Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics

Journal of Advanced Research 25 (2020) 205–216

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

Numerical evaluation of fractional Tricomi-type model arising from
physical problems of gas dynamics
O. Nikan a, J.A. Tenreiro Machado b, Z. Avazzadeh c,d,⇑, H. Jafari e
a

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
Department of Electrical Engineering, ISEP-Institute of Engineering, Polytechnic of Porto, Porto, Portugal
c
Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam
d
Faculty of Natural Sciences, Duy Tan University, Da Nang 550000, Vietnam
e
Department of Mathematical Sciences, University of South Africa, UNISA 0003, South Africa
b

h i g h l i g h t s

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

 The fractional Tricomi-type model is

adopted for describing the anomalous
process of nearly sonic speed gas
dynamics.

 A new hybrid scheme based LRBF-FD
method is formulated to solve the
model.
 The LRBF-FD method useful for
irregular domains with good accuracy
is proposed.
 The stability and convergence of the
proposed method are analyzed using
the energy method.

a r t i c l e

i n f o

Article history:
Received 15 April 2020
Revised 3 June 2020
Accepted 21 June 2020
Available online 23 June 2020
Keywords:
Caputo fractional derivative
Time fractional Tricomi-type model
LRBF-FD
Stability analysis

a b s t r a c t
This paper deals with approximating the time fractional Tricomi-type model in the sense of the Caputo
derivative. The model is often adopted for describing the anomalous process of nearly sonic speed gas
dynamics. The temporal semi-discretization is computed via a finite difference algorithm, while the spatial discretization is obtained using the local radial basis function in a finite difference mode. The local
collocation method approximates the differential operators using a weighted sum of the function values

over a local collection of nodes (named stencil) through a radial basis function expansion. This technique
considers merely the discretization nodes of each subdomain around the collocation node. This leads to
sparse systems and tackles the ill-conditioning produced of global collocation. The theoretical convergence and stability analyses of the proposed time semi-discrete scheme are proved by means of the discrete energy method. Numerical results confirm the accuracy and efficiency of the new approach.
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
This paper proposes an efficient numerical formulation for solving the time fractional Tricomi-type model (TFTTM), that can be
written as
⇑ Corresponding author.
E-mail addresses: (O. Nikan), (J.A.T.
Machado), (Z. Avazzadeh), jafari.usern@gmail.
com (H. Jafari).

@ a uðx; t Þ
À t 2c Duðx; t Þ ¼ f ðx; t Þ;
@ta

/>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 ( />
x ¼ ðx; yÞ 2 X & R2 ;

0 < t 6 T;
ð1Þ


206

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

along with the initial and Dirichlet boundary conditions given by

uðx; 0Þ ¼ g ðxÞ;
@uðx; 0Þ
¼ wðxÞ;
@t
uðx; tÞ ¼ hðx; t Þ;

 ¼ X [ @ X;
x2X
 ¼ X [ @ X;
x2X
x 2 @ X;

t > 0;

ð2Þ
ð3Þ
ð4Þ

where T is the final time, D denotes the Laplace operator with
respect to space variables x; c is real non-negative number, X is a
bounded domain in R2 , and @ X represents the boundary of X. The
fractional derivative @ a uðx; t Þ=@t a of order 1 < a < 2 is defined in
the Caputo sense as

@ a uðx; t Þ
1
¼
@t a
Cð2 À aÞ

Z
0

t

@ 2 uðx; sÞ
ðt À sÞ1Àa ds;
@s2

where CðÁÞ represents the Euler’s gamma function [1,2].
During the 20s, Tricomi [3] started the work on the linear partial differential equations of variable type with boundary condition. Later, Frankl [4] showed that the gas flows with nearly
sonic speeds could be described by the Tricomi model. For the
numerical solution of the TFTTM, we find some works published
during the last years. Zhang et al. [5] formulated a local discontinuous Galerkin finite element, Zhang et al. [6] used the finite element scheme and Liu et al. [7] applied the reduced-order finite
element technique to approximate the TFTTM. More recently, Dehghan and Abbaszadeh [8] adopted the element-free Galerkin technique and Ghehsareh et al. [9] implemented the local Petrov–
Galerkin formulation.
Numerical techniques are extensively applied to approximate
partial differential equation (PDE) and we can mention the finite
element, finite difference, finite volume, and pseudo-spectral
methods. However, usually these techniques are defined on data
point meshes meaning that a grid generation is often required,
which in turn increases the computation time. Moreover, these
schemes have insufficient accuracy over irregular and nonsmooth domains because they provide the problem solution only
on mesh points. As a result, meshless techniques have been developed to overcome these problems. One important meshless technique is the radial basis function (RBF) method. The RBF is a very
efficient instrument for interpolating a scattered set of points
and, due to these characteristics, has received attention during last
years [10–13]. Indeed, the RBF approximation is a powerful tool
that is particularly relevant for high-dimensional problems. Rolland Hardy [14] proposed the RBF technique in 1971 by introducing the multiquadric (MQ) algorithm as a meshless interpolation
method using the MQ radial function. Richard Franke [15] popularized this approach in 1982 with a review of the 32 most used interpolation techniques. Franke performed a set of comprehensive

tests and concluded that the MQ method had the best overall performance. Furthermore, he advanced that the interpolation matrix
related to the MQ radial function has unconditional nonsingularity. Later, in 1986, Micchelli [16] proved this using
research from the 30s and 40s by Schoenberg. Kansa [17,18] considered the MQ method for approximating elliptic and parabolic
PDE. Nonetheless, the well conditioned of the RBF interpolation
matrix and good accuracy are not verified simultaneously. This is
known as the Uncertainty Principle following the work of Schaback
[19]. Fornberg and Larsson [20] implemented this technique to
elliptic PDE. The existence, uniqueness, and convergence of the
RBF approximation were discussed in detail in several works
[21,16,22].
Hereafter, this paper shows that the local RBF is an efficient
computational technique to numerically approximate the TFTTM
with high accuracy and low computational complexity. Following
these ideas, this paper is arranged as follows. Section 2 formulates
the temporal discretization via finite difference and discusses its

convergence and error analysis. Section 3 applies the local RBFfinite difference (LRBF-FD) for space discretization. Section 4 illustrates the method with three numerical examples that show its
efficiency and verify the theoretical analysis. Finally, Section 5 concludes with a summary of the key conclusions.
Temporal discretization
To apply the numerical scheme for the solution of Eq. (1), let
dt ¼ T=M; tk ¼ kdt; k ¼ 1; . . . ; M, for a positive integer M. Therefore,
the time domain ½0; T Š is covered by temporal discretization points
tk . The following lemmas will be used in the derivation of the time
difference scheme [23].
Lemma 1. ([23].) If 1 < a < 2 and g ðtÞ 2 C 2 ½0; T Š, then it follows

Z

tn

g 0 ðsÞðtn À sÞ1Àa ds ¼

0

where

 n
R  6




1
1
dt 3Àa max jg 00 ðt Þj:
þ
06t6tn
2ð2 À aÞ 2

Lemma
bk ¼

Z
n
X
g ðt k Þ À g ðt kÀ1 Þ tk
ðt n À sÞ1Àa ds þ Rn ;
dt
tkÀ1
k¼1

dt 2Àa
2Àa

2. ([23].) Suppose that 1 < a < 2 a0 ¼ dtCð12ÀaÞ
h
i
2Àa
2Àa
; k ¼ 0; 1; 2; . . .. Then it follows
ðk þ 1Þ
À ðkÞ

and


 1 R tn 0
1Àa

Cð2ÀaÞ 0 g ðsÞðt n À sÞ ds À a0

!
nÀ1

À Á
P
 b0 g ðt n Þ À
ðbnÀkÀ1 À bnÀk Þg t j À bnÀ1 g ð0Þ 
k¼1



3Àa
1
1
1
6 Cð2À
max jg 00 ðt Þj:
aÞ 2ð2ÀaÞ þ 2 dt
06t6t n

h
i
2Àa
2Àa
2Àa
;
Lemma 3. ([23].) If 1 < a < 2 and bk ¼ dt2Àa ðk þ 1Þ
À ðkÞ
k ¼ 0; 1; 2; . . . ; then it follows

b0 > b1 > b2 > . . . > bk ! 0; as k ! 1:
We introduce the following notation:

À Á2c

uk À ukÀ1
;
dt

1

dt ukÀ2 ¼

lk ¼ tk

;

f

kÀ12

k

¼

f þf
2

kÀ1

Á

ð5Þ

Let us consider

v ðx; tÞ ¼

@uðx; t Þ
;

@t

Vðx; tk Þ ¼

1
Cð2 À aÞ

ð6Þ
Z

t
0

@ v ðx; sÞ
ðt À sÞ1Àa ds:
@s

ð7Þ

From (6) it results that Taylor expansion at t ¼ t kÀ1 can be written
2

as:

v kÀ

1
2

kÀ1

1

¼ dt ukÀ2 þ R1 2 ;

ð8Þ

where

 1
 kÀ2 
R1  6 C 1 dt2 :

ð9Þ

Based on Lemma 2, we have

"
V ¼ a0 b0 v
k

n

#
kÀ1
X
À
Á j
0
À

bkÀjÀ1 À bkÀj v À bkÀ1 v þ Rk2
j¼1


207

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

and also

"

VkÀ2 ¼ a0 b0 v kÀ2 À
1

1

kÀ1
X
À

bkÀjÀ1 À bkÀj

Á

where L2 ðXÞ represents the space of measurable functions whose
square is Lebesgue integrable in X and a ¼ ða1 ; . . . ; ad Þ denotes a
P
d-tuple of non-negative integer with jaj ¼ di¼1 ai . Let us consider

#

v jÀ

1
2

À bkÀ1 v 0 þ R2 2 ;
kÀ1

j¼1

Da v ¼

where

 1
 kÀ2 
R2  6 C 2 dt 3Àa :



v kÀ ; q
1
2



"

¼ a0 b0 v kÀ2 À
1

kÀ1
X
À

bkÀjÀ1 À bkÀj

Á

1
2

À bkÀ1 q :

1

!12

L ðXÞ

jaj6m

:

Now, let us examine the analysis of stability and the error estimates
for the difference algorithm.

 1 

Vk þ VkÀ1
kÀ1
¼ a0 P v kÀ2 w þ R2 2 ;
2

ð11Þ

where v 0 ðxÞ ¼ v ðx; 0Þ ¼ wðxÞ ¼ w. If we substitute (8) into (11), we
obtain

VkÀ2

d

X  a 2
D v  2

ð10Þ

Using Lemma 2, the expression (10) can be written as
1

2

jjv jjHm ðXÞ ¼

#

v jÀ

j¼1

VkÀ2 ¼

1

The norm kv km of the space Hm ðXÞ can be written as

We define the operator

P

@ jaj v

Á
@xa @xa . . . @xa



 kÀ1 
1
kÀ1
¼ a0 P dt ukÀ2 ; w þ a0 P R1 2 ; 0 þ R2 2 :

Corollary 1. (Poincaré inequality [24]). Suppose that 1 6 p 6 1
and that X is a bounded open set. Then, there exists a constant CX
(depending on X and p) such that

 k 



g  6 CX $gk :

ð12Þ
Lemma 4. ([23].) For any G ¼ fG1 ; G2 ; . . .g and q, we have

Substituting the above result (12) into (1) yields



1
1
kÀ1
kÀ1
a0 P dt ukÀ2 ; w ¼ DukÀ2 þ f 2 þ R2 2 ;

ð13Þ

M
M
a
X
À
Á
t 1Àa X
t 2À
M
P Gj ; q Gj P M dt G2j À
q2 :
2

2ð2 À aÞ
j¼1
j¼1

where

 kÀ1 
n
o
1
kÀ1
RkÀ2 ¼ À a0 P R1 2 ; 0 þ R2 2 :
Based on Lemma 2, 3 and inequalities (9) we can write

( "

kÀ12

R

kÀ1
a0 b0 R1 2

6

þ

bkÀjÀ1 À bkÀj

j¼1

( "
6

kÀ1
PÀ

a0 b0 C 1 dt2 þ

kÀ1
PÀ

Á

#

vj À

bkÀjÀ1 À bkÀj

j¼1

Á

kÀ1
bkÀ1 R1 2

þ

Lemma 5. ([25].) If xn is nonnegative sequence and the sequence

yn fulfills

8
>
< y0 6 d0 ;

)

kÀ1
R2 2

#

v j À bkÀ1 C 1 dt2

)
þ C 2 dt 3Àa

È Â
Ã
É
¼ a0 b0 C 1 dt 2 þ ðb0 À bkÀ1 ÞC 1 dt 2 þ C 2 dt3Àa
È Â
Ã
É
6 a0 2b0 C 1 dt 2 þ C 2 dt3Àa
n
h
i
o

2Àa
¼ dtCð12ÀaÞ 2 dt2Àa C 1 dt2 þ C 2 dt 3Àa


1
þ C 2 dt3Àa :
6 ð2Àa2C
ÞCð2ÀaÞ

u
by its numerical approximation U
semi-discrete recursive algorithm:

, leads to the following



kÀ1
1
1
kÀ1
a0 P dt U kÀ2 ; w ¼ l 2 DU kÀ2 þ f 2 ;

a0 b0 U À
þa0 dt

1
dt
2

j¼1

k

l DU ¼ a0 b0 U

kÀ1
PÀ

xk yk ;

k¼0

then yn satisfies

8
>
< y1 6 d0 ð1 þ x0 Þ þ y0 ;
nÀ2
nÀ2
nÀ1
Q
Q
P
>
ð 1 þ xk Þ þ
zk
ð1 þ xs Þ þ znÀ1 ;
: yn 6 d0 þ
k¼0

n P 2:

s¼kþ1

yn 6

!
!
nÀ1
nÀ1
X
X
d0 þ
zk exp
xk :
k¼0

k¼0

Making use of these lemmas, we can derive the following result
of stability.
Theorem 1. If U k 2 H10 ðXÞ, then the difference formula (14) is
unconditional stable with respect to the H1 -norm.

kÀ1

kÀ1

þ

l DU


Á
1
k
kÀ1
bjÀkÀ1 À bjÀk dt U jÀ2 þ a0 bkÀ1 dtw þ 12 dt f þ f
:
k

k¼0

nÀ1
P

ð14Þ

or, equivalently, we get
k

zk þ

Moreover, if d0 P 0 and zn P 0 for n P 0, then it holds

1

kÀ12

nÀ1

P

k¼0

Dropping the error term RkÀ2 and approximating the exact value
kÀ12

>
: yn 6 d0 þ

1
dt
2

kÀ1

Proof. The following variational weak formulation will be
obtained by multiplying both sides of Eq. (14) by m and integrating
over X

D

Theoretical analysis of the time discretization scheme
We star by defining some functional spaces that will be used in
the subsequent discussion. Let us define the functional space
endowed with the standard norms and inner products
n
o
H1 ðXÞ ¼ v 2 L2 ðXÞ; ddxv 2 L2 ðXÞ ;
n

o
H10 ðXÞ ¼ v 2 H1 ðXÞ; v j@X ¼ 0 ;
n
o
Hm ðXÞ ¼ v 2 L2 ðXÞ; Da v 2 L2 ðXÞ; for all positive integerjaj 6 m ;


 E
D
E
1
1
1
a0 P dt nkÀ2 ; w ; m ¼ lkÀ2 DnkÀ2 ; m ;

ð15Þ

$ 1
À
Á
1
1
where nkÀ2 ¼ U kÀ2 À U kÀ2 denotes the perturbation at the k À 12 th
$

1

1

time level, so that U kÀ2 and U kÀ2 are the exact and approximate solutions of Eq. (14), respectively.

Using the divergence theorem

Z
X

rv rx ¼

where

Z
@X

v

@x
À
@n

Z
X

v Dx ;


208

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

@x @x
@x

¼
n1 þ
n2
@n
@x
@y

Theorem 2. Let uk and U k be the solutions of (13) and (14), respec-

is the normal derivative, that is, representing the derivative in the
outward normal direction to the boundary @ X, we get

(
)
kÀ1
D
E X
E
À
ÁD
1
1
a0 b0 dt nkÀ2 ; m À
bkÀjÀ1 À bkÀj dt nkÀ2 ; m

Proof. Taking the inner product of Eqs. (13) and (14) with m on the
both sides, we obtain their corresponding variational weak form as
follows:

j¼1

D
E
1
1
¼ ÀlkÀ2 rnkÀ2 ; rm Á
Letting

(

ð16Þ

kÀ1
D
E X
E
À
ÁD
1
1
1
1
a0 b0 dt nkÀ2 ; dt nkÀ2 À
bkÀjÀ1 À bkÀj dt njÀ2 ; dt nkÀ2

)

j¼1

E

1

1

¼ ÀlkÀ2 rnkÀ2 ; rdt nkÀ2 Á
Summing on k; k ¼ 1; . . . ; M, and applying Cauchy–Schwarz inequality, we deduce that

(

)


M
kÀ1
X
Á jÀ1  kÀ1 
 kÀ12 2 X À
2
2
a0
b0 dt n  À
bkÀjÀ1 À bkÀj dt n dt n 


 E
D
D
E D 1 E
1
1

1
kÀ
a0 P dt U kÀ2 ; w ; m ¼ lkÀ2 DU kÀ2 ; m þ f 2 ; m ;

M
X

l

kÀ12

2dt

k¼1

ð17Þ

1

ð18Þ

1

1

where fkÀ2 ¼ ukÀ2 À U kÀ2 . Subtracting Eq. (17) from Eq. (18) and
using the divergence theorem again, we arrive at
(
)
kÀ1

D
E X
E
D
E D 1 E
À
ÁD
1
1
1
1
a0 b0 dt fkÀ2 ; m À
bkÀjÀ1 À bkÀj dt fkÀ2 ; m
¼ ÀlkÀ2 rfkÀ2 ; rm þ RkÀ2 ; m Á ð19Þ
j¼1

Setting

1

m ¼ dt fkÀ2 in Eq. (19) yields

(
)
D
E kP
E
À1 À
ÁD
1

1
1
1
a0 b0 dt fkÀ2 ; dt fkÀ2 À
bkÀjÀ1 À bkÀj dt fjÀ2 ; dt fkÀ2

j¼1

k¼1

6


 E
D
D
E D 1 E D 1 E
1
1
1
kÀ
a0 P dt ukÀ2 ; w ; m ¼ lkÀ2 DukÀ2 ; m þ f 2 ; m þ RkÀ2 ; m
and

1

m ¼ dt nkÀ2 in Eq. (16), we obtain

D
1

tively, such that both belong to H10 ðXÞ. Then, the difference formula
À
Á
(14) has convergence order O dt 3Àa .




 
 kÀ1 2  k 2
rn  À rn  :

¼ Àl

kÀ12

j¼1

D

kÀ12

rf

E D 1
E
1
1
; rdt fkÀ2 þ RkÀ2 ; dt fkÀ2 Á

Now, we sum from k ¼ 1 to M to get

Making use of Lemma 4, we can conclude that
1 



 
M 
M
a
X
t 1À
 kÀ12 2 X lkÀ2  kÀ1 2  k 2
m
06
d
n
6
r
n
À
r
n
 t






2Cð2 À aÞ k¼1
2dt
k¼1

a0

M
P

(






kÀ1
P
1 2
1 
1 


b0 dt fkÀ2  À
bkÀjÀ1 À bkÀj Þdt fjÀ2 dt fkÀ2 

)
6

j¼1

k¼1


2 
2  P


1
M
M 
P



 kÀ12  kÀ12 
lkÀ2 
À
rfk  À rfkÀ1  þ
R dt f :
2dt
k¼1

ð20Þ

k¼1

and then



2

lMÀ2 rnM  6
1

M
X



2

lkÀ2 rnk  6
1

k¼1

M
X



2

lkÀ2 rnkÀ1 
1

k¼1


2
1
þ lMÀ2 dt 2aþ1 rn0  :

If we change the index from M to k, then we arrive at



2

lkÀ2 rnk  6
1

k
X



2

ljÀ2 rnj  6 T 2a
1

j¼0


k 
X


2
1
 j 2
rn  þ lkÀ2 dt 2aþ1 rn0  :
j¼0

This expression can be rewritten as:
2a 



k

2
 2
 k 2 X T  j 2
rn  6
rn  þ dt2aþ1 rn0  6 dt2aþ1 n0 
kÀ1
j¼0

þ

k
X

l

2

j¼0

After applying the discrete Gronwall’s lemma to this inequality, it
yields



k h

i
 2  X
 2
 k 2
1 þ T 2a dt À2a ¼ dt2a n0 
rn  6 dt2aþ1 n0 
þT

 
n

ing # ¼ 2Cð2ÀaÞ, we deduce that





M 
M 
M 

a
X
X
t 1À
 kÀ12  kÀ12  Cð2 À aÞ X  kÀ12 2
 kÀ12 2
M
R
þ
d
f

R dt f  6



 :


t
a
4Cð2 À aÞ k¼1
t 1À
M
k¼1
k¼1
ð21Þ
Inserting Eq. (21) into Eq. (20), it follows that





M 
M
a
X
X
t1À
lkÀ12  k 2  kÀ1 2
 kÀ12 2
M
dt f  6 À
rf  À rf 
2Cð2 À aÞ k¼1
2dt
k¼1
M 
Cð 2 À a Þ X


a
t 1À
M

and using the Poincaré inequality, we obtain the desired result

  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi 
 k 

2cþ1  0 
n :
n  6 exp T

Hence the proof is complete.
The convergence order of the time-discrete approach is given in
the following theorem.

R

2
 þ

kÀ12 

k¼1


M 
a
X
t 1À
 kÀ12 2
M
dt f  :
4Cð2 À aÞ k¼1

ð22Þ

Multiplying Eq. (22) by 2dt, changing M to k, and simplifying results

in




k 
k 
P
P
 k 2
 1 2
 jÀ1 2
2
rf  6 2dtL Cð2 À aÞt ak À1 fjÀ2  þ 2dtCð2 À aÞt ak À1 R 2 
j¼1

j¼1


 1 2
k 
P
 1 2
 
6 2dtL2 Cð2 À aÞtak À1 fjÀ2  þ 2kdtCð2 À aÞtak À1 max fjÀ2  :

j¼0


 2


 2
6 dt 2a T 2cþ1 þ T 2cþ1 n0  6 exp T 2cþ1 n0 

r22 , by choos-

a
t 1À
M

þ


2


T 2a dtÀ2a rnj  :

2aþ1  0 2

2

In virtue of the Young’s inequality, jr1 r2 j 6 2#12 r21 þ #2

j¼1

16j6k

Employing a similar technique to the one adopted in the previous
theorem yields




2
k
 2 X
 k 2


T 2a dtÀ2a rfj 
rf  6 dt 2aþ1 f0  þ
j¼0

þ

k
X
j¼1

 1 2


2kdt Cð2 À aÞt kaÀ1 max RjÀ2  :
16j6k


209

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

Noticing that f0 ¼ 0, we get





 1 2
k
 k 2 X 2a À2a  j 2


T dt rf  þ 2kdt Cð2 À aÞtkaÀ1 max RjÀ2  ;
rf  6
16j6k

j¼0

and applying the Poincaré inequality results in

 2
k 

X
Á2
 k 
 jÀ12 2
2À
2
f  6 C2X dtL Cð2 À aÞt kaÀ1

f  þ C2X Cð2 À aÞTC dt 3Àa :
j¼1

ð23Þ
Using the discrete Gronwall inequality, the expression (23) can be
rewritten as the following form

Fig. 1. Schematic diagram of a stencil used for approximating the differential
operator on a non-uniform nodes.

Fig. 2. The computational domains fX1 ; X2 ; X3 ; X4 g.
Table 1
Numerical errors L1 and temporal accuracy C dt with h ¼ 1=10 and a ¼ 1:3 on X1 at T ¼ 1.
TPS-RBF r 4 lnðr Þ

dt

1=10
1=20
1=40
1=80
1=160
1=320
1=640
1=1280

MQ-RBF

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ e2 r 2

L1

C dt

L1

C dt

4:9210e À 03
1:7509e À 03
5:8335e À 04
1:9061e À 04
6:1506e À 05
1:9502e À 05
6:0741e À 06
1:8520e À 06

–
1:4909
1:5857
1:6137
1:6318
1:6571
1:6829
1:7136

4:0081e À 03
1:4613e À 03
5:1632e À 04

1:6843e À 04
5:3772e À 05
1:6807e À 05
5:1996e À 06
1:5644e À 06

–
1:4557
1:5009
1:6161
1:6472
1:6778
1:6926
1:7328

Table 2
Numerical errors L1 and spatial accuracy C h on X1 .
h

1=4
1=8
1=16
1=8
1=16

dt

1=4
1=40
1=400

1=8
1=80

a ¼ 1:45
L1

Ch

3:3808e À 01
1:6495e À 02
5:6526e À 04
2:6559e À 02
1:6845e À 03

–
4:3573
4:8670
–
3:9788

h

dt

a ¼ 1:65
L1

Ch

1=4

1=8
1=16
1=8
1=16

1=4
1=48
1=576
1=8
1=96

3:2919e À 01
7:9865e À 03
4:9705e À 04
2:0182e À 02
2:1633e À 03

–
5:3652
4:0061
–
3:2218


210

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

Table 3
The absolute errors of the LRBF-FD with a ¼ 1:8; N ¼ 801 and dt ¼ 1=100 at T ¼ 1 on

X4 .
NI

L1

CPU (s)

51
71
91
101

5:4671e À 04
5:5065e À 04
5:6804e À 04
5:7043e À 04

38.61
45.32
52.17
65.83

Table 4
The obtained condition number and the CPU time for the GRBF and LRBF-FD with
N ¼ 381 and dt ¼ 1=200 at T ¼ 1.
Domain

Method

CPU (s)

Condition number

X2

GRBF
LRBF-FD
GRBF
LRBF-FD
GRBF
LRBF-FD

53:42
40:21
60:04
48:51
57:21
45:34

8:6071e þ 06
3:4948e þ 02
6:7234e þ 07
5:5923e þ 02
4:6629e þ 07
5:5911e þ 02

X3
X4

 2


kÀ1
À
Á2
P 2 2
 k 
2
CX L Cð2 À aÞtkaÀ1
f  6 TC C2X Cð2 À aÞ dt 3Àa exp
j¼0



À
Á2
6 TC CX Cð2 À aÞ dt3Àa exp C2X L2 Cð2 À aÞkdtt kaÀ1


À
Á2
¼ TC 2 C2X Cð2 À aÞ dt 3Àa exp C2X L2 Cð2 À aÞt ak


À
Á2
6 TC 2 C2X Cð2 À aÞ dt 3Àa exp C2X L2 Cð2 À aÞT 2
À
Á2
6 C ðT; a; CX Þ dt3Àa :
2

2

As a result, we obtain:

 
 k 
f  6 C ðT; a; CX Þdt 3Àa :

The proof is completed

Fig. 3. The absolute error with dt ¼ 1=100; N ¼ 151 and a ¼ 1:5, at T 2 f0:25; 0:5; 075; 1g on the rectangular domain X1 .

ð24Þ


O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

211

Fig. 4. Sparsity pattern of the coefficient matrix when N ¼ 400.

Fig. 5. The approximated solutions and their corresponding absolute errors with dt ¼ 1=100, at T ¼ 1 on X3 .

Spatial discretization by the local radial basis function in a
finite difference mode
È
É
Given a set of distinct nodes X C ¼ xc1 ; . . . ; xcN # Rd and the corresponding function values uðxi Þ; i ¼ 1; 2; . . . ; N, the RBF interpolant
is represented in the form

uðxÞ ’ SðxÞ ¼

N
X

aj /j ðx; eÞ;

Luðxi Þ ’
ð25Þ

j¼1



where /j ðx; eÞ ¼ / kx À xcj k2 ; e ; j ¼ 1; . . . ; N; is a RBF corresponding
th

center with shape parameter e [26]. The expansion coeffiÈ ÉN
cients aj j¼1 , can be obtained by enforcing the interpolation condiÀ Á
tion S xci ¼ uci ; i ¼ 1; . . . ; N; at a set of nodes that usually coincides
with the N centers. It is worth to mention that the associated matrix

the j

/ is a non-singular and invertible for any arbitrarily set of distinct
scattered point [16,27].
Kansa [17,18] adopted the linear partial differential operator L
on the interpolation (25) to approximate Lu at the N scatter nodes,
namely

N
X
bj L/j ðxi ; eÞ:

ð26Þ

j¼1

The relation (26) defines a global RBF (GRBF) approximation, i.e. for
approximating L at reference point xi , all points in the domain are
involved. The GRBF meshless methods have the disadvantage of
dense and ill-conditioned interpolation matrices, but, on the other
hand, the sparse matrices of these techniques have better condition
numbers. Nonetheless, the differentiation matrices associated with
local meshless methods, that are used for solving PDE, require the
multiplication of the interpolation matrix by its inverse. This results


212

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

Fig. 6. The approximated solutions and their corresponding absolute errors with dt ¼ 1=100 and N ¼ 451 at T ¼ 1 on X4 .

Table 5
Numerical errors L1 and temporal accuracy C dt with h ¼ 1=10 and a ¼ 1:7 on X1 at T ¼ 1.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ e2 r 2

dt

TPS-RBF r 4 lnðrÞ
L1

C dt

L1

C dt

1=10
1=20
1=40
1=80
1=160
1=320
1=640
1=1280

6:1551e À 03
3:8996e À 03
1:7470e À 03
7:6426e À 04
3:2383e À 04
1:3565e À 04
5:6581e À 05
2:3258e À 05

–
1:1460

1:1301
1:1919
1:2021
1:2214
1:2652
1:3164

6:7029e À 02
3:0289e À 03
1:3839e À 03
6:0576e À 04
2:6329e À 04
1:1292e À 04
4:6981e À 05
1:8865e À 05

–
1:1460
1:1301
1:1919
1:2021
1:2214
1:2652
1:3164

MQ-RBF

Table 6
Numerical errors L1 and spatial accuracy C h on X1 .
h

1=4
1=8
1=16
1=8
1=16

dt

1=4
1=40
1=400
1=8
1=80

a ¼ 1:65

a ¼ 1:45

L1

Ch

h

dt

L1

Ch

4:7480e À 01
9:4222e À 03
7:7252e À 04
1:2036e À 01
4:3565e À 03

–
5:6551
3:6084
–
4:7880

1=4
1=8
1=16
1=8
1=16

1=4
1=48
1=576
1=8
1=96

5:0888e À 01
1:0732e À 02
9:5886e À 04
1:3354e À 01
5:2140e À 03

–
5:5673
3:4845
–
4:7227

in dense matrices again, and one may use the generalized inverse to
solve this limitation. Nonetheless, we must note that the discussion
of this subject falls outside the scope of the present work [28–31].
An innovative method named the LRBF-FD has been proposed in
[32] to overcome this issue. The new technique was also brought
up and examined more extensively in [32–37]. The discretization
in LRBF-FD (as a local meshless method) is obtained for a set of local
differentiation matrices and adding them up forms a large, sparse
system matrix. In order to calculate the differentiation matrix at
each point, merely the neighboring points are taken into
consideration.
Let us now discuss the proposed method in more detail. For
each node N ¼ fx1 ; . . . ; xN g # Rd in space, we consider a subset
n
o
ðiÞ
ðiÞ
SI ¼ x1 ; . . . ; xNI # N consisting of N I À 1 surrounding nodes
and

ðiÞ
xk

itself, and we define it as a stencil. Fig. 1 illustrates the

influence domain of every reference point xi . In the LRBF-FD, the
derivatives of a function in a node requires to be only a list of its
nearest stencil. The approximation of an operator L at the central
node xi is obtained as a weighted sum of function values of u at the
N I stencil nodes

Luðxi Þ ’

NI
 
X
ð iÞ
ð iÞ
wj u xj :

ð27Þ

j¼1

È
ÉNI
Following [32,33], by using a set of RBF /j ðx; eÞ j¼1
centered at SI
n
oNI
ðiÞ
for obtaining the LRBF-FD weights, wj
, in Eq. (27)

j¼1

L/k ðxi ; eÞ ¼

NI
X

ðiÞ

wj /j ðxk ; eÞ;

j¼1

k ¼ 1; . . . ; NI :

ð28Þ


O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

213

Fig. 7. The approximated solutions and their corresponding absolute errors with dt ¼ 1=100; N ¼ 200 and a ¼ 1:5 at T ¼ 1 on the rectangular domain X1 .

Fig. 8. The approximated solutions and their corresponding absolute errors with dt ¼ 1=100, at T ¼ 1 on X2 .

The unknown weights of LRBF-FD can be determined by solving the
system of linear equations in the following form:

UwI ¼ ½LUŠI ;

ð29Þ

where the coefficient matrix UNI ÂNI has entries /kj ¼ /j ðxk ; eÞ, wI
n
oN I
ðiÞ
represents the NI Â 1 vector of differential weights wj
, called
j¼1

algorithm can be used to determine the NI À 1 closest neighboring
points in the computation of the differentiation weights for the
stencils. We find the kd-tree algorithm named knnsearch in
the statistical toolbox of MATLAB. Additionally, the algorithm by
Sarra [38] is used to find the optimal shape parameter.
Results and discussion

I

LRBF-FD weights, and ½LUŠ is the N I  1 vector for the values
L/k ðxi ; eÞ; k ¼ 1; . . . ; NI . Due to the nonsingularity of the matrix U
[27], we calculate the weights vector wI given by

wI ¼ UÀ1 ½LUŠI :

ð30Þ

The derivatives are approximated in the LRBF-FD as for the classical
FD method. In brief, the derivatives are discretized at any node via

the RBF interpolation by means of a small collection of neighboring
nodes forming a stencil similar to those obtained with the FD. In the
n
oNI
ðiÞ
FD the weights wj
in the node xi are obtained on its stencil
j¼1

values, with the difference that in the LRBF-FD instead of polynomials, the RBF interpolation are used. A fast and effective kd-tree

This section investigates three problems to highlight the high
efficiency of the proposed method and to illustrate the theoretical
analysis established in the previous section for different values of h
and dt. The rate of convergence in time and space [39] are calculated by using the formulae:



Þjj
C dt ¼ log2 jjLjjL11ðð2dt;h
;
dt;hÞjj
Á
 À 4
jjL1 23Àa dt;2h jj
;
C h ¼ log2
jjL1 ðdt;hÞjj
 À
Á

À
Á
where L1 ¼ max U xj ; T À u xj ; T . All numerical results are
16j6NÀ1

obtained using MATLAB 2016a.


214

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216

Fig. 9. The approximated solutions and their absolute errors with dt ¼ 1=100 and N ¼ 451 at T ¼ 1 on X4 .

Example 1. Consider the following TFTTM:

NI

L1

CPU (s)

11
15
21
31

5:0349e À 03
5:1589e À 03
5:1635e À 03

5:1670e À 03

15.1
17.4
18.3
19.5

Àt

Method

CPU (s)

Condition number

X2

GRBF
LRBF-FD
GRBF
LRBF-FD
GRBF
LRBF-FD

828:69
502:35
751:14
463:29
678:47
352:07

3:3064e þ 10
1:3069e þ 03
3:4601e þ 10
1:3442e þ 03
4:8933e þ 10
1:3468e þ 03

X3
X4

Fig. 2 shows the computational domains in with two kinds of
distribution points that are considered in the follow-up. The
domain X1 ¼ ½0; 1Š2 denotes a rectangular domain with uniformly
distributed points. The irregular domain X2 is created using the
relation rðhÞ ¼ 0:8 þ 0:1ðsinð6hÞ þ sinð3hÞÞ with uniformly distributed points. The relation rðhÞ ¼ 1 À 14 cosð4hÞ; 0 6 h 6 2p, produces the irregular domain X3 which is covered by Halton
distributed points [40]. The domain X4 represents a set of Halton
2

ÀÀ
4

points in the unit circle in ½À1; 1Š including Halton non-uniform
points.

The LRBF-FD is applied here with several values for h; dt and a,
at T on X1 ; X2 ; X3 and X4 . The main results are presented in Tables
1–4 and Figs. 3–6. Tables 1 and 2 report the values achieved for the
absolute error and the convergence rates for several values dt and h
when T ¼ 1 on X1 . From Table 1, one can conclude that the

obtained computational orders support the theoretical order.
Table 3 lists the absolute errors L1 of the LRBF-FD for various values of local points N I . Table 4 exhibits the achieved condition number and CPU time for the GRBF and LRBF-FD on the irregular
domains. It is observed that coefficient matrix of LRBF-FD collocation procedure is more well-posed than the coefficient matrix of
GRBF method. Fig. 4 shows the sparsity pattern of the matrix associated with the LRBF-FD. Fig. 3 includes the graphs of the absolute

Table 10
Numerical errors EkU and temporal accuracy C dt with h ¼ 1=15 on X1 at T ¼ 1.
dt

1=10
1=20
1=40
1=80

a ¼ 1:2

a ¼ 1:8

EkU

C dt

EkU

C dt

1:0866e À 02
5:6681e À 03
2:3676e À 03
9:5950e À 04

–
0:9389
1:2594
1:3031

8:3954e À 03
3:7239e À 03
1:4702e À 03
4:9881e À 04

–
1:1728
1:3408
1:5595

Table 9
Comparison of the absolute error in the solution for several values of h; dt and a at T ¼ 1 on X1 .

a

h

dt

LRBF-FD

Ref. [9]

Ref. [7]

1:2

1=10
1=20
1=40
1=10
1=20
1=40

1=10
1=20
1=40
1=10
1=20
1=40

4:4471e À 03
9:0283e À 04
5:6298e À 04
3:6245e À 03
5:6761e À 04
1:9843e À 04

2:7161e À 03
8:1844e À 04
3:1115e À 04
2:5158e À 03
6:3084e À 04
9:8684e À 05

8:63934e À 02
2:2133e À 02
5:5200e À 03
8:7656e À 02
2:2912e À 02
5:9500e À 03

1:99

ð31Þ

The initial and boundary conditions corresponding to this example
can
be
calculated
from
an
exact
solution
À
ÁÀ
Á
uðx; y; t Þ ¼ t 2þa x4 À x2 y4 À y2 .

Table 8
The obtained condition number and the CPU time for the GRBF and LRBF-FD with
N ¼ 1781 and dt ¼ 1=1024 at T ¼ 1.
Domain

À 4
ÁÀ
Á
x À x2 y 4 À y 2
Á
À
Á
À
Á
À
Á
Á
12x2 À 2 y4 À y2 À 12y2 À 2 x4 À x2 ; x;y 2 X; 0 < t 6 T:

2
@ a uðx;y;tÞ
À t2 Duðx;y;t Þ ¼ Cð6t
4ÀaÞ
@ta

Table 7
The absolute errors of the LRBF-FD with N ¼ 401and dt ¼ 1=80 at T ¼ 1 on X2 .


215

O. Nikan et al. / Journal of Advanced Research 25 (2020) 205–216
Table 11
Numerical errors L1 and spatial accuracy C h with on X1 at T ¼ 1.
h

a ¼ 1:3

a ¼ 1:7

L1

Ch

L1

Ch

2=4; 2=8
2=8; 2=16
2=16; 2=32

2:3540e À 03
8:7995e À 04
3:5412e À 04

–
1:1460
1:1301

2:2039e À 03
6:4571e À 04
2:3653e À 04

–

1:7711
1:4489

errors by choosing h ¼ 1=10; dt ¼ 1=100 and a ¼ 1:5, when T ¼ 1
on X1 . Fig. 5 depicts the approximate solutions and their absolute
errors with a ¼ 1:15; N ¼ 451 and dt ¼ 1=100 when T ¼ 1 on X3 .
Finally, Fig. 6 shows the approximate solutions and their absolute
errors with a ¼ 1:75; N ¼ 353 and dt ¼ 1=100, when T ¼ 1 on X4 .

where the error at dt is evaluated as a difference between the solutions U dt and U 2dt for time steps dt and 2dt at T given by
Edt ¼ kU dt À U 2dt k. The following predictor–corrector procedure
and norm are illustrated to obtain the error in spatial variable

Example 2. We consider the following TFTTM:

where U h1 and U h2 are the approximate solutions with respect to h1
and h2 , respectively. The spatial convergence rate can be calculated
as:

a

@ uðx; y; tÞ
À tDuðx; y; t Þ
@t a

¼ 2t2 sinð2pxÞsinð2pyÞ


t Àa
þ 4p2 t ; x; y 2 X; 0 < t 6 T:

Cð3 À aÞ
ð32Þ

The initial and boundary conditions corresponding to this example
can be obtained from the exact solution uðx; y; tÞ ¼
t 2 sinð2pxÞ sinð2pyÞ.
The LRBF-FD is formulated here for different quantities of dt; h
and a, at T on X1 ; X2 ; X3 and X4 . The results are illustrated in Tables
5 and 8 and Figs. 7–9. Tables 5 and 6 report the values achieved for
the absolute error and the convergence rates for several values dt
and h when T ¼ 1 on X1 . From Table 5, one can conclude that the
computational order is in good agreement with the theoretical
order. Table 7 illustrates the absolute errors L1 of the LRBF-FD
for several values of local points N I . Table 8 reports the obtained
condition number and the CPU time for the GRBF and LRBF-FD
on the irregular domains. It can be mentioned that coefficient
matrix of the LRBF-FD collocation procedure is more smaller than
the coefficient matrix of the GRBF. Table 9 compares the absolute
errors of the RBF-FD and those from [7,9] with several values of
dt and h. Fig. 7 represents the resulting absolute errors by choosing
h ¼ 1=10; dt ¼ 1=100 and a ¼ 1:5, when T ¼ 1 on X1 . Fig. 8 includes
the absolute errors achieved with N ¼ 451; dt ¼ 1=100 for a ¼ 1:65,
when T ¼ 1 on X2 . Finally, Fig. 9 displays the approximate solutions and their absolute errors with a ¼ 1:45; N ¼ 353 and
dt ¼ 1=100, when T ¼ 1 on X4 .

Eh1 ;h2 ¼ kU h1 À U h2 k;


C h ¼ log2


Eh;h=2
;
Eh=2;h=4

ð34Þ

ð35Þ

where Eh;h=2 and Eh=2;h=4 are the absolute error between the solutions
with mesh sizes fh; h=2g and fh=2; h=4g, respectively.
We apply the LRBF-FD to obtain the numerical results for several quantities of dt; h and a. Table 10 reports the achieved errors
and convergence orders with respect to the temporal domain.
Table 11 lists the achieved errors and convergence orders with
respect to the spatial domain.
Concluding remarks
This paper presented a novel method for finding an approximate solution of the TFTTM. One of the key results that emerges
from this work is that the method is robust and has a reliable accuracy even for a complex domain using irregular nodal distributions.
It ought to be said here that the irregularly nodal distribution and
complex domain lead to considerable difficulties for standard techniques. The proposed algorithm includes two parts, namely, a first
one where the problem is discretized based on finite difference
scheme in the temporal direction, and a second one where the
LRBF-FD is used for the spatial approximation. The stability and
convergence of the semi-discrete scheme are rigorously investigated. Numerical results highlight the efficiency of the method.
Declaration of Competing Interest

Example 3. Lastly, we consider the following TFTTM:

@ a uðx; y; t Þ
À t Duðx; y; tÞ ¼ 0;

@ta

x; y 2 X;

The authors declare that there is no conflict of interests regarding the publication of this manuscript.

0 < t 6 T;

with the initial conditions

uðx; y; 0Þ ¼ 0;

@uðx; y; 0Þ
¼ 0;
@t

Compliance with Ethics Requirements

x; y 2 X

This manuscript does not contain any studies with human participants or animals performed by any of the authors.

and boundary condition

uðx; y; t Þ ¼ t 2þa cosð2pxÞ cosð2pyÞ;

Acknowledgement

x; y 2 @ X:

Since the analytical solution of the above problem is unknown, we
apply the relation presented by Kamranian et al. [41] for the criterion convergence of the solution:

EkU ¼

kU kþ1 À U k k
kU kþ1 k

;

ð33Þ

The authors are very grateful to the reviewers for their valuable
comments on the manuscript that led to many improvements.
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