupled Burgers’ equations in 1-D
as follows:
1 ABC

@
Dat uðt; xÞ þ k1 uðt; xÞux ðt; xÞ þ b1 @x
ðuðt; xÞv ðt; xÞÞ ¼ quxx ðt; xÞ;

1 ABC

@
Dat v ðt; xÞ þ k2 v ðt; xÞv x ðt; xÞ þ b2 @x
ðuðt; xÞv ðt; xÞÞ ¼ qv xx ðt; xÞ;

1

1Àa
t

r

H

jÀ1
X

jÀp

v jþ1Àp
Àv i
i

uð4tÞ

jþ1

uiþ1 À2ui

jþ1

þuiÀ1

wð4xÞ2

where (j ¼ 0; 1; 2; . . . ; N;

1-D coupled Burgers’ equations

a
r1À
t

Hp;j

ð15Þ

rt
1

; 0 < rt 6
:
l1
l1

a
r1À
t

uijþ1Àp ÀuijÀp



jÀ1
uiþ1
ÀujÀ1
ujÀ1
À2ujÀ1
þujÀ1
jÀ1
i
iþ1
i
iÀ1
¼ 0:
þ
l
u
À
q

þh k1 ujþ1
1 i
i
wð4xÞ
wð4xÞ2

Using Eq. (10), the particular case can be obtained when q ¼ k1 ¼ 0,
a



jþ1
jþ1
uiþ1
Àuijþ1
ujþ1 À2uijþ1 þuiÀ1
þð1 À hÞ k1 ujþ1
þ l1 ujþ1
À q iþ1 wð4xÞ
2
i
i
wð4xÞ


jÀ1
jÀ1
jÀ1
jÀ1
jÀ1

uiþ1 Àui
uiþ1 À2ui þuiÀ1
jÀ1
¼ R:
þ
l
u
À
q
þh k1 ujþ1
2
1
i
i
wð4xÞ
wð4xÞ

uð4tÞ

p¼0

Dat uðt; xÞ þ k1 uðt; xÞux ðt; xÞ þ l1 uðt; xÞ À quxx ðt; xÞ ¼ 0:

ABC

Hp;j

Where (j ¼ 0; 1; 2; . . . ; N; i ¼ 0; 1; 2; . . . ; M) and R is the
truncation error. Neglecting the truncation error, the resulting
computable difference scheme takes the form:

ð10Þ

1

jÀp

Àui

uð4tÞ

ð14Þ

1

put a ¼ ðl þ kiÞ, then we can write (9) as follows:

jþ1Àp

ui

i ¼ 0; 1; 2; . . . ; M).



Hp;j þ ð1 À hÞ ðk2 v jþ1
þ b2 ujþ1
i
i Þ

p¼0

þb2 v jþ1
i
jÀ1
jÀ1
v iþ1
Àv i

wð4xÞ

jþ1

jþ1

uiþ1 Àui

wð4xÞ

Àq

þ b2 v jÀ1
i

jÀ1


jþ1

jþ1

v jþ1
À2v i þv iÀ1
iþ1

wð4xÞ2
jÀ1

uiþ1 Àui

Àq

wð4xÞ

wð4xÞ

h
þ h ðk2 v jÀ1
þ b2 ujÀ1
i
i Þ

jÀ1
jÀ1
jÀ1
v iþ1
À2v i þv iÀ1

a 2 C;

jþ1

v iþ1
Àv jþ1
i

wð4xÞ



2

¼ Rj2;i :
ð16Þ

ð13Þ

Where Rj1;i and Rj2;i are the truncation errors. Neglecting the trunca-

with the initial conditions:

uðt0 ; xÞ ¼ g 1 ðxÞ;

v ðt0 ; xÞ ¼ g2 ðxÞ;

tion errors, the resulting computable difference scheme takes the
form:

L0 6 x 6 L;

and the boundary conditions:

uðt; L0 Þ ¼ uðt; LÞ ¼ f 1 ðtÞ;

v ðt; L0 Þ ¼ v ðt; LÞ ¼ f 2 ðtÞ;

1

t > 0:

Where k1 ; k2 ; b1 and b2 are constants, uðt; xÞ and v ðt; xÞ are the velocity components, g 1 ðxÞ; g 2 ðxÞ,
f 1 ðt; xÞ and f 2 ðt; xÞ are known functions and t 0 is the initial time.
This coupled equation found in [15] when k ¼ 0.

H
r1Àa
t

jÀ1
X
p¼0

jþ1Àp

ui

jÀp

Àui
uð4tÞ



Hp;j þ ð1 À hÞ ðk1 ujþ1
þ b1 v jþ1
i
i Þ


jþ1

jþ1

uiþ1 Àui

wð4xÞ

h
þ h ðk1 ujÀ1
þ b1 v jÀ1
i
i Þ

jÀ1
jÀ1
jÀ1
jÀ1
jÀ1
jÀ1
jÀ1
uiþ1 Àui
v iþ1

Àv i
u À2ui þuiÀ1
¼ 0;
þ b1 ujÀ1
À q iþ1 wð4xÞ
2
i
wð4xÞ
wð4xÞ
þb1 ujþ1
i

v jþ1
Àv ijþ1
iþ1
wð4xÞ

Àq

jþ1
ujþ1
À2uijþ1 þuiÀ1
iþ1
2

wð4xÞ


22

N.H. Sweilam et al. / Journal of Advanced Research 25 (2020) 19–29

1

1Àa
t

r

H



jÀ1
jÀp
X
v jþ1Àp
Àv i
i

Hp;j þ ð1 À hÞ ðk2 v jþ1
þ b2 ujþ1
i
i Þ

uð4tÞ

p¼0

where,

jþ1
v iþ1
Àv ijþ1

wð4xÞ



h
þ b2 ujÀ1
þ
h
ðk2 v jÀ1
i
i Þ
wð4xÞ

jÀ1
jÀ1
jÀ1
jÀ1
jÀ1
jÀ1
jÀ1
v iþ1 Àv i
uiþ1 Àui
v À2v i þv iÀ1
¼ 0:
þ b2 v jÀ1

À q iþ1 wð4xÞ
2
i
wð4xÞ
wð4xÞ
þb2 v jþ1
i

jþ1
uiþ1
Àuijþ1

wð4xÞ

Àq

jþ1
v iþ1
À2v ijþ1 þv jþ1
iÀ1
2

ð17Þ
Stability analysis for the WANSFDM for solving Burgers’ models
Stability analysis for the WANSFDM for solving 1-D Burgers’ equation
In the following, we used the idea of Jon von Neumann technique to claim the stability of (15), ([25,26]). This idea will be
applied after linearizing (10). Assume that uji ¼ nj eicq4x , where
pffiffiffiffiffiffiffi
c ¼ À1, the requirement is jnðqÞj 6 1, then (15) will be written
as follows:

1

1Àa
t

r

H

jÀ1
X

njþ1Àp eicq4x ÀnjÀp eicq4x
uð4tÞ

p¼0

þð1 À hÞ

h

Hp;j

jÀ1
X
nÀp ðgÀ1Þ

uð4tÞ

p¼0

ð18Þ

h

i

i
2q
þgÀ1 h l1 À wð4xÞ
2 ðcosðq 4 xÞ À 1Þ ¼ 0:

jÀ1
jÀ1
X
X
ðnÀp dÞ=uð4tÞ ¼
ðgÀp dÞ=uð4tÞ ¼ Ao ;

a
r1À
t

HAo g À

a
r1À
t

HAo þ Bg þ C g

1

a
r1À
t



HAo þ B g2 À

¼ 0;

ð20Þ

h

1
a
r1À
t

HAo g þ C ¼ 0; jgj

i

and

i

"

Hp;j þ ð1 À hÞ Àq

jþ1
X jþ1
þ X jþ1
iþ1 À 2X i
iÀ1

uð4tÞ
p¼0


jÀ1
jÀ1
jÀ1
X iþ1 À2X i þX iÀ1
þh Àq
¼ 0;
2
wð4xÞ

wð4xÞ2

#

ð23Þ

X ji ¼ nj Ç eicq4x ;

pffiffiffiffiffiffiffi
where c ¼ À1; Ç 2 R2Â1 and n 2 R2Â2 is the amplification matrix.
By substituting into (23) and using the Euler formula, we have:

ð24Þ

rt

where,
I

is the unit matrix,
PjÀ1 Àp
B1 ¼ p¼0 ðn dÞ=uð4tÞ, and.

2q
A1 ¼ ð1 À hÞ wð4xÞ
2 ðcosðq 4 xÞ À 1Þ,



rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


1


2
1

1

À 1Àa HB1 þ ð 1À


HB1 Þ À 4C 1 ðA1 À r1À
a HB1 Þ


rt
rt a
t




jn1 j ¼




1

2ðA1 À r1À
a HB1 Þ


1;

t

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi




2




1
1
1