an efficient class of fdm based on hermite formula for solving fractional reaction-subdiffusion...

7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION

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International Journal of Mathematics and Computer

Applications Research (IJMCAR)

ISSN 2249-6955Vol. 2 Issue 4 Dec 2012 61-75

TJPRC Pvt. Ltd.,

AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FORSOLVING

FRACTIONALREACTION-SUBDIFFUSIONEQUATIONS

N. H. SWEILAM1, M. M. KHADER

2& M. ADEL

1

1Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

2Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

ABSTRACT

In this article, a numerical study is introduced for the fractional reaction-subdiffusion equations by using an

efficient class of finite difference methods (FDM). The proposed scheme is based on Hermite formula. The stability

analysis and the convergence of the proposed methods are given by a recently proposed procedure similar to the standard

John von Neumann stability analysis. Simple and accurate stability criterion valid for different discretization schemes of

the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked

numerically. Finally, numerical examples are carried out to confirm the theoretical results.

KEYWORDS: Finite Difference Methods, Hermite Formula, Fractional Reaction-Subdiffusion Equation, Stability and

Convergence Analysis

INTRODUCTION

In the last few years, there are many studies for the fractional differential equations (FDEs), because of their

important applications in many areas like physics, medicine and engineering, and this field allows us to study fractal

phenomena which can not be studied by the ordinary case. There are many applications of the FDEs see (Chang Chen et al.

(2007) - Hilfer (2000), Liu (2004), Metzler (2000)), the studied models have received a great deal of attention like in the

fields of viscoelastic materials (Bagley (1999)), control theory (Podlubny (1999)), advection and dispersion of solutes in

natural porous or fractured media (Benson et al. (2000)), anomalous diffusion. Most FDEs do not have exact solutions, so

approximate and numerical techniques must be used. Recently, several numerical methods to solve FDEs have been given

such as, variational iteration method (Sweilam et al.(2007)), homotopy perturbation method (Khader (2012)), Adomian

decomposition method (Yu (2008)), homotopy analysis method (Sweilam and Khader (2011)), collocation method (Khader

(2007), Sweilam and Khader (2010)) and finite difference method (Sweilam et al. (2011), Sweilam et al. (2012)).

In this section, we employed the Caputo definitions of fractional derivative operatorD and its properties (Liu et

al. (2007a), Metzler (2000), Podlubny (1999)) .

DEFINITION 1

The Riemann-Liouville derivative of order of the function )(xy is defined by

0,>0,>,)(

)(

)(

1=)(

10

xd

x

y

dx

d

nxyD

n

x

n

n

x +

where n is the smallest integer exceeding and (.) is the Gamma function. If Nm= , then the above definition

coincides with the classicalthm derivative ).(

)(xy m


2/15

62 N. H. Sweilam, M. M. Khader & M. Adel

FRACTIONAL REACTION-SUBDIFFUSION EQUATION

The standard meanfield model for the evolution of the concentrations ),( txa and ),( txb ofA and B particles

is given by the reaction-diffusion equations

),,(),(),(=),( 2

2

txbtxatxaxDtxat

(1)

),,(),(),(=),(2

2

txbtxatxbx

Dtxbt

(2)

where D is the diffusion coefficient assumed in this paper to equal for species and is the rate constant for the

bimolecular reaction.

In order to generalize the reaction-diffusion problem to a reaction-subdiffusion problem, we must deal with the

subdiffusive motion of the particles. Seki et al. (2003) and Yuste et al. (2004) replaced Eqs. (1) and (2) with the set of

reaction-subdiffusion equations in which both the motion and the reaction terms are affected by the subdiffusive character

of the process

)],,(),(),([=),(2

21 txbtxatxa

xkDtxa

tt

(3)

)],,(),(),([=),(2

21 txbtxatxb

xkDtxb

tt

(4)

where k is the generalized diffusion coefficient andtD is the Riemann-Liouville fractional partial derivative of order

with respect to t. The fractional reaction-subdiffusion equations (3) and (4) are decoupled, which are equivalent to

solve the following fractional reaction-subdiffusion equation

,


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An Efficient Class of FDM Based on Hermite Formula forSolving 63FractionalReaction-SubdiffusionEquations

FINITE DIFFERENCE SCHEME OF THE FRACTIONAL REACTION-SUBDIFFUSION

EQUATION

In this section, we introduce an efficient class of FDM and use it to obtain the discretization finite difference

formula of the reaction-subdiffusion equation (6). For some positive constant numbers Nand M , we use the notations

x and t , at space-step length and time-step length, respectively. The coordinates of the mesh points are xjxj =

)0,1,...,=( Nj , and tmtm = , ( = 0,1,..., )m M and the values of the solution ),( txu on these grid points are

;UmJ

m

jmj utxu ),( whereN

Lhx == ,

M

Tt == .

For more details about discretization in fractional calculus see (Lubich (1986), Richtmyer et al. (1967), Yuste and

Acedo (2005)).

In the first step, the ordinary differential operators are discretized as follows (Zhang (2006)).

),(

)2

1(1

1=| ,

O

u

t

u

t

mjt

mt

jx +

+

+

(9)

where+

t denotes the backward difference operator with respect to t. Now, using the formula (9), we can derive an

efficient approximate formula of the fractional derivative for ),( txut

, 1)


4/15


The above formula can be rewritten as

1

,

=1

| = ( ),1

(1 )2

m rmt j

x t rj m

rt

uuA O

t

+ +

+

+

(10)

where)(2

=

A and 11 1)(= rrr .

We must note thatr

satisfy the following facts

(1) 0>r , 1,2,...=r .

(2) 1> +rr , 1,2,...=r .

Now, we are going to obtain the finite difference scheme of the fractional reaction-subdiffusion equation (6). In our study

we take 1== k .

To achieve this aim we evaluate this equation at the points of the grid ),(mj

tx ,

).,(),(),(

=),(2

2

mjmj

mj

mj txgtxux

txutxu

t+

(11)

Using Eqs.(10) and (11), we have

).(),(),(

)2

1(1

=),(

1

1=2

2

Otxgtxu

uA

x

txumjmj

t

rm

jt

r

m

r

mj++

+

++

(12)

In order to get two additional equations, replace j by 1j and 1+j , respectively, in the above equation, so we have

),(),(),(

)2

1(1

=),(

11

1

1

1=2

1

2

Otxgtxu

uA

x

txumjmj

t

rm

jt

r

m

r

mj++

+

+

+

(13)

and

).(),(),(

)2

1(1

=),(

11

1

1

1=2

1

2

Otxgtxu

uA

x

txumjmj

t

rm

jt

r

m

r

mj++

++

+

+

+

+

+ (14)

The Hermite formula with two-order derivatives at the grid point ),( mj tx is

).(=)),(),(2),((12),(),(

10),(

4

1122

1

2

2

2

2

1

2

hOtxutxutxuhx

txu

x

txu

x

txumjmjmj

mjmjmj

+

++

+

+

(15)

Substitute from Eqs.(12)-(14) into Eq.(15), and denote ),( mj txu bym

ju , ),( mj txg bym

jg , then with neglect the high

order terms, we have


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,)(12)52(12)(12=]

)21(1

[

]

)2

1(1

[10]

)2

1(1

[

1

22

1

2

1

1

1

1=

2

1

1=

2

1

1

1

1=

2

m

j

m

j

m

j

m

j

t

rm

jt

r

m

r

m

j

t

rm

jt

r

m

r

m

j

t

rm

jt

r

m

r

uhuhuhgu

Ah

gu

Ahgu

Ah

+++

+

+

+

+

++

+

+

+

++

+

+

(16)

under some simplifications, we can obtain the following form

.)12

1(1)

6

5(2)

12

1(1

)12

1

(1)6

5

(2)12

1

(1=)(12

1

)(6

5)(

12

1)(

6

1

)(3

5)(

6

1

)12

1

6

5

12

1(2)

12

1

6

5

12

1(2

1

1

2121

1

2

1

22

1

21

11

2

121

11

2

1

1

12=

2

1

2=

2

1

1

12=

2

1

1

11

1

2

11

2

+

+

++

+

+

+

+

+

+

+

++

++++

++

++

+++++

m

j

m

j

m

j

m

j

m

j

m

j

m

j

m

j

m

j

m

j

m

j

m

j

rm

j

rm

jr

m

r

rm

j

rm

jr

m

r

rm

j

rm

jr

m

r

m

j

m

j

m

j

m

j

m

j

m

j

uhuhuh

uhuhuhggh

gghgghuuAh

uuAhuuAh

uuuAhuuuAh

(17)

Letm

jU be the approximate solution, and letm

j

m

j

m

j UuT = , Nj 1,2,...,= , Mm 1,2,...,= be the error, so

we have the error formula

.)12

1(1)

6

5(2)

12

1(1

)12

1

(1)6

5

(2)12

1

(1=)(6

1

)(3

5)(

6

1

)12

1

6

5

12

1(2)

12

1

6

5

12

1(2

1

1

2121

1

2

1

22

1

2

1

1

12=

2

1

2=

2

1

1

12=

2

1

1

11

1

2

11

2

+

+

+

+

+

+

+

+

+

++

+++

++

+++++

m

j

m

j

m

j

m

j

m

j

m

j

rm

j

rm

jr

m

r

rm

j

rm

jr

m

r

rm

j

rm

jr

m

r

m

j

m

j

m

j

m

j

m

j

m

j

ThThTh

ThThThTTAh

TTAhTTAh

TTTAhTTTAh

(18)

with

.1,2,...,=0,==0 MmTTm

N

m(19)

PROPOSITION 1

Assuming that the solution of (18) has the formjh

m

m

j eT i= , then

2 2 2

12 2 2

2

12 2 2 =2

5 1 1 5( ( )) (1 ) ( ) (1 )6 6 12 12=5 1 5 1

( ( )) (1 ) (1 ) ( )6 6 12 12

5 1( ( ))6 6 ( ),

5 1 5 1( ( )) (1 ) (1 ) ( )6 6 12 12

m m

m

r m r m r

r

cos h h A h cos h h

cos h h A h h cos h

h A cos h

cos h h A h h cos h

+

+ + +

+ + +

+

+ + +

(20)

where m 2= .


6/15


PROOF

Substitute in (18) byjh

m

m

j eT i= and divide by jhe i , we get

0.=)(6

1)(

3

5

)(6

1])

12

1(1)

6

5(2

)12

1[(1])

12

1(1)

6

5(2)

12

1(1[

)212

1

6

5

12

1()2

12

1

6

5

12

1(

i

12=

2

12=

2

i

12=

2

1

i22

i2i22i2

1

2ii2ii

h

rmrmr

m

rrmrmr

m

r

h

rmrmr

m

rm

h

h

m

hh

m

hh

m

hh

eAhAh

eAhehh

ehehheh

AheeAhee

++

+

++

+++

++++

(21)

Using some trigonometric formulae and some simplifications we can obtain

0.=)())(6

1

6

5(

)12

5(1)()

12

1(1))(

6

1

6

5(

)()12

1(1)

12

5(1))(

6

1

6

5

12=

2

1

222

222

rmrmr

m

r

m

m

hcosAh

hhcoshAhhcos

hcoshhAhhcos

+

+

+++

+++

(22)

from which we can obtain the required formula and this completes the proof.

STABILITY ANALYSIS

In this section, we use the John von Neumann method to study the stability analysis of the finite difference scheme (17).

PROPOSITION 2

Assume that

,)(2)(6

22

2

+

h

h(23)

then

1.

)()12

1(1)

12

5(1))(

6

1

6

5(

)12

5(1)()

12

1(1))(

6

1

6

5(

0222

222

+++

+++

hcoshhAhhcos

hhcoshAhhcos

(24)

PROOF

Since )(2=

A , then )(2

1

=

A , and by our assumption that )(2)(6

22

2

+ h

h

, so we can

obtain

,6

212

2

h

h

A +

so,

,3

1

6

1

2

12

2222

AhAhhh +

and since 1)(2

hsin , so we have


7/15


,2

1)()

3

1

6

1(2 22222 hAhhsinAhh +

i.e.,

0,)()

3

1)

12

1(2(1

2

1 22222+ hsinAhhhAh

0,)(3

1)()

12

12(1)

12

5(1)

12

1(1 2222222 ++ hsinAhhsinhhhAh

so,

0,))(2(12

1)

12

12(1)

12

5(1)

12

1(1))](2(1

2

1

3

1[1 2222 ++ hcoshhhhcosAh

i.e.,

0,)]12

5(1))(2)

12

1[(1))(2

6

1

6

5( 222 +++ hhcoshhcosAh

and since

),12

5(1))(2)

12

1(1))(2)

12

1(1)

12

5(1 2222 hhcoshhcoshh ++

we have that

0,)])(2)12

1(1)

12

5[(1))(2

6

1

6

5( 222 +++ hcoshhhcosAh

which completes the proof.

PROPOSITION 3

Assume that m , be the solution of (20), with the condition (23), then |||| 0 m , )1,2,...,=( Mm .PROOF

If we take 1=m in Eq.(20), we obtain

.||

)()12

1(1)

12

5(1))(

6

1

6

5(

)12

5(1)()

12

1(1))(

6

1

6

5(

|=| 0222

222

1

hcoshhAhhcos

hhcoshAhhcos

+++

+++

(25)

From Proposition 2, we have that |||| 01 .

Suppose that |||| 0 s , 11,2,...,= ms . For 1>m

.|

)()12

1(1)

12

5(1))(

6

1

6

5(

)())(6

1

6

5(

)()12

1(1)

12

5(1))(

6

1

6

5(

)12

5(1)()

12

1(1))(

6

1

6

5(

|=||

222

1

2=

2

1222

222

hcoshhAhhcos

hcosAh

hcoshhAhhcos

hhcoshAhhcos

rmrmr

m

r

mm

+++

+

+++

+++

+

(26)


8/15


Now, since

,)(=)( 011

2=121

2=

mrmrrm

rmrmrmr

m

r

+

+(27)

we have

.|

)()121(1)

125(1))(

61

65(

))(6

1

6

5(

)()12

1(1)

12

5(1))(

6

1

6

5(

)())(6

1

6

5(

)()12

1(1)

12

5(1))(

6

1

6

5(

)12

5(1)()

12

1(1)(1))(

6

1

6

5(

|=||

222

0

2

222

1

1

2=

2

1222

22

2

2

hcoshhAhhcos

hcosAh

hcoshhAhhcos

hcosAh

hcoshhAhhcos

hhcoshAhhcos

m

rmrr

m

r

mm

+++

+

+

+++

+

+

+++

+++

+

(28)

Using triangle inequality

.|||

)()12

1(1)

12

5(1))(

6

1

6

5(

))(6

1

6

5(

|

|||)()12

1(1)12

5(1))(6

1

6

5(

)())(6

1

6

5(

|

|||

)()12

1(1)

12

5(1))(

6

1

6

5(

)12

5(1)()

12

1(1)(12))(

6

1

6

5(

|||

0222

2

222

1

1

2=

2

1222

22

2

2

hcoshhAhhcos

hcosAh

hcoshhAhhcos

hcosAh

hcoshhAhhcos

hhcoshAhhcos

m

rm

rr

m

r

mm

+++

+

+

+++

+

+

+++

+++

+

(29)

Now, since |||| 01 m , |||| 0 rm , and also since mrrmr +

2112= =)( , so the above inequality takes the

form

|,||

)()12

1(1)

12

5(1))(

6

1

6

5(

)125(1)()

121(1))(

61

65(

||| 0222

222

hcoshhAhhcos

hhcoshAhhcos

m

+++

+++

(30)

and using Proposition 2 directly we complete the proof.

We know that

.|)(|= 2

=

2

2NT m

N

(31)


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THEOREM 1

The difference scheme (17) is stable under the condition (23).

PROOF

From Proposition 3, (31) and the formula2

0

2TTm , Mm 1,2,..,= , which means that the difference scheme is

stable.

By the Lax equivalence theorem (Richtmyer et al. (1967)) we can show that the numerical solution converges to the exact

solution as 0, h .

NUMERICAL RESULTS

In this section, we will test the proposed method by considering the following numerical examples.

EXAMPLE 1

Consider the initial-boundary value problem of fractional reaction-subdiffusion equation

,01,


10/15


Figure 2: The Behavior of the Exact Solution and the Numerical Solution of (32) by Means of the Proposed Method

at 0.3= ,60

1=h ,

70

1= and 1=T .


at 0.5= ,60

1=h ,

20

1= , and 2=T .

Figure 4: The Behavior of the Numerical Solution of (32) by Means of the Proposed Method at

20

1=h ,

20

1= , and 0.5=T , with different values of


11/15


Figure 5: The Behavior of the Numerical Solution of (32) By Means of the Proposed Method at

40

1=h ,

40

1= , 0.7= , with different values of T

EXAMPLE 2

Consider the following initial-boundary problem of the fractional reaction-subdiffusion equation

1,


12/15



at 0.2= ,60

1=h ,

20

1= , and 0.2=T

Figure 8: The Behavior of the Exact Solution and the Numerical Solution of (33) by Means Of The Proposed

Method at 0.9= ,50

1=h ,

150

1= and 0.01=T

Figure 9: The Behavior of the Numerical Solution of the Proposed Problem (33) by Means of the Proposed Metho

at 0.4= ,40

1=h ,

100

1= , and 1=T


13/15


From the previous figure 9, we can see that the numerical solution is unstable, since the stability condition (32) is not

satisfied.

The following two tables 1 and 2 show the magnitude of the maximum error between the numerical solution and the exact

solution obtained by using the proposed finite difference scheme with different values ofh

,

and the final time

T.

Table 1: The Maximum Error with Different Values of h , at 0.5= , and 0.2=T

h MaximumError

50

1

50

1

0.01602

70

1

80

1

0.01303

120

1

100

1

0.01216

140

1

150

1

0.01118

160

1

180

1

0.01091

Table 2: The Maximum Error with Different Values of h , at 0.2= , and 0.1=T

h MaximumError

10

1

10

1

0.01013

10

1

20

1

0.00290

20

1

20

1

0.00283

20

1

20

1

0.00281

30

1

30

1

0.00023

CONCLUSIONS AND REMARKS

This paper presented a class of numerical methods for solving the fractional reaction-subdiffusion

equations. This class of methods depends on the finite difference method based on Hermite formula. Special attention is

given to study the stability and convergence of the fractional finite difference scheme. To execute this aim we have

resorted to the kind of fractional John von Neumann stability analysis. From the theoretical study we can conclude that,

this procedure is suitable for the proposed fractional finite difference scheme and leads to very good predictions for the

stability bounds. Numerical solution and exact solution of the proposed problem are compared and the derived stability

condition is checked numerically. From this comparison, we can conclude that, the numerical solutions are in excellent

agreement with the exact solution. All computations in this paper are running using the Matlab programming.


14/15


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an efficient class of fdm based on hermite formula for solving fractional reaction-subdiffusion...

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