Stable numerical results to a class of time-space fractional partial differential equations via spectral method

In present time significant attention has been given to study non-integer order partial differential equations. The current article is devoted to find numerical solutions to the following class of time–space fractional partial differential.

Stable numerical results to a class of time-space fractional partial

differential equations via spectral method

Kamal Shaha, Fahd Jaradb, Thabet Abdeljawadc,d,e,⇑

a

Department of Mathematics, University of Malakand, Chakdara Dir (lower), Khyber Pakhtunkhawa, Pakistan

b

Çankaya University, Department of Mathematics, 06790 Etimesgut, Ankara, Turkey

c

Department of Mathematics and General Sciences, Prince Sultan University, P.O Box 66833, Riyadh 11586, Saudi Arabia

d

Department of Medical Research, China Medical University, Taichung 40402, Taiwan

e Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

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 6 April 2020

Revised 2 May 2020

Accepted 20 May 2020

Available online 19 June 2020

Keywords:

Fractional partial differential equations

Caputo fractional derivative

Shifted Jacobin polynomials

Operational matrices

Numerical solution

Stability

a b s t r a c t

In this paper, we are concerned with finding numerical solutions to the class of time–space fractional par-tial differenpar-tial equations:

Dp

tuðt; xÞ þjDpuðt; xÞ þsuðt; xÞ ¼ gðt; xÞ; 1 < p < 2; ðt; xÞ 2 ½0; 1  ½0; 1;

under the initial conditions

uð0; xÞ ¼ hðxÞ; utð0; xÞ ¼ /ðxÞ;

and the mixed boundary conditions

uðt; 0Þ ¼ uxðt; 0Þ ¼ 0;

where Dp

t is the arbitrary derivative in Caputo sense of order p corresponding to the variable time t Further, Dpis the arbitrary derivative in Caputo sense with order p corresponding to the variable space

x Using shifted Jacobin polynomial basis and via some operational matrices of fractional order integra-tion and differentiaintegra-tion, the considered problem is reduced to solve a system of linear equaintegra-tions The

https://doi.org/10.1016/j.jare.2020.05.022

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

q Peer review under responsibility of Cairo University.

⇑ Corresponding author at: Department of Mathematics and General Sciences, Prince Sultan University, P O Box 66833, Riyadh 11586, Saudi Arabia.

E-mail addresses: kamalshah408@gmail.com (K Shah), fahd@cankaya.edu.tr (F Jarad), tabdeljawad@psu.edu.sa (T Abdeljawad).

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

used method doesn’t need discretization A test problem is presented in order to validate the method Moreover, it is shown by some numerical tests that the suggested method is stable with respect to a small perturbation of the source data gðt; xÞ Further the exact and numerical solutions are compared via 3D graphs which shows that both the solutions coincides very well

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

In present time significant attention has been given to study

non-integer order partial differential equations In fact, it was

shown that in many situations, derivatives of non-integer order

are very effective for the description of many physical phenomena

find numerical solutions to the following class of time–space

frac-tional partial differential equations:

Dptuðt; xÞ þjDpxuðt; xÞ þsuðt; xÞ ¼ gðt; xÞ; 1 < p < 2;

ðt; xÞ 2 ½0; 1  ½0; 1; ð1Þ

under the initial conditions

uð0; xÞ ¼ hðxÞ; utð0; xÞ ¼ /ðxÞ; ð2Þ

and the mixed boundary conditions

uðt; 0Þ ¼ uxðt; 0Þ ¼ 0; ð3Þ

whereðs;jÞ 2 R R; Dp

tdenotes the Caputo fractional derivative of

fractional derivative of order p with respect to the variable space

x; utis the derivative of u with respect to the variable time t; uxis

the derivative of u with respect to the variable space x, and

h; / : ½0; 1 ! R; g : ½0; 1  ½0; 1 ! R are given functions

The modeling of some real world problems by using differential

equations is a warm area of research in last many years Here we,

remark that partial differential equations have important

applica-tions in many branches of science and engineering For instance

heat transfer is a very important branch of mechanical and

aero-space engineering analyses because many machines and devices

in both these engineering disciplines are vulnerable to heat An

engineer can predict about with possible shape changes of the

plate in vibrations from the simulation results of the aforesaid

equations Many engineering problems fall into such category by

nature, and the use of numerical methods will to find their

solu-tions are important for engineers In particularly time–space one

dimensional equation has many applications For concerned

Conventionally, numerous techniques were developed to find

approximate solutions to different classes of fractional partial

method have their own advantages and disadvantages in

applica-tion point of view For example, homotopy methods depend on a

small parameter which restricted these methods Similarly the

methods that are involving integral transform also are limited in

applications In last few decades, some interesting numerical

schemes based on radial basis functions (RBFs) and meshless

tech-niques were introduced These methods require collocation and

Recently, numerical schemes based on operational matrices have

attracted the attention of many researches The mentioned

tech-niques provide highly accurate numerical solutions to both linear

and nonlinear ordinary as well as partial differential equations of

classical and fractional order In the mentioned schemes, some

operational matrices of fractional order integration and differenti-ation are constructed, which play central roles to find approximate solutions for the considered problems In the most existing works, the mentioned matrices are obtained using a certain polynomial

[4,12,13,21,24,25,30]) However, in these methods, discretization

is required, which needs extra memory Further for discretization and collocation extra amount of memory should be utilized To

operational matrices without discretizing the data and omitting collocation method to compute numerical solutions for both ordi-nary as well as partial fractional differential equations

Motivated by the above cited works, in this paper, a numerical

basis and some operational matrices of fractional order integration and differentiation without actually discretizing the problem The Jacobi polynomials are more general polynomials and including

‘‘Legendre polynomials, Gegenbauer polynomials, Zernike and Che-byshev polynomials” as special cases The concerned polynomials have numerous applications in Quantum physics, fluid mechanics and solitary theory of waves, see detail[16]

equations of the form given by

HTk2A¼ B;

where the matrix H is the unknown which may be determined

dimension k2 k2

and 1 k2

respectively Here it is remarkable that the obtained system of algebraic equations is then solved by Gauss elimination method through Matlab for the unknown matrix H Fur-ther we demonstrate that by computational software, the solution

is easily obtained up to better accuracy The computations in our work are performed using Matlab-16.‘‘The paper is organized as fol-lows In Section 2, we recall briefly some necessary definitions and mathematical preliminaries about fractional calculus In Section 3,

we recall some basic properties on Jacobi polynomials, which are required for establishing main results In Section 4, The shifted Jacobi operational matrices of fractional derivatives and fractional integrals are obtained Section 5 is devoted to the numerical scheme, which is based on operational matrices In Section 6, numerical experiments are presented Also in the same section,

we study the stability of the method with respect to a perturbation

of the source data Conclusion is made in Section 7.”

Basic materials Some fundamentals notions, definitions and results are recalled here from[6,17,18]

Cl;l2 R, if and only if, there exists a real number m >lsuch that

fðxÞ ¼ xmgðxÞ; x > 0;

Definition 2 A real function fðxÞ; x > 0 is said to be in space

Cnp;l2 R; n 2 N0¼ N [ 0, if and only if, fn2 Cl

Definition 3 Corresponding to arbitrary order p> 0, for a function

f2 Cl;lP 1, arbitrary order integral is recalled as

IpfðxÞ ¼ 1

CðpÞ

Rx

0ðx lÞp1fðlÞdl;

I0fðxÞ ¼ f ðxÞ:

For f2 Cl;lP 1; p; q P 0andc> 1, we have

IpIqfðxÞ ¼ IpþqfðxÞ;

and

Ipxc¼ Cðcþ 1Þ

p2 ðn  1; n; n ¼ ½p þ 1, the arbitrary derivative in Caputo sense

is provided by

ðDpfÞðxÞ ¼ InpfðnÞðxÞ:

For a power function for order p2 ðn  1; n; n ¼ ½p þ 1, the

arbi-trary derivative in Caputo sense, one has

Dpxk¼ 0 if 06 k 6 ½p;

Cðkþ1Þ

Cð1þkpÞxpþ c if kP ½p þ 1;

(

ð5Þ

where k2 N0

We have the following properties

l Then

DpIpfðxÞ ¼ f ðxÞ;

and

IpDpfðxÞ ¼ f ðxÞ Xn1

i¼0

fðiÞð0þÞxi i!; x P 0: ð6Þ

Derivation of Shifted Jacobi polynomials from fundamental

Jacobi Polynomials

Here we provide fundamental characteristic of the Jacobi

poly-nomials The famous Jacobi polynomialsPð - ; x Þ

the interval½1; 1 as

Pð-; x Þ

i ðyÞ ¼ð-þ x þ2i1Þ½-2  x 2 þyð-þ x þ2i2Þð-þ x þ2i2Þ

2ið-þ x þiÞð-þ x þ2i2Þ Pði1-;xÞðyÞ

ð-þi1Þð x þi1Þð-þ x þ2iÞ

ið-þ x þiÞð-þ x þ2i2Þ Pði2-;xÞðyÞ; i ¼ 2; 3; ;

where Pð-; x Þ

0 ðyÞ ¼ 1; Pð-; x Þ

1 ðyÞ ¼-þ x þ2

2 yþ- x

2 :

ð7Þ

By means of the substitutionyþ12 ¼t

L, we get a revised version of the concerned polynomials called the shifted Jacobi polynomials

over the interval½0; L A general term Qð- ; x Þ

L;i ðtÞ of degree i of the sug-gested polynomials on½0; L, with-> 1;x> 1 is as:

Qð-; x Þ

L ;i ðtÞ ¼Xi

n¼0

ð1ÞinCði þxþ 1ÞCði þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCði þ-þxþ 1Þði  nÞ!n!Lntn; ð8Þ

where

Qð-; x Þ

L ;i ð0Þ ¼ ð1ÞiCði þxþ 1Þ

Cðxþ 1Þi! ;

and

Qð-; x Þ L;i ðLÞ ¼Cði þ-þ 1Þ

Cð-þ 1Þi! :

Result regarding orthogonality of the said polynomials is

Z L 0

Qð-; x Þ

L ;i ðtÞQð-; x Þ

L ;j ðtÞWð-; x Þ

L ðtÞdt ¼ Rð-; x Þ

andxð-;x Þ

L ðtÞ ¼ ðL  tÞ-txis the weight function, and

Rð-; x Þ

0 if i– j;



such that

hi¼ L

-þ x þ1Cði þ-þ 1ÞCði þxþ 1Þ ð2i þ-þxþ 1Þi!Cði þ-þxþ 1Þ : ð10Þ

Here for the readers we provide few special cases from shifted Jacobi basis as:

ðiÞ LL;iðtÞ ¼ Qð0;0Þ

sit-ting-¼x¼ 0 in(8) ðiiÞ TL;iðtÞ ¼Cðiþ1ÞCðiþC1 Þð1ÞQð12 ; 1

2 Þ

2 in(8) ðiiiÞ In same line one has UL;iðtÞ ¼Cðiþ2ÞCðiþC3 Þð1ÞQðL;i1;1ÞðtÞ, is known as

(8)

ðivÞ Also if we sit-¼xin(8)we get shifted Gegenbauer (Ultra-spherical) polynomials as

G

-L ;iðtÞ ¼Cði þ 1ÞCða þ1Þ

Cði þ a þ1Þ Qð-1;x  1 Þ

ðvÞ Further if one sit-¼1; x¼1

shifted Chebyshev polynomials as

VL ;iðtÞ ¼ðCð2i þ 1ÞÞ!

ðCð2i  1ÞÞ!Q

ð 1 ; 1

2 Þ

L ;i ðtÞ:

ðviÞ sitting -¼1

Chebyshev polynomials as

WL ;iðtÞ ¼ðCð2i þ 1ÞÞ!

ðCð2i  1ÞÞ!Q

ð 1

2 ; 1 Þ

L ;i ðtÞ:

Here we claim that performing numerical computation with shifted Jacobi polynomials means that the above special cases are also considered Some time the shifted jacobi polynomials are also called hypergeometric polynomials which constitute a big class of orthogonal polynomials These polynomials are orthogonal with respect to some weight function, for more detail (see[19])

Assume that UðtÞ is a square integrable function with respect to the weight functionxð-;x Þ

terms of shifted Jacobi polynomials as

UðtÞ ¼X1 j¼0

DjQð-; x Þ

L ;j ðtÞ;

shifted Jacobi polynomials of two variable instead of one (see[2])

Definition 5 Let fQð-;xÞ

L ;i ðtÞg1

shifted Jacobi polynomials on½0; L The notions fQð-;xÞ

L ;i;j ðt; xÞg1i;j¼0for two variable shifted Jacobi polynomials which are defined on

½0; L  ½0; L by

Qð-; x Þ

L ;i;j ðt; xÞ ¼ Qð-; x Þ

L ;i ðtÞQð-; x Þ

L ;j ðxÞ; ðt; xÞ 2 ½0; L  ½0; L:

The family fQð - ; x Þ

L;i;j ðt; xÞg1i;j¼0 is orthogonal with respect to the weighted function

Wð-; x Þ

L ðt; xÞ ¼ Wð-; x Þ

L ðtÞWð-; x Þ

L ðxÞ; ðt; xÞ 2 ½0; L  ½0; L:

RL

0

RL

0Qð - ; x Þ L;i;j ðt; xÞQð - ; x Þ

L;k;l ðt; xÞWð - ; x Þ

L ðt; xÞdtdx

¼RL

0

RL

0Qð - ; x Þ

L;i ðtÞQð - ; x Þ

L;j ðxÞQð - ; x Þ L;k ðtÞQð - ; x Þ L;l ðxÞWð - ; x Þ

L ðtÞWð - ; x Þ

¼ RL

0Qð - ; x Þ

L;i ðtÞQð - ; x Þ

L;k ðtÞWð - ; x Þ

 RL

0Qð - ; x Þ L;j ðxÞQð - ; x Þ L;l ðxÞWð - ; x Þ

¼ Rð - ; x Þ L;k Rð - ; x Þ L;l ; where

Rð-; x Þ

L ;k Rð-; x Þ

L ;l ¼ hihj if ði; jÞ ¼ ðk; lÞ;

0 otherwise:



assume that a square integrable function Uðt; xÞ with respect to the

weight functionWð - ; x Þ

considered polynomials as

Uðt; xÞ ¼X1

i¼0

X1

j¼0

Di;jQð-; x Þ

where the notions Di;jare Jacobi coefficients provided by

Di;j¼ 1

hihj

Z L

0

Z L

0

Qð-; x Þ

L;i;j ðt; xÞUðt; xÞWð-; x Þ

L ðt; xÞdtdx: ð12Þ

expressed as:

Uðt; xÞ ’ Ukðx; yÞ ¼Xk1

i¼0

Xk1 j¼0

Di ;jQð-; x Þ

L ;i;j ðt; xÞ ¼ HT

k 2Uk2ðt; xÞ;

where

HTk2¼ ðD0 ;0; D0 ;1; ; D0 ;k1; ; Dk 1;0; Dk 1;1; ; Dk 1;k1Þ

and

Uk 2ðt; xÞ ¼ Qð-; x Þ

L;0;0ðt; xÞ; Qð-; x Þ

L;0;1ðt; xÞ; ; Qð-; x Þ

L ;0;k1ðt; xÞ; ;



Qð-; x Þ

L ;k1;0ðt; xÞ; Qð-; x Þ

L ;k1;1ðt; xÞ; ; Qð-; x Þ

L ;k1;k1ðt; xÞT:

ð13Þ

Construction of required matrices corresponding to arbitrary

order derivatives and integrals

Here in this part, letN¼ f0; 1; ; k  1g, some results are: For

p> 0 and i; j; a; b 2N, let

dj; b¼ 1 if b¼ j;

0 if b– j



and

Wa ;bði; jÞ ¼ uma

n¼0Da;n;pGi ;j;b;

where

an

Cða þxþ 1ÞCða þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCða þ-þxþ 1Þða  nÞ!Cðp þ n þ 1Þ

and

Gi;j;b¼ di;b

Xi l¼0

ð1ÞilCði þ l þ-þxþ 1Þ

Cðl þxþ 1Þði  lÞ Cðn þ p þ l þxþ 1Þ

Cðn þ p þ l þxþ-þ 2Þ

ð2i þ-þxþ 1Þi!Lp

Cði þ-þ 1Þ :

Keeping in mind the above definitions, notions, one has the results presented here as:

IptðUk2ðt; xÞÞ ’ Mp

k2k 2Uk2ðt; xÞ; ðt; xÞ 2 ½0; L  ½0; L; ð14Þ

where Ip

t is the Riemann–Liouville fractional integral of order p> 0 with respect to the variable time t, and Mp

k2k 2is the square matrix

of size k2, given by

Mp

k2k 2¼ ðMp

v;rÞ16v;r6k2;

with

Mp

v;r¼ Wa;bði; jÞ; v¼ ka þ b þ 1; r ¼ ki þ j þ 1; i; j; a; b 2N:

Proof

Letða; bÞ be a fixed pair of positive integers such that a; b 2N Then

Ip

tQð-; x Þ L;a;b ðt; xÞ ¼ Ip

tQð-; x Þ

L ;a ðtÞ

Qð-; x Þ L;b ðxÞ:

On the other hand, we have

Ip

tQð-; x Þ L;a ðtÞ ¼Xa

n¼0

ð1ÞanCða þxþ 1ÞCða þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCða þ-þxþ 1Þða  nÞ!n!Ln:

FromProperty (4), we obtain

Ipttn¼ n!

Cðp þ n þ 1Þtnþp;

which yields

IptQð - ; x Þ L;a ðtÞ ¼Xa n¼0

ð1ÞanCða þxþ 1ÞCða þ n þ-þxþ 1Þ Cðn þxþ 1ÞCða þ-þxþ 1Þða  nÞ!Ln

Cðp þ n þ 1Þtnþp: Therefore, we have

Qð-; x Þ L;a;b ðt; xÞ ¼X

a

n¼0

Da ;n;p

Ln tnþpQð-; x Þ

Approximating tnþpQð - ; x Þ

tnþpQð-; x Þ

L ;b ðxÞ ’Xk1

i¼0

Xk1 j¼0

Si ;j;bQð-; x Þ L;i ðtÞQð-; x Þ

where

Si;j;b¼ 1

hihj

Z L 0

Z L 0

Qð-; x Þ

L ;i;j ðt; xÞtnþpQð-; x Þ

L ;b ðxÞWð-; x Þ

L ðt; xÞdtdx:

On the other hand, we have

Si;j;b¼ 1

hihj

ZL 0

tnþpQð - ; x Þ L;i ðtÞWð - ; x Þ

L ðtÞdt

0 Qð - ; x Þ L;j ðxÞQð - ; x Þ L;b ðxÞWð - ; x Þ

L ðxÞdx

Si ;j;b¼ dj;b h

Z L

tnþpQð-; x Þ

L ;i ðtÞWð-; x Þ

L ðtÞdt

:

Further, we have

ZL

0

tnþpQð - ; x Þ

L;i ðtÞWð - ; x Þ

L ðtÞdt ¼Xi

l¼0

ð1ÞilCði þxþ 1ÞCði þ l þ-þxþ 1Þ Cðl þxþ 1ÞCði þ-þxþ 1Þði  lÞ!l!Ll



ZL 0

tnþpþlþ xðL  tÞ-dt:

L, we obtain

RL

0tnþpþlþxðL  tÞ-dt¼ Lnþpþlþ x þ1RL

0sðnþpþlþx þ - þ1Þ1ð1  sÞð - þ1Þ1

ds

¼ Lnþpþlþ x þ - þ1Bðn þ p þ l þxþ 1;-þ 1Þ;

where B is the beta function Next, using the property

Bðx; yÞ ¼CðxÞCðyÞ

Cðx þ yÞ ; x> 0; y > 0;

we obtain

Z L

0

tnþpþlþxðL  tÞ-dt¼ Lnþpþlþ x þ-þ1

Cðn þ p þ l þxþ 1ÞCð-þ 1Þ

Cðn þ p þ l þxþ-þ 2Þ :

Hence,

RL

0tnþpQð-; x Þ

L ;i ðtÞWð-; x Þ

L ðtÞdt ¼ Xi

l¼0

ð1Þ il Cðiþ x þ1ÞCðiþlþ-þ x þ1Þ Cðlþ x þ1Þ Cðiþ-þ x þ1ÞðilÞ!l!

Cðnþpþlþ x þ1ÞCð-þ1Þ Cðnþpþlþ x þ-þ2Þ Lnþpþlþxþ-þ1;

which yields

Si ;j;b¼ d i;b

h i

Xi

l¼0

ð1Þ il Cðiþ x þ1ÞCðiþlþ-þ x þ1Þ

Cðlþ x þ1ÞCðiþ-þ x þ1ÞðilÞ!l!

Cðnþpþlþ x þ1ÞCð-þ1Þ

Cðnþpþlþ x þ-þ2Þ Lnþpþlþxþ-þ1:

Si;j;b¼ Ln

Gi;j;b:

IptQð-; x Þ

L ;a;b ðt; xÞ ’X

a

n¼0

Da ;n;p

Xk1 i¼0

Xk1 j¼0

Gi ;j;bQð-; x Þ

L ;i ðtÞQð-; x Þ

L ;j ðxÞ;

that is,

Ip

tQð-; x Þ

L ;a;b ðt; xÞ ’Xk1

i¼0

Xk1 j¼0

Xa ;bði; jÞQð-; x Þ

L ;i;j ðt; xÞ;

which yields(16)

For p> 0 and i; j; a; b 2N, let

di ; a¼ 1 if a¼ i;

0 if a– i



and

X

a ;bði; jÞ ¼X

b

n¼0

Db;n;pGi;j;a;

where

bn

Cðb þxþ 1ÞCðb þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCðb þ-þxþ 1Þðb  nÞ!Cðp þ n þ 1Þ

and

Gi;j;a¼ di;a

Xj

l¼0

ð1Þ jl Cðjþlþ-þ x þ1Þ

Cðlþ x þ1ÞðjlÞ!l!

Cðnþpþlþ x þ1ÞCð-þ1Þ

Cðnþpþlþ x þ-þ2Þ ð2jþ-þ x þ1Þj!L p

Cðjþ-þ1Þ :

obtain the following result

Lemma 3 LetUk 2ðt; xÞ be the vectorial function defined by (13) Then

IpxðUk2ðt; xÞÞ ’ Np

k 2 k 2Uk2ðt; xÞ; ðt; xÞ 2 ½0; L  ½0; L; ð17Þ

where Ipis the Riemann–Liouville fractional integral of order p> 0 with respect to the variable time x, and Np

k 2 k 2is the square matrix

of size k2, given by

Npk2 k 2¼ ðNp

v;rÞ16v;r6k2;

with

Npv;r¼X

a ;bði; jÞ; v¼ ka þ b þ 1; r ¼ ki þ j þ 1; 0 6 i; j; a; b

6 k  1:

For p> 0 and i; j; a; b 2N, let

Wa ; bði; jÞ ¼

0 if a¼ 0; 1; ; ½p;

Xa n¼½pþ1

if a¼ ½p þ 1; ½p þ 2; ; k  1;

8

>

>

where

anCða þxþ 1ÞCða þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCða þ-þxþ 1Þða  nÞ!Cð1 þ n  pÞ

and

Ii;j;b¼ dj;b

Xi l¼0

ð1Þ il Cðiþlþ-þ x þ1Þ Cðlþ x þ1ÞðilÞ!l! Cðnpþlþ x þ1Þ Cð-þ1Þ

Cðnpþlþ x þ-þ2Þ ð2iþ-þ x þ1Þi!

Cðiþ-þ1ÞL p :

The following result holds

Lemma 4 LetUk 2ðt; xÞ be the vectorial function defined by(13) Then

DptðUk2ðt; xÞÞ ’ Rp

k2k 2Uk2ðt; xÞ; ðt; xÞ 2 ½0; L  ½0; L; ð18Þ

k2k 2is the square matrix of size k2, given by

Rp

k2k 2¼ ðRp

v;rÞ16v;r6k2;

with

Rpv;r¼ Wa ;bði; jÞ; v¼ ka þ b þ 1; r ¼ ki þ j þ 1;

06 i; j; a; b 6 k  1:

Proof

a; b 2 f0; 1; ; k  1g Then

Dp

tQð-; x Þ L;a;b ðt; xÞ ¼ Dp

tQð-; x Þ

L ;a ðtÞ

Qð-; x Þ L;b ðxÞ:

On the other hand, we have

Dp

tQð-; x Þ

L ;a ðtÞ ¼Xa

n¼0

ð1ÞanCða þxþ 1ÞCða þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCða þ-þxþ 1Þða  nÞ!n!LnDp

ttn:

We consider two cases

Case.1 a¼ 0; 1; ; ½p In this case, from(1), we have

Dptn¼ 0; n ¼ 0; 1; 2; 3; ; a:

DptQð-; x Þ

Case.2 a¼ ½p þ 1; ½p þ 2; ; k  1 In this case, from (1), we

have

Dp

ttn¼ 0; n ¼ 0; 1; 2; 3; ; ½p

and

Dpttn¼ Cðn þ 1Þ

Cð1 þ n  pÞtnp; n ¼ ½p þ 1; ½p þ 2; ; a:

Therefore,

Dp

tQð - ; x Þ

L;a;b ðt; xÞ

¼ Xa

n¼½pþ1

ð1ÞanCða þxþ 1ÞCða þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCða þ-þxþ 1Þða  nÞ!Cð1 þ n  pÞLntnpQð - ; x Þ

L;b ðxÞ:

Then, we obtain

DptQð-; x Þ

L ;a;b ðt; xÞ ¼ X

a

n¼½pþ1

Da;n;p

Ln t

npQð-; x Þ

Approximating tnpQð - ; x Þ

one has

tnpQð-; x Þ

L ;b ðxÞ ’Xk1

i¼0

Xk1 j¼0

Si ;j;bQð-; x Þ

L ;i ðtÞQð-; x Þ

where

Si ;j;b¼ 1

hihj

Z L

0

Z L

0 Qð-; x Þ

L ;i;j ðt; xÞtnpQð-; x Þ

L ;b ðxÞWð-; x Þ

L ðt; xÞdtdx:

On the other hand, we have

Si;j;b¼ 1

hihj

ZL

0

tnpQð - ; x Þ

L;i ðtÞWð - ; x Þ

L ðtÞdt

0

Qð - ; x Þ L;j ðxÞQð - ; x Þ L;b ðxÞWð - ; x Þ

L ðxÞdx

Si ;j;b¼ dj ;b

hi

Z L

0

tnpQð-; x Þ

L;i ðtÞWð-; x Þ

L ðtÞdt

:

Further, we have

ZL

0

tnpQð - ; x Þ

L;i ðtÞWð - ; x Þ

L ðtÞdt ¼Xi

l¼0

ð1ÞilCði þxþ 1ÞCði þ l þ-þxþ 1Þ Cðl þxþ 1ÞCði þ-þxþ 1Þði  lÞ!l!Ll



ZL 0

tnpþlþxðL  tÞ-dt:

L, one has

Z L

0

tnpþlþxðL  tÞ-dt¼ Lnpþlþ x þ-þ1

Cðn  p þ l þxþ 1ÞCð-þ 1Þ

Cðn  p þ l þxþ-þ 2Þ :

Hence,

RL

0tnpQð-; x Þ

L ;i ðtÞWð-; x Þ

L ðtÞdt ¼ Xi

l¼0

ð1Þ il Cðiþ x þ1ÞCðiþlþ-þ x þ1Þ Cðlþ x þ1ÞCðiþ-þ x þ1ÞðilÞ!l!

Cðnpþlþ x þ1ÞCð-þ1Þ Cðnpþlþ x þ-þ2Þ Lnpþlþxþ-þ1;

which yields

Si ;j;b¼ d i;b

h i

Xi

l¼0

ð1Þ il Cðiþ x þ1ÞCðiþlþ-þ x þ1Þ

Cðlþ x þ1ÞCðiþ-þ x þ1ÞðilÞ!l!

Cðnpþlþ x þ1ÞCð-þ1Þ

Cðnpþlþ x þ-þ2Þ Lnpþlþxþ-þ1:

Si;j;b¼ LnIi;j;b:

DptQð-; x Þ

L ;a;b ðt; xÞ ’X

k1 i¼0

Xk1 j¼0

Xa n¼½pþ1

Da ;n;pIi ;j;bQð-; x Þ

L ;i;j ðt; xÞ;

that is,

DptQð-; x Þ

L ;a;b ðt; xÞ ’X

k1 i¼0

Xk1 j¼0

Wa ;bði; jÞQð-; x Þ

L ;i;j ðt; xÞ: ð22Þ

Finally,(19) and (22)yield(18) For p> 0 and i; j; a; b 2N, let

la ;bði; jÞ ¼

0 if b¼ 0; 1; ; ½p;

Xb n¼½p

Db;n;pIj;i;a if b¼ ½p þ 1; ½p þ 2; ; k  1;

8

>

>

where

Db ;n;p¼ ð1ÞbnCðb þxþ 1ÞCðb þ n þ-þxþ 1Þ

Cðn þxþ 1ÞCðb þ-þxþ 1Þðb  nÞ!Cð1 þ n  pÞ

and

Ii ;j;a¼ di ;a

Xj l¼0

ð1Þ jl Cðjþlþ-þ x þ1Þ Cðlþ x þ1ÞðjlÞ!l!

Cðnpþlþ x þ1Þ Cð-þ1Þ Cðnpþlþ x þ-þ2Þ

ð2jþ-þ x þ1Þj!

Cðjþ-þ1ÞL p :

Following the same arguments used in the proof of Lemma, we obtain the following result

Lemma 5 LetUk2ðt; xÞ be the vectorial function defined by(13) Then

DpxðUk2ðt; xÞÞ ’ Sp

k2k 2Uk2ðt; xÞ; ðt; xÞ 2 ½0; L  ½0; L; ð23Þ

where Sp

k 2 k 2is the square matrix of size k2, given by

Sp

k2k 2¼ ðSp

v;rÞ16v;r6k2;

with

Sp

v;r¼la ;bði; jÞ; v¼ ka þ b þ 1; r ¼ ki þ j þ 1; 0 6 i; j; a; b 6 k  1:

General algorithm for numerical results

In this section, using the previous obtained results, the problem

a certain algebraic equation Let 1< p < 2 We write Dp

tuðt; xÞ in the form:

Dp

tuðt; xÞ ¼ HT

k 2Uk 2ðt; xÞ; ð24Þ

where function vectorUk2ðt; xÞ is given in(13)and unknown matrix

HT

k2with size 1 k2

Thus one has

IptðDp

tuðt; xÞÞ ¼ HT

k2IptðUk 2ðt; xÞÞ:

uðt; xÞ ¼ uð0; xÞ þ tutð0; xÞ þ HT

k 2Mp

k2k 2Uk2ðt; xÞ;

uðt; xÞ ¼ hðxÞ þ t/ðxÞ þ HT

k 2Mp

k2k 2Uk2ðt; xÞ:

form:

hðxÞ þ t/ðxÞ ¼ ZTk2Mk2;

where ZT

k2is a matrix of size 1 k2

The coefficients of the matrix ZT

k2

uðt; xÞ ¼ ðHT

k2Mpk2 k 2þ ZT

k2ÞUk 2ðt; xÞ: ð25Þ

Similarly, we may write gðt; xÞ in the form:

gðt; xÞ ¼ QT

k 2Uk2ðt; xÞ; ð26Þ

k 2is a matrix of size 1 k2

Now, using(1), (24), (25), and (26), we obtain

Dpuðt; xÞ ¼1

s Q

T

2Uk 2ðt; xÞ j HT

2Mp

k 2 k 2þ ZT

2

Uk 2ðt; xÞ  HT

2Uk 2ðt; xÞ

; that is,

Dpxuðt; xÞ ¼1

s Q

T

k2j HTk2Mp

k2k 2þ ZT

k2

 HT

k2

Uk2ðt; xÞ;

Next, we obtain

IpxðDp

xuðt; xÞÞ ¼1

s Q

T

k2j HTk2Mpk2 k 2þ ZT

k2

 HT

k2

IpxðUk 2ðt; xÞÞ

uðt; xÞ ¼ uðt; 0Þ þ uxðt; 0Þx

þ1

s Q

T

k2j HTk2Mp

k2k 2þ ZT

k2

 HT

k2

Np

k2k 2Uk2ðt; xÞ;

uðt; xÞ ¼1

s Q

T

k 2j HT

k2Mp

k 2 k 2þ ZT

k2

 HT

k2

Np

k 2 k 2Uk 2ðt; xÞ: ð27Þ

HT

k 2Mp

k2k 2þ ZT

k 2¼1

s Q

T

k 2j HT

k 2Mp

k2k 2þ ZT

k 2

 HT

k 2

Np

k2k 2;

which yields the algebraic equation

HT

A¼ Mp

k2k 2þ1

sðjM

p

k2k 2þ Ik2 k 2ÞNp

k2k 2

and B is the matrix of size 1 k2

given by

B¼1

sðQ

T

k 2jZTk2ÞNp

k2k 2 ZT

k 2:

Here, Ik2 k 2denotes the identity matrix of size k2 The algebraic Eq

(28)is equivalent to a system of k2linear equations with k2

vari-ables, which can be solved using Matlab Finally, after solving

(28), the numerical solution to(1)–(3)can be computed using(25)

Numerical experiments

This portion is devoted to present a test problem Therefore,

consider the given problem as

D1t:5uðt; xÞ þ D1 :5

x uðt; xÞ ¼ gðt; xÞ; ðt; xÞ 2 ½0; 1  ½0; 1; ð29Þ

under the initial conditions

uð0; xÞ ¼ utð0; xÞ ¼ 0 ð30Þ

and the mixed boundary conditions

uðt; 0Þ ¼ uxðt; 0Þ ¼ 0; ð31Þ

where the source term gðt; xÞ is given by

gðt; xÞ ¼Cð1:5Þ2 ðx2

ffiffi t

p

þ t2 ffiffiffi x

p Þ; ðt; xÞ 2 ½0; 1  ½0; 1: ð32Þ

uðt; xÞ ¼ t2x2; ðt; xÞ 2 ½0; 1  ½0; 1:

Forðt; xÞ 2 ½0; 1  ½0; 1, we denote by Eðt; xÞ the absolute error at the pointðt; xÞ, that is,

Eðt; xÞ ¼ juðt; xÞ  uðt; xÞj; ðt; xÞ 2 ½0; 1  ½0; 1:

ð-;xÞ ¼ ð0; 0Þ are shown inTable 1 The absolute errors at different pointsðt; xÞ in the case k ¼ 4 and

ð-;xÞ ¼ ð0:5; 1Þare shown inTable 2 Observe that in both cases, at

equal to the exact solution with a negligible amount of absolute error

Next, we fixð-;x; kÞ ¼ ð0; 0; 4Þ, we compare our result with the

t¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ¼ 0:75, and display the result inFig 1

is shown byFig 2, the obtained result is satisfactory

Now, in order to check the stability of the approximated solu-tion, a perturbation term is introduced in the source function

perturbed source gðt; xÞ given by

gðt; xÞ ¼ gðt; xÞ þtx; ðt; xÞ 2 ½0; 1  ½0; 1; ð33Þ

Table 1 Absolute errors in the case ð-;x; kÞ ¼ ð0; 0; 4Þ.

Table 2 Absolute errors in the case ð-;x; kÞ ¼ ð0:5; 1; 4Þ.

where> 0 We denote by u the numerical solution of the

per-turbed problem Forðt; xÞ 2 ½0; 1  ½0; 1, we denote by Eðt; xÞ the

absolute error at the pointðt; xÞ, that is

Eðt; xÞ ¼ juðt; xÞ  uðt; xÞj; ðt; xÞ 2 ½0; 1  ½0; 1;

where u is the approximate solution without noise (the

The absolute errors Eðt; xÞ for¼ 0; 0:1 at different points ðt; xÞ

absolute errors Eðt; xÞ for ¼ 0; 0:05 at different points ðt; xÞ in the case k¼ 4 and ð-;xÞ ¼ ð0; 0Þ are shown inTable 4

pointsðt; xÞ, we have Eðt; xÞ <, which confirms the stability of the method with respect to a perturbation of the source data

ðt ¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ¼ 0:75Þ in the case ð-;x; kÞ ¼ ð0; 0; 4Þ Similarly inFig 2, the exact and approximate solutions at different values of t that is ðt ¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ¼ 0:75Þ in the case

Fig 1 Exact and approximate solutions at different values of t that is ðt ¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ¼ 0:75Þ in the case ð-;x; kÞ ¼ ð0; 0; 4Þ.

Fig 2 Exact and approximate solutions at different values of t that is ðt ¼ 0:1; t ¼ 0:25; t ¼ 0:5; t ¼ 0:75Þ in the case ð-;x; kÞ ¼ ð0; 0:1; 4Þ.

Table 3

Absolute errors in the case ð-;x; k;Þ ¼ ð0; 0; 4; 0:01Þ.

Table 4 Absolute errors in the case ð-;x; k;Þ ¼ ð0; 0; 4; 0:05Þ.

ð-;x; kÞ ¼ ð0; 0:1; 4Þ are presented In both cases the effect of time

and the parameters values have testified At takingð-;xÞ ¼ ð0; 0Þ

for parameters, we get the solution more precise as compare to

ð-;xÞ ¼ ð0; 0:1Þ at same scale k ¼ 4 Further for more explanation,

we give comparison between exact and approximate solution in

Fig 3by usingð-;xÞ ¼ ð0; 0:1Þ at same scale k ¼ 4, to the given

problem We see that both surfaces coincide very well which

illus-trate the accuracy of the considered method

Conclusion

The suggested method provides an easy way to solve

numeri-cally the class of fractional partial differential Eqs.(1)–(3) Using

shifted Jacobi polynomial basis, the considered problem is reduced

to a system of linear algebraic equations which has been solved by

Matlab using Gauss elimination method for the unknown

coeffi-cient matrix which then used to obtained the required numerical

solution of the considered problem Moreover, from numerical

experiments, we observed that the method is stable with respect

to a perturbation of the source data In future, the method can be

easily extended to solve other types of fractional partial differential

equations from physics and other fields of science

Compliance with Ethics Requirements

Our research work does not contain any studies with human or

animal subjects

Declaration of Competing Interest

The authors declare that there are no conflicts of interest

regarding the publication of this paper

Acknowledgments

We are thankful to the reviewer for their nice suggestions

which improved this paper very well

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