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

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.

Numerical evaluation of fractional Tricomi-type model arising from

physical problems of gas dynamics

O Nikana, J.A Tenreiro Machadob, Z Avazzadehc,d,⇑, H Jafarie

a

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

b

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

h i g h l i g h t s

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

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

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 spa-tial 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 conver-gence and stability analyses of the proposed time semi-discrete scheme are proved by means of the dis-crete 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 (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Introduction This paper proposes an efficient numerical formulation for solv-ing the time fractional Tricomi-type model (TFTTM), that can be written as

@au x; tð Þ

@ta  t2 cDu x; tð Þ ¼ f x; tð Þ; x ¼ x; yð Þ 2X R2; 0 < t 6 T;

ð1Þ

https://doi.org/10.1016/j.jare.2020.06.018

2090-1232/Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University.

⇑ Corresponding author.

E-mail addresses: omidnikan77@yahoo.com (O Nikan), jtm@isep.ipp.pt (J.A.T.

Machado), zakiehavazzadeh@duytan.edu.vn (Z Avazzadeh), jafari.usern@gmail.

com (H Jafari).

Contents lists available atScienceDirect Journal of Advanced Research

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j a r e

along with the initial and Dirichlet boundary conditions given by

u x; 0ð Þ ¼ g xð Þ; x2
X¼X[ @X; ð2Þ

@u x; 0ð Þ

@t ¼ w xð Þ; x2
X¼X[ @X; ð3Þ

u x; tð Þ ¼ h x; tð Þ; x2 @X; t > 0; ð4Þ

where T is the final time, D denotes the Laplace operator with

respect to space variables x;cis real non-negative number,Xis a

bounded domain inR2, and@Xrepresents the boundary ofX The

fractional derivative @au xð ; tÞ=@ta of order 1<a< 2 is defined in

the Caputo sense as

@au x; tð Þ

@ta ¼ 1

Cð2aÞ

Z t 0

@2

u x; sð Þ

@s2 ðt sÞ1 ads;

whereCð Þ represents the Euler’s gamma function [1,2]

During the 20s, Tricomi[3]started the work on the linear

par-tial differenpar-tial equations of variable type with boundary

condi-tion 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

discontin-uous Galerkin finite element, Zhang et al.[6]used the finite

ele-ment scheme and Liu et al [7]applied the reduced-order finite

element technique to approximate the TFTTM More recently,

Deh-ghan and Abbaszadeh[8]adopted the element-free Galerkin

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

non-smooth domains because they provide the problem solution only

on mesh points As a result, meshless techniques have been

devel-oped to overcome these problems One important meshless

tech-nique 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

Rol-land Hardy[14]proposed the RBF technique in 1971 by

introduc-ing the multiquadric (MQ) algorithm as a meshless interpolation

method using the MQ radial function Richard Franke[15]

popular-ized this approach in 1982 with a review of the 32 most used

inter-polation techniques Franke performed a set of comprehensive

tests and concluded that the MQ method had the best overall

per-formance Furthermore, he advanced that the interpolation matrix

related to the MQ radial function has unconditional

non-singularity Later, in 1986, Micchelli [16] proved this using

research from the 30s and 40s by Schoenberg Kansa[17,18]

con-sidered 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 RBF-finite difference (LRBF-FD) for space discretization Section 4 illus-trates the method with three numerical examples that show its efficiency and verify the theoretical analysis Finally, Section 5 con-cludes 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; tk¼ kdt; k ¼ 1; ; M, for a positive integer M Therefore, the time domain 0½ ; T 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 tð Þ 2 C2

0; T

½ , then it follows

Z t n

0

g0ð Þ tsðn sÞ1 ads¼Xn

k¼1

g tð Þ  g tk ðk1Þ dt

Z t k

tk1

tn s

ð Þ1 adsþ Rn;

where

Rn

  6 1

2 2ð aÞþ

1 2

dt3amax

0 6t6t n

jg00ð Þj:t

Lemma 2 ([23].) Suppose that 1<a< 2 a0¼ 1

dt C ð 2 a Þ and

bk¼dt2a 2 ahðkþ 1Þ2 a kð Þ2 ai

; k ¼ 0; 1; 2; Then it follows



 1

C 2 ð a Þ

Rt n

0 g0ð Þ tsðn sÞ1 ads a0

 b0g tð Þ n n1P

k¼1ðbnk1 bnkÞg tj

 

 bn1g 0ð Þ



6 1

C 2 ð a Þ 2 2 ð1a Þþ1

dt3amax 06t6t n

jg00ð Þj:t

Lemma 3 ([23].) If 1<a< 2 and bk¼dt2a

2 ahðkþ 1Þ2 a kð Þ2 ai

;

k¼ 0; 1; 2; ; then it follows

b0> b1> b2> > bk! 0; as k ! 1:

We introduce the following notation:

dtuk1¼uk uk1

dt ; lk¼ t k 2 c

; fk1¼f

k

þ fk1

Let us consider

vðx; tÞ ¼ @u x; tð Þ

V x; tð kÞ ¼ 1

Cð2aÞ

Z t 0

@vðx; sÞ

@s ðt sÞ1ads: ð7Þ

From(6)it results that Taylor expansion at t¼ tk1can be written as:

vk 1

¼ dtuk1þ Rk11; ð8Þ

where

Rk11



  6 C

Based onLemma 2, we have

Vk¼ a0 b0vnXk1

j¼1

bkj1 bkj

vj bk1v0

þ Rk 2

and also

Vk 1

¼ a0 b0vk 1

Xk1 j¼1

bkj1 bkj

vj 1

 bk1v0

þ Rk21;

where

Rk2 1



  6 C

2dt3a:

We define the operator

P vk 1

; q

¼ a0 b0vk 1

Xk1 j¼1

bkj1 bkj

vj 1

 bk1q

: ð10Þ

UsingLemma 2, the expression(10)can be written as

Vk 1

¼ V

kþ Vk1

2 ¼ a0P vk 1

w

þ Rk21; ð11Þ

wherev0ð Þ ¼x vðx; 0Þ ¼ w xð Þ ¼ w If we substitute(8)into(11), we

obtain

Vk 1

¼ a0P d tuk1; w þ a0P R k11; 0 þ Rk2 1: ð12Þ

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

a0P d tuk1; w ¼Duk1þ fk 1

þ Rk21; ð13Þ

where

Rk1¼  an 0P R k11; 0 þ Rk2 1o

:

Based onLemma 2, 3and inequalities(9)we can write

Rk16 a0 b0Rk1 1þk1P

j¼1

bkj1 bkj

vj bk1Rk1 1

þ Rk2 1

6 a0 b0C1dt2þk1P

j¼1 bkj1 bkj

vj bk1C1dt2

þ C2dt3a

¼ a0b0C1dt2þ bð 0 bk1ÞC1dt2

þ C2dt3a

6 a02b0C1dt2

þ C2dt3a

dtC 2 ð a Þ 2dt2a

2 aC1dt2

þ C2dt3a

6 2C 1

2 a

ð ÞC 2 ð a Þþ C2

dt3a:

Dropping the error term Rk1 and approximating the exact value

uk 1

by its numerical approximation Uk1, leads to the following

semi-discrete recursive algorithm:

a0P d tUk1; w ¼lk12

DUk1þ fk 1

or, equivalently, we get

a0b0Uk1dtlkDUk¼ a0b0Uk1þ1dtlk1DUk1

þa0dtk1P

j¼1bjk1 bjk

dtUj1þ a0bk1dtw þ1dt f kþ fk1

:

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

H1ð Þ ¼X v2 L2ð Þ;X dv

dx2 L2ð ÞX

;

H10ð Þ ¼X nv2 H1ð Þ;X vj@X¼ 0o

;

Hmð Þ ¼X nv2 L2ð Þ; DX av2 L2ð Þ; for all positive integerjX aj 6 mo;

where L2ð Þ represents the space of measurable functions whoseX

square is Lebesgue integrable in Xanda¼ða1; ;adÞ denotes a d-tuple of non-negative integer withjaj ¼Pd

i¼1ai Let us consider

Dav¼ @j a jv

@xa

1@xa

2 @xa

d



The normkvkmof the space Hmð Þ can be written asX

v

j j

j jHm ð Þ X ¼ X

j a j6m

Dav

 

 2

L 2 ð Þ X

!1

:

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

Corollary 1 (Poincaré inequality [24]) Suppose that 16 p 6 1 and thatXis a bounded open set Then, there exists a constantCX

(depending onXand p) such that

gk

 

  6 CX $ gk:

Lemma 4 ([23].) For any G¼ Gf 1; G2; g and q, we have

XM j¼1

P G j; qGjPt1Ma

2 dt

XM j¼1

G2j  t2Ma

2 2ð aÞq

2:

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

ynfulfills

y06 d0;

yn6 d0þn1P

k¼0

zkþn1P k¼0

xkyk;

8

>

>

then ynsatisfies

y16 d0ð1þ x0Þ þ y0;

yn6 d0þn2Q

k¼0

1þ xk

ð Þ þn2P

k¼0

zk Q n1 s¼kþ1

1þ xs

ð Þ þ zn1; n P 2:

8

>

>

Moreover, if d0P 0 and znP 0 for n P 0, then it holds

yn6 d0þXn1

k¼0

zk

! exp Xn1 k¼0

xk

! :

Making use of these lemmas, we can derive the following result

of stability

Theorem 1 If Uk2 H1ð Þ, then the difference formulaX (14) is unconditional stable with respect to the H1-norm

Proof The following variational weak formulation will be obtained by multiplying both sides of Eq.(14)bymand integrating overX

a0P d tnk1; w ;m

¼lk 1

Dnk1;m

where nk1¼ Uk 1

 Uk 1

denotes the perturbation at the k 1

th time level, so that Uk1and Uk 1

are the exact and approximate solu-tions of Eq.(14), respectively

Using the divergence theorem

Z

Xr v r x¼

Z

@Xv@@nx

Z

Xv D x;

where

@n¼ @@xxn1þ @x

@yn2

is the normal derivative, that is, representing the derivative in the

outward normal direction to the boundary@X, we get

a0 b0Ddtnk1;mEXk1

j¼1

bkj1 bkj

dtnk1;m

¼ lk 1

rnk1;r m

Lettingm¼ dtnk1in Eq.(16), we obtain

a0 b0 dtnk1; dtnk1

Xk1 j¼1

bkj1 bkj

dtnj1; dtnk1

¼ lk 1

rnk1;rdtnk1



Summing on k; k ¼ 1; ; M, and applying Cauchy–Schwarz

inequal-ity, we deduce that

a0

XM

k¼1

b0dtnk12

Xk1 j¼1

bkj1 bkj

dtnj1



 



  d

tnk1



 



 

6X

M

k¼1

lk 1

2dt rnk12

rnk2

:

Making use ofLemma 4, we can conclude that

06 t1ma

2Cð2aÞ

XM

k¼1

dtnk1



 



 2

6X M

k¼1

lk 1

2dt rnk12

rnk2

and then

lM 1

rnM

 

 2

6X

M

k¼1

lk 1

rnk



 



 2

6X M

k¼1

lk 1

rnk1



 



 2

þlM 1

dt2aþ1 rn02

:

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

lk 1

rnk



 



 2

6Xk

j¼0

lj 1

rnj



 



 2

6 T2 aXk j¼0

rnj



 



 2

þlk 1

dt2 a þ1 rn02

:

This expression can be rewritten as:

rnk



 



 2

6X

k

j¼0

T2 a

lk 1rnj2

þ dt2 a þ1 rn02

6 dt2 a þ1  n0 2

þXk

j¼0

T2 adt2arnj2

:

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

yields

rnk



 



 2

6 dt 2 a þ1  n0 2 k

j¼0

1þ T2 a

dt2a

¼ dt2 a  n0 2

þ T2 a þ1  n0 2

6 dt 2 aT2 c þ1þ T2 c þ1

n0

 

 2

6 exp T 2 c þ1

n0

 

 2

and using the Poincaré inequality, we obtain the desired result

nk



 



  6rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexp T 2 c þ1

n0

 

 :

Hence the proof is complete

The convergence order of the time-discrete approach is given in

the following theorem

Theorem 2 Let ukand Ukbe the solutions of(13)and(14), respec-tively, such that both belong to H1ð Þ Then, the difference formulaX

(14)has convergence orderO dt 3 a

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

a0P d tuk1; w ;m

¼lk 1

Duk1;m

þ fk 1

;m

þ Rk 1

;m

ð17Þ

and

a0P d tUk1; w ;m

¼lk 1

DUk1;m

þ fk 1

;m

where fk1¼ uk 1

 Uk 1

Subtracting Eq (17) from Eq (18) and using the divergence theorem again, we arrive at

a 0 b 0Dd t fk1;mE X k1

j¼1

b kj1 b kj

d t fk1;m

¼ lk 1

rfk1;rm

þ R k 1

;m

 ð19Þ Settingm¼ dtfk1in Eq.(19)yields

a0 b0 dtfk1; dtfk1

k1P j¼1

bkj1 bkj

dtfj1; dtfk1

¼ lk 1

rfk1;rdtfk1

þ Rk 1

; dtfk1



Now, we sum from k¼ 1 to M to get

a0

PM k¼1 b0dtfk12

k1P j¼1bkj1 bkjÞ d tfj1 d

tfk1



 



 

6

PM k¼1

lk12

2dt rfk2

rfk12

þPM k¼1Rk1 d

tfk1



 



 : ð20Þ

In virtue of the Young’s inequality,jr1r2j 6 1

2# 2r2þ# 2

2r2, by choos-ing# ¼ t1a

M

2 C ð 2 a Þ, we deduce that

XM k¼1

Rk1



 



  d

tfk1



 



  6C 2 ð aÞ

t1a

M

XM k¼1

Rk1



 



 2

þ t1Ma

4C 2 ð aÞ

XM k¼1

dtfk1



 



 2

: ð21Þ

Inserting Eq.(21)into Eq.(20), it follows that

t1a

M

2Cð2aÞ

XM k¼1

dtfk1



 



 2

6 X M

k¼1

lk 1

2dt rfk2

rfk12

þCð2aÞ

t1a

M

XM k¼1

Rk1



 



 2

þ t1Ma

4Cð2aÞ

XM k¼1

dtfk1



 



 2

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

rfk



 



 2

6 2dtL2Cð2aÞta 1

k

Pk j¼1fj12

þ 2dtCð2aÞta 1

k

Pk j¼1

Rj1



 



 2

6 2dtL2Cð2aÞta 1

k

Pk j¼1fj12

þ 2kdtCð2aÞta 1

16j6kfj12

:

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

rfk



 



 2

6 dt2 a þ1  f0 2

þXk j¼0

T2 adt2arfj2

þXk j¼1 2kdtCð2aÞta 1

k max 16j6kRj12

:

Noticing that f0¼ 0, we get

rfk



 



 2

6X k

j¼0

T2adt2arfj2

þ 2kdtCð2aÞta 1

k max 16j6kRj12

;

and applying the Poincaré inequality results in

fk



 



 2

6 C2

XdtL2Cð2aÞta 1

k

Xk j¼1

fj1



 



 2

þ C2

XCð2aÞTC2

dt3a

 2 : 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 L 1 and temporal accuracy Cdtwith h ¼ 1=10 anda¼ 1:3 onX1 at T ¼ 1.

1 þe2 r 2

p

Table 2

Numerical errors L 1 and spatial accuracy C h onX1



 



 2

6 TC2

C2

XCð2aÞ dt 3 a2

exp k1P j¼0 C2

XL2Cð2aÞta 1

k

6 TC2

C2

XCð2aÞ dt 3 a2

exp C2

XL2Cð2aÞkdtta 1

k

¼ TC2C2

XCð2aÞ dt 3 a2

exp C2

XL2Cð2aÞta

k

6 TC2

C2

XCð2aÞ dt 3 a2

exp C2

XL2Cð2aÞT2

6 C T;ð a; CXÞ dt 3 a2

:

As a result, we obtain:

fk



 



  6 C T;a; C

X

The proof is completed

Fig 3 The absolute error with dt ¼ 1=100; N ¼ 151 anda¼ 1:5, at T 2 0:25; 0:5; 075; 1 f g on the rectangular domainX.

Table 3

The absolute errors of the LRBF-FD witha¼ 1:8; N ¼ 801 and dt ¼ 1=100 at T ¼ 1 on

X4

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.

Spatial discretization by the local radial basis function in a

finite difference mode

Given a set of distinct nodes XC¼ xc

1; ; xc N



# Rdand the cor-responding function values u xð Þ; i ¼ 1; 2; ; N, the RBF interpolanti

is represented in the form

u xð Þ ’ S xð Þ ¼XN

j¼1

where /jðx;eÞ ¼ / kx  xc

jk2;e

; j ¼ 1; ; N; is a RBF corresponding the jth center with shape parametere[26] The expansion

coeffi-cients aj

N

j¼1, can be obtained by enforcing the interpolation

condi-tion S xc

i

 ¼ uc

i; 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

/ is a non-singular and invertible for any arbitrarily set of distinct scattered point[16,27]

Kansa[17,18]adopted the linear partial differential operatorL

on the interpolation(25)to approximateLu at the N scatter nodes, namely

Lu xð Þ ’i XN

j¼1

bjL/jðxi;eÞ: ð26Þ

The relation(26)defines a global RBF (GRBF) approximation, i.e for approximatingL 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

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 onX3

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 N¼ xf 1; ; xNg# Rd in space, we consider a subset

SI¼ xð Þ i

1; ; xð Þ i

N I

#N consisting of NI 1 surrounding nodes and xð Þ i

itself, and we define it as a stencil.Fig 1illustrates 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 operatorL at the central node xiis obtained as a weighted sum of function values of u at the

NIstencil nodes

Lu xð Þ ’i XN I

j¼1

wð Þjiu xð Þji : ð27Þ

Following[32,33], by using a set of RBF /jðx;eÞNI

j¼1centered atSI for obtaining the LRBF-FD weights, wn ð ÞjioN I

j¼1, in Eq.(27)

L/kðxi;eÞ ¼XNI

j¼1

wð Þji/jðxk;eÞ; k ¼ 1; ; NI: ð28Þ

Fig 6 The approximated solutions and their corresponding absolute errors with dt ¼ 1=100 and N ¼ 451 at T ¼ 1 onX4

Table 5

Numerical errors L 1 and temporal accuracy Cdtwith h ¼ 1=10 anda¼ 1:7 onX1 at T ¼ 1.

1 þe2 r 2

p

Table 6

Numerical errors L 1 and spatial accuracy C h onX1

The unknown weights of LRBF-FD can be determined by solving the

system of linear equations in the following form:

UwI¼ LU½ I

where the coefficient matrix UN I NI has entries /kj¼ /jðxk;eÞ, wI

represents the NI 1 vector of differential weights wð Þ i

j

n oN I

j¼1, called LRBF-FD weights, and ½LUI

is the NI 1 vector for the values L/kðxi;eÞ; k ¼ 1; ; NI Due to the nonsingularity of the matrix U

[27], we calculate the weights vector wIgiven by

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

FD the weightsnwð ÞjioN I

j¼1in the node xiare obtained on its stencil values, with the difference that in the LRBF-FD instead of

polynomi-als, the RBF interpolation are used A fast and effective kd-tree

algorithm can be used to determine the NI 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 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 calcu-lated by using the formulae:

Cdt¼ log2 jjL 1 ð 2dt;h Þjj

jjL 1 ð dt;h Þjj

;

Ch¼ log2

jjL 1 2 4a dt;2h

jj jjL 1 ð dt;h Þjj

;

16j6N1U x j; T u x j; T All numerical results are obtained using MATLAB 2016a

Fig 7 The approximated solutions and their corresponding absolute errors with dt ¼ 1=100; N ¼ 200 anda¼ 1:5 at T ¼ 1 on the rectangular domainX1

Fig 8 The approximated solutions and their corresponding absolute errors with dt ¼ 1=100, at T ¼ 1 onX2

Fig 2shows the computational domains in with two kinds of

distribution points that are considered in the follow-up The

domainX1¼ 0; 1½ 2denotes a rectangular domain with uniformly

distributed points The irregular domainX2 is created using the

relation r hð Þ ¼ 0:8 þ 0:1 sin 6hð ð Þ þ sin 3hð ÞÞ with uniformly

dis-tributed points The relation r hð Þ ¼ 1 1cos 4hð Þ; 0 6 h 6 2p,

distributed points[40] The domainX4represents a set of Halton

points in the unit circle in½1; 12

including Halton non-uniform points

Example 1 Consider the following TFTTM:

@ a u x;y;t ð Þ

@t a t2Du x;y;tð Þ ¼ 6t 2

C 4 ð a Þx4x2

y4y2

t4 12x22

y4y2

 12y 22

x4x2

; x;y 2X; 0 <t 6 T: ð31Þ

The initial and boundary conditions corresponding to this example

u xð ; y; tÞ ¼ t2þ ax4 x2

y4 y2

The LRBF-FD is applied here with several values for h; dt anda,

at T onX1;X2;X3andX4 The main results are presented inTables 1–4andFigs 3–6.Tables 1 and 2report the values achieved for the absolute error and the convergence rates for several values dt and h

obtained computational orders support the theoretical order

Table 3lists the absolute errors L1of the LRBF-FD for various val-ues of local points NI.Table 4exhibits the achieved condition num-ber and CPU time for the GRBF and LRBF-FD on the irregular domains It is observed that coefficient matrix of LRBF-FD colloca-tion procedure is more well-posed than the coefficient matrix of GRBF method.Fig 4shows the sparsity pattern of the matrix asso-ciated with the LRBF-FD.Fig 3includes the graphs of the absolute

Fig 9 The approximated solutions and their absolute errors with dt ¼ 1=100 and N ¼ 451 at T ¼ 1 onX4

Table 7

The absolute errors of the LRBF-FD with N ¼ 401and dt ¼ 1=80 at T ¼ 1 onX2

Table 9

Comparison of the absolute error in the solution for several values of h; dt andaat T ¼ 1 onX1

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.

Table 10 Numerical errors E k

U and temporal accuracy Cdtwith h ¼ 1=15 onX1 at T ¼ 1.

E k

1=20 5:6681e  03 0:9389 3:7239e  03 1:1728 1=40 2:3676e  03 1:2594 1:4702e  03 1:3408 1=80 9:5950e  04 1:3031 4:9881e  04 1:5595

The convergence order of the time-discrete approach is given in

the following theorem

Theorem Let ukand Ukbe the solutions of( 13)and(14),