an efficient class of fdm based on hermite formula for solving fractional reaction-subdiffusion...
Post on 04-Apr-2018
217 Views
Preview:
TRANSCRIPT
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
1/15
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
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
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
,
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
3/15
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)
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
4/15
64 N. H. Sweilam, M. M. Khader & M. Adel
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
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
5/15
An Efficient Class of FDM Based on Hermite Formula forSolving 65FractionalReaction-SubdiffusionEquations
,)(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= .
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
6/15
66 N. H. Sweilam, M. M. Khader & M. Adel
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
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
7/15
An Efficient Class of FDM Based on Hermite Formula forSolving 67FractionalReaction-SubdiffusionEquations
,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)
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
8/15
68 N. H. Sweilam, M. M. Khader & M. Adel
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)
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
9/15
An Efficient Class of FDM Based on Hermite Formula forSolving 69FractionalReaction-SubdiffusionEquations
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,
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
10/15
70 N. H. Sweilam, M. M. Khader & M. Adel
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 .
Figure 3: The Behavior of the Exact Solution and the Numerical Solution of (32) by Means of the Proposed Method
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
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
11/15
An Efficient Class of FDM Based on Hermite Formula forSolving 71FractionalReaction-SubdiffusionEquations
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,
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
12/15
72 N. H. Sweilam, M. M. Khader & M. Adel
Figure 7: The Behavior of the Exact Solution and the Numerical Solution of (33) by Means of the Proposed Method
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
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
13/15
An Efficient Class of FDM Based on Hermite Formula forSolving 73FractionalReaction-SubdiffusionEquations
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.
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
14/15
74 N. H. Sweilam, M. M. Khader & M. Adel
REFERENCES
1. Bagley, R. L., & Calico, R. A. (1999). Fractional-order state equations for the control of viscoelastic damped
structures.J. Guidance Control Dynamics, 14(2), 304-311.
2. Benson, D. A., & Wheatcraft, S. W., & Meerschaert, M. M. (2000). The fractional-order governing equation ofL e vy motion. Water Resour, Res., 36(6), 1413-1424.
3. Chang Chen, M. & Liu, F.& Turner I., & Anh, V. (2007). Fourier method for the fractional diffusion equation
describing sub-diffusion.J. Comp. Phys., 227(2) , 886-897.
4. Gorenflo, R., & Mainardi, F. (1998). Random walk models for space-fractional diffusion processes. Fractional
Calculus Applied Analysis, 1, 167-191.
5. Henry, B. I.,& Wearne, S. L. (2000). Fractional reaction-diffusion. Physica, A 276, 448-455.
6. Hilfer, R. (2000).Applications of Fractional Calculus in Physics. World Scientific, Singapore.
7. Khader, M. M. (2011). On the numerical solutions for the fractional diffusion equation. Communications in
Nonlinear Science and Numerical Simulations, 16, 2535-2542.
8. Khader, M. M. (2012). Introducing an efficient modification of the homotopy perturbation method by using
Chebyshev polynomials.Arab Journal of Mathematical Sciences, 18, 61-71.
9. Khader, M. M., & Sweilam, N. H., & Mahdy, A. M. S. (2011). An efficient numerical method for solving the
fractional diffusion equation.Journal of Applied Mathematics and Bioinformatics, 1, 1-12.
10. Kilbas, A. A., & Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential
Equations. Elsevier, San Diego.
11. Liu, F., & Anh, V., & Turner, I. (2004). Numerical solution of the space fractional Fokker-Planck equation. J.
Comp. Appl. Math. 166, 209-219.
12. Liu, Q., & Liu, F., & Turner, I., & Anh, V. (2007). Approximation of the Le'vy-Feller advection-dispersion
process by random walk and finite difference method.J. Comp. Phys., 222, 57-70.
13. Liu, F., &Zhuang, P., & Anh, V., &Turner, I. , & Burrage, K. (2007). Stability and convergence of the difference
methods for the space-time fractional advection-diffusion equation,Appl. Math. Comp., 191(1), 12-20.
14. Lubich, C. (1986). Discretized fractional calculus. SIAM J. Math. Anal., 17, 704-719.
15. Metzler, R., & Klafter, J. (2000). The random walks guide to anomalous diffusion: a fractional dynamics
approach, Phys. Rep., 339, 1-77.
16. Podlubny, I. (1999). Fractional Differential Equations, Academic Press. San Diego.
17. Richtmyer, R. D., & Morton, K. W. (1967). Difference Methods for Initial-value Problems[M]. Interscience
Publishers, New York.
18. Seki, K., & Wojcik, M., & Tachiya, M. (2003). Fractional reaction-diffusion equation,J. Chem. Phys., 119, 2165-
2174.
19. Sweilam, N. H., & Khader, M. M. (2010). A Chebyshev pseudo-spectral method for solving fractional integro-
-
7/30/2019 AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION
15/15
An Efficient Class of FDM Based on Hermite Formula forSolving 75FractionalReaction-SubdiffusionEquations
differential equations.ANZIAM, 51, 464-475.
20. Sweilam, N. H., & Khader, M. M., & M. Adel. (2012). On the stability analysis of weighted average finite
difference methods for fractional wave equation. Fractional Differential Calculus, 2(1), 17-29.
21. Sweilam, N. H., & Khader, M. M., & Al-Bar, R. F. (2007). Numerical studies for a multi-order fractional
differential equation. Physics Letters A, 371, 26-33.
22. Sweilam, N. H., & Khader, M. M., & Nagy, .A. M. (2011). Numerical solution of two-sided space- fractional
wave equation using finite difference method, Journal of Computational and Applied Mathematics, 235 , 2832-
2841.
23. Sweilam, N. H., & Khader, M. M. (2011). Semi exact solutions for the bi-harmonic equation using homotopy
analysis method.World Applied Sciences Journal, 13, p.(1-7).
24. Yu, Q., & Liu, F., & Anh, V., & Turner, I. (2008). Solving linear and nonlinear space-time fractional reaction-
diffusion equations by Adomian decomposition method, Int. J. Numer. Meth. Eng., 47(1), 138-153.
25. Yuste, S. B. , & Acedo, L. (2005). An explicit finite difference method and a new Von Neumann-type stability
analysis for fractional diffusion equations. SIAM J. Numer. Anal., 42, 1862-1874.
26. Yuste, S. B., & Acedo, L., & Lindenberg, K. (2004). Reaction front in an CBA + reaction-subdiffusion
process, Phys. Rev. E, 69, 036126.
27. Zhang, W. S. (2006). Finite Difference Methods for Partial Differential Equations in Science Computation[M] .
Higher Education Press, Beijin.
28. Zhuang, P., & Liu, F. , & Anh, V., & Turner, I. (2008). New solution and analytical techniques of the implicit
numerical methods for the anomalous sub-diffusion equation, SIAM J. Numer. Anal., 46(2), 1079-1095.
top related