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International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 2 Issue 4 Dec 2012 61-75 © TJPRC Pvt. Ltd.,

AN EFFICIENT CLASS OF FDM BASED ON HERMITE FORMULA FOR SOLVING FRACTIONAL REACTION-SUBDIFFUSION EQUATIONS N. H. SWEILAM1, M. M. KHADER2 & M. ADEL1 1 2

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

Department 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)). α

In this section, we employed the Caputo definitions of fractional derivative operator D and its properties (Liu et al. (2007a), Metzler (2000), Podlubny (1999)) .

DEFINITION 1 The Riemann-Liouville derivative of order

α

of the function y (x) is defined by

1 dn x y (τ ) Dx y ( x ) = dτ , x > 0, α > 0, n ∫0 Γ(n − α ) dx ( x − τ )α −n +1 where n is the smallest integer exceeding α and Γ (.) is the Gamma function. If α = m ∈ N , then the above definition α

coincides with the classical m

th

derivative

y ( m ) ( x).


62

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

FRACTIONAL REACTION-SUBDIFFUSION EQUATION The standard meanfield model for the evolution of the concentrations a ( x, t ) and b ( x, t ) of A and B particles is given by the reaction-diffusion equations

∂ ∂2 a ( x, t ) = D 2 a ( x, t ) − εa ( x, t )b( x, t ), ∂t ∂x

(1)

∂ ∂2 b( x, t ) = D 2 b( x, t ) − εa ( x, t )b( x, t ), (2) ∂t ∂x where D is the diffusion coefficient assumed in this paper to equal for species and ε 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 a ( x, t ) = Dt1−α [kα 2 a ( x, t ) − εa ( x, t )b( x, t )], ∂t ∂x

(3)

∂ ∂2 (4) b( x, t ) = Dt1−α [kα 2 b( x, t ) − εa ( x, t )b( x, t )], ∂t ∂x γ where kα is the generalized diffusion coefficient and Dt is the Riemann-Liouville fractional partial derivative of order

γ 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

∂ ∂2 1−α u ( x, t ) = Dt [kα 2 u ( x, t ) − εu ( x, t )] + f ( x, t ), 0 < t ≤ T , 0 < x < L, ∂t ∂x

(5)

which can be rewritten as

∂α ∂2 u ( x , t ) = k u ( x, t ) − εu ( x, t ) + g ( x, t ), 0 < t ≤ T , 0 < x < L, α ∂t α ∂x 2 α −1 where 0 < α < 1 and g ( x, t ) = Dt f ( x, t ) .

(6)

We assume Dirichlet boundary conditions for this problem as follows

u (0, t ) = φ (t ), u ( L, t ) = ψ (t ), 0 < t ≤ T ,

(7)

with the initial condition (8) u ( x,0) = ω ( x ), 0 ≤ x ≤ L. In the last few years appeared many papers to study the proposed model (6)-(8) (Henry et al. (2000), Liu et al. (2007), Liu et al. (2007b), Seki et al. (2003), Yuste et al. (2004)). In this paper, we study the time fractional case and use an efficient class of finite difference methods based on Hermite formula to solve this model. The plan of the paper is as follows; In section 3, we give some approximate formulae of the fractional derivatives and numerical finite difference scheme are given. In section 4, we study the stability and the accuracy of the presented scheme. In section 5, we present numerical solutions of fractional reaction-subdiffusion problem. The paper ends with some conclusions in section 6.


63

An Efficient Class of FDM Based on Hermite Formula for Solving Fractional Reaction-Subdiffusion Equations

FINITE DIFFERENCE EQUATION

SCHEME

OF

THE

FRACTIONAL

REACTION-SUBDIFFUSION

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 N and M , we use the notations

∆x and ∆t , at space-step length and time-step length, respectively. The coordinates of the mesh points are x j = j∆x ( j = 0,1,..., N ) , and t m = m∆t , ( m = 0,1,..., M ) and the values of the solution u ( x, t ) on these grid points are

u ( x j , t m ) ≡ u mj ≡ UmJ ; where ∆x = h =

L T , ∆t = τ = . N M

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)). + m ∂u 1 δt u j | x ,t = + O (τ ), ∂t j m τ (1 − 1 δ + ) t 2

where

(9)

δ t+ denotes the backward difference operator with respect to t . Now, using the formula (9), we can derive an ∂α u ( x, t ) , (0 < α ≤ 1) at the points ( x j , t m ) as follow ∂t α

efficient approximate formula of the fractional derivative for

+ r +1 m −1 ( r +1)τ t m ∂u ( x j , ζ ) ∂α u 1 1 1 δt u j −α | = ( t − ) d = [ + O (τ 2 )](mτ − ζ ) −α dζ ζ ζ ∑ m α x j ,t m ∫ ∫ τ r 0 1 Γ(1 − α ) ∂ζ Γ(1 − α ) r = 0 τ (1 − δ + ) ∂t t 2 + r + 1 m −1 ( r +1)τ 1 1 δt u j = [ + O (τ 2 )](mτ − ζ ) −α dζ ∑ ∫ τ r Γ (1 − α ) r = 0 τ (1 − 1 δ + ) t 2 + r +1 τ 1−α m −1 1 δ t u j = [ + O (τ 2 )][(m − r )1−α − ( m − r − 1)1−α ] ∑ 1 Γ (2 − α ) r = 0 τ (1 − δ t+ ) 2

=

τ −α

m −1

∑ Γ (2 − α ) r =0

=

=

δ t+ u rj +1 1 (1 − δ t+ ) 2

τ −α

m

Γ(2 − α )

τ −α

m

∑ Γ(2 − α ) r =1

∑ r =1

[(m − r )1−α − (m − r − 1)1−α +

δ t+ u mj −r +1 1 (1 − δ t+ ) 2

δt+u mj − r +1 1 (1 − δt+ ) 2

[r 1−α − (r − 1)1−α ] +

[r 1−α − ( r − 1)1−α ] +

m −1 1 [(m − r )1−α − ( m − r − 1)1−α ]O (τ 2 −α )] ∑ Γ (2 − α ) r =0

m 1 [r 1−α − (r − 1)1−α ]O (τ 2−α ) ∑ Γ(2 − α ) r =1

m 1−α O (τ ). Γ(2 − α )


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

The above formula can be rewritten as m δt+u mj − r +1 ∂α u τ | = A χ + O (τ ), ∑ x , t r α 1 ∂t α j m r =1 (1 − δt+ ) 2

where

Aατ =

τ −α

We must note that (1) (2)

and

Γ(2 − α )

χr

(10)

χ r = r 1−α − (r − 1)1−α .

satisfy the following facts

χ r > 0 , r = 1,2,... . χ r > χ r +1 , r = 1,2,... .

Now, we are going to obtain the finite difference scheme of the fractional reaction-subdiffusion equation (6). In our study we take kα = ε = 1 . To achieve this aim we evaluate this equation at the points of the grid ( x j , t m ) ,

∂ 2u ( x j , t m ) ∂α u ( x j , tm ) = − u ( x j , t m ) + g ( x j , t m ). ∂t α ∂x 2

(11)

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

∂ 2u( x j , t m ) ∂x 2

δ t+ u mj − r +1

m

∑χ

τ α

=A

r

1 (1 − δ t+ ) 2

r =1

+ u ( x j , t m ) − g ( x j , t m ) + O (τ ).

(12)

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

∂ 2 u ( x j −1 , t m ) ∂x 2

τ α

=A

δ t+ u mj−−1r +1

m

∑χ

r

r =1

1 (1 − δ t+ ) 2

+ u ( x j −1 , t m ) − g ( x j −1 , t m ) + O (τ ),

(13)

+ u ( x j +1 , t m ) − g ( x j +1 , t m ) + O (τ ).

(14)

and

∂ 2 u ( x j +1 , t m ) ∂x 2

τ α

=A

δ t+ u mj+−1r +1

m

∑χ r =1

r

1 (1 − δ t+ ) 2

The Hermite formula with two-order derivatives at the grid point ( x j , t m ) is

∂ 2 u ( x j −1 , t m )

∂ 2u ( x j , t m )

∂ 2 u ( x j +1 , t m )

12 (15) (u ( x j −1 , t m ) − 2u ( x j , t m ) + u ( x j +1 , t m )) = O ( h 4 ). 2 ∂x ∂x ∂x h m m Substitute from Eqs.(12)-(14) into Eq.(15), and denote u ( x j , t m ) by u j , g ( x j , t m ) by g j , then with neglect the high 2

+ 10

order terms, we have

2

+

2


65

An Efficient Class of FDM Based on Hermite Formula for Solving Fractional Reaction-Subdiffusion Equations

m

h 2 [ Aατ ∑ χ r

δ t+ u mj−−1r +1

m

− g mj−1 ] + 10h 2 [ Aατ ∑ χ r

δ t+ u mj −r +1

− g mj ] + 1 + 1 + r =1 (1 − δ t ) (1 − δ t ) 2 2 + m − r +1 m δ u t j + 1 h 2 [ Aατ ∑ χ r − g mj+1 ] = (12 − h 2 )u mj−1 − 2(12 + 5h 2 )u mj + (12 − h 2 )u mj+1 , 1 + r =1 (1 − δ t ) 2 r =1

(16)

under some simplifications, we can obtain the following form

1 5 1 1 5 1 2h 2 Aατ ( u mj−1 + u mj + u mj+1 ) − 2h 2 Aατ ( u mj−−11 + u mj −1 + u mj+−11 ) + 12 6 12 12 6 12 1 2 τ m 5 2 τ m m − r +1 m−r m − r +1 m−r h Aα ∑ χ r (u j −1 − u j −1 ) + h Aα ∑ χ r (u j −uj )+ r =2 r =2 6 3 1 2 τ m 1 5 (17) h Aα ∑ χ r (u mj+−1r +1 − u mj+−1r ) − h 2 ( g mj−1 + g mj−−11 ) − h 2 ( g mj + g mj −1 ) − r =2 6 12 6 1 2 m 1 5 1 h ( g j +1 + g mj+−11 ) = (1 − h 2 )u mj−1 − (2 + h 2 )u mj + (1 − h 2 )u mj+1 + 12 12 6 12 1 2 m −1 5 2 m −1 1 2 m −1 (1 − h )u j −1 − (2 + h )u j + (1 − h )u j +1 . 12 6 12 m m m m Let U j be the approximate solution, and let T j = u j − U j , j = 1,2,..., N , m = 1,2,..., M be the error, so we have the error formula

1 m 5 m 1 m 1 5 1 T j −1 + T j + T j +1 ) − 2h 2 Aατ ( T jm−1−1 + T jm −1 + T jm+1−1 ) + 12 6 12 12 6 12 m 1 2 τ m 5 h Aα ∑ χ r (T jm−1− r +1 − T jm−1− r ) + h 2 Aατ ∑ χ r (T jm − r +1 − T jm − r ) + r =2 r =2 6 3 1 2 τ m 1 2 m 5 1 m − r +1 m−r h Aα ∑ χ r (T j +1 − T j +1 ) = (1 − h )T j −1 − (2 + h 2 )T jm + (1 − h 2 )T jm+1 + r =2 6 12 6 12 1 2 m −1 5 2 m −1 1 2 m−1 (1 − h )T j −1 − (2 + h )T j + (1 − h )T j +1 . 12 6 12

(18)

T0m = TNm = 0, m = 1,2,..., M .

(19)

2h 2 Aατ (

with

PROPOSITION 1

Assuming that the solution of (18) has the form

T jm = ξ m eiβjh , then

5 1 1 5 ( + cos ( β h ))h 2 Aατ + (1 − h 2 )cos ( β h ) − (1 + h 2 ) 6 6 12 12 ξm = ξ m −1 5 1 5 1 ( + cos ( β h ))h 2 Aατ + (1 + h 2 ) − (1 − h 2 )cos ( β h ) 6 6 12 12 1 2 τ 5 h Aα ( + cos ( β h )) m 6 6 − χ r (ξ m − r +1 − ξ m − r ), ∑ 5 1 5 1 ( + cos ( β h ))h 2 Aατ + (1 + h 2 ) − (1 − h 2 )cos ( β h ) r =2 6 6 12 12 where β = 2πm .

(20)


66

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

PROOF

Substitute in (18) by

T jm = ξ meiβjh and divide by e iβjh , we get

1 5 1 1 5 1 ( e −iβh + + e iβh )2h 2 Aατ ξ m −( e −iβh + + e iβh )2h 2 Aατ ξ m −1 12 6 12 12 6 12 1 2 −iβh 5 2 1 2 iβ h 1 + [−(1 − h )e + (2 + h ) − (1 − h )e ]ξ m − [(1 − h 2 )e −iβh 12 12 12 6 5 2 1 2 iβh 1 2 τ m − (2 + h ) + (1 − h )e ]ξ m −1 − h Aα ∑ χ r (ξ m − r +1 − ξ m − r )e −iβh r =2 6 12 6 5 2 τ m 1 2 τ m − h Aα ∑ χ r (ξ m− r +1 − ξ m− r ) − h Aα ∑ χ r (ξ m −r +1 − ξ m − r )e iβh = 0. r =2 r =2 3 6

(21)

Using some trigonometric formulae and some simplifications we can obtain

5 2 1 2 5 1  2 τ  6 + 6 cos( βh))h Aα + (1 + 12 h ) − (1 − 12 h )cos( βh)ξ m   5 1 1 5   −  ( + cos( β h))h 2 Aατ + (1 − h 2 )cos( β h) − (1 + h 2 )ξ m −1 6 12 12 6   m 1 2 τ 5 − h Aα ( + cos( β h)) ∑ χ r (ξ m − r +1 − ξ m − r ) = 0. r =2 6 6

(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

τα ≤

2h 2 , (6 + h 2 )Γ(2 − α )

(23)

then

5 1 1 5 ( + cos( β h))h 2 Aατ + (1 − h 2 )cos( β h) − (1 + h 2 ) 12 12 0≤ 6 6 ≤ 1. 5 1 5 1 2 τ 2 2 ( + cos( β h))h Aα + (1 + h ) − (1 − h )cos( β h) 6 6 12 12

(24)

PROOF

τ −α

τ

Since Aα =

Γ(2 − α )

, then

2h 2 1 α , and by our assumption that ≤ , so we can τ τ = τ Aα Γ (2 − α ) (6 + h 2 )Γ(2 − α ) α

obtain

1 2h 2 ≤ , Aατ 6 + h 2 so,

1 1 1 2 + h 2 − h 2 ≤ h 2 Aατ − h 2 Aατ , 2 6 3 2

and since sin ( β h) ≤ 1 , so we have


An Efficient Class of FDM Based on Hermite Formula for Solving Fractional Reaction-Subdiffusion Equations

67

1 1 1 (2 − h 2 + h 2 Aατ ) sin 2 ( β h) ≤ h 2 Aατ − h 2 , 6 3 2 i.e.,

1 1 1 h 2 Aατ − h 2 −(2(1 − h 2 ) + h 2 Aατ ) sin 2 ( β h) ≥ 0, 2 12 3 h 2 Aατ + (1 −

1 2 5 1 1 h ) − (1 + h 2 ) − 2(1 − h 2 ) sin 2 ( β h) − h 2 Aατ sin 2 ( β h) ≥ 0, 12 12 12 3

so,

h 2 Aατ [1 −

11 1 5 1 1 (1 − cos (2 β h))] + (1 − h 2 ) − (1 + h 2 ) − 2(1 − h 2 ) (1 − cos (2 β h)) ≥ 0, 32 12 12 12 2

i.e.,

5 1 1 5 h 2 Aατ ( + cos (2 β h)) + [(1 − h 2 )cos (2 β h)) − (1 + h 2 )] ≥ 0, 6 6 12 12 and since

(1 +

5 2 1 1 5 h ) − (1 − h 2 )cos (2 β h)) ≥ (1 − h 2 )cos (2 β h)) − (1 + h 2 ), 12 12 12 12

we have that

5 1 5 1 h 2 Aατ ( + cos (2 β h)) + [(1 + h 2 ) − (1 − h 2 )cos (2 β h))] ≥ 0, 6 6 12 12 which completes the proof.

PROPOSITION 3 Assume that PROOF If we take

ξ m , be the solution of (20), with the condition (23), then | ξ m |≤| ξ 0 | , (m = 1,2,..., M ) .

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

5 1 1 5 ( + cos( βh))h 2 Aατ + (1 − h 2 )cos( βh) − (1 + h 2 ) 12 12 | ξ1 |= 6 6 | ξ0 | . 5 1 5 2 1 2 2 τ ( + cos( βh))h Aα + (1 + h ) − (1 − h )cos( βh) 6 6 12 12 From Proposition 2, we have that | ξ1 |≤| ξ 0 | .

(25)

Suppose that | ξ s |≤| ξ 0 | , s = 1,2,..., m − 1 . For

m >1 5 1 1 5 ( + cos( β h))h 2 Aατ + (1 − h 2 )cos( βh) − (1 + h 2 ) 12 12 | ξ m |=| 6 6 ξ m −1 5 1 5 1 ( + cos( β h))h 2 Aατ + (1 + h 2 ) − (1 − h 2 )cos( β h) 6 6 12 12 m 1 2 τ 5 h Aα ( + cos( β h))∑χ r (ξ m − r +1 − ξ m − r ) 6 6 r =2 − |. 5 1 5 2 1 2 2 τ ( + cos( β h))h Aα + (1 + h ) − (1 − h )cos( β h) 6 6 12 12

(26)


68

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

Now, since m

m −1

r =2

r =2

∑ χ r (ξ m − r +1 − ξ m −r ) = χ 2ξ m−1 − ∑ ( χ r − χ r +1 )ξ m −r − χ mξ 0 ,

(27)

5 1 1 5 ( + cos( βh))h 2 Aατ (1 − χ 2 ) + (1 − h 2 )cos( βh) − (1 + h 2 ) 12 12 | ξ m |=| 6 6 ξ m −1 5 1 5 2 1 2 2 τ ( + cos( βh))h Aα + (1 + h ) − (1 − h )cos( βh) 6 6 12 12 m −1 1 2 τ 5 h Aα ( + cos( βh))∑( χ r − χ r +1 )ξ m − r 6 6 r =2 + 5 1 5 2 1 2 τ ( + cos( βh))h Aα + (1 + h ) − (1 − h 2 )cos( βh) 6 6 12 12 1 2 τ 5 h Aα ( + cos( β h)) χ mξ 0 6 6 + |. 5 1 5 1 2 τ ( + cos( β h))h Aα + (1 + h 2 ) − (1 − h 2 )cos( β h) 6 6 12 12

(28)

we have

Using triangle inequality

5 5 1 1 ( + cos( βh))h 2 Aατ (12 − χ 2 ) + (1 − h 2 )cos ( βh) − (1 + h 2 ) 12 12 || ξ m −1 | | ξ m |≤| 6 6 5 1 5 1 2 τ 2 2 ( + cos( βh))h Aα + (1 + h ) − (1 − h )cos( β h) 12 6 6 12 m −1 1 2 τ 5 h Aα ( + cos( βh))∑( χ r − χ r +1 ) 6 6 r =2 +| || ξ m − r | 5 1 5 1 ( + cos( β h))h 2 Aατ + (1 + h 2 ) − (1 − h 2 )cos ( βh) 6 6 12 12 5 1 h 2 Aατ ( + cos( β h)) χ m 6 6 || ξ 0 | . +| 5 1 5 2 1 2 2 τ ( + cos( β h))h Aα + (1 + h ) − (1 − h )cos ( βh) 6 6 12 12 −1 Now, since | ξ m −1 |≤| ξ 0 | , | ξ m −r |≤| ξ 0 | , and also since ∑ m r = 2 ( χ r − χ r +1 ) =

(29)

χ 2 − χ m , so the above inequality takes the

form

5 1 1 5 ( + cos ( β h)) h 2 Aατ + (1 − h 2 )cos ( β h) − (1 + h 2 ) 12 12 || ξ 0 |, | ξ m |≤| 6 6 5 1 5 1 2 τ 2 2 ( + cos ( β h)) h Aα + (1 + h ) − (1 − h )cos ( βh) 6 6 12 12

(30)

and using Proposition 2 directly we complete the proof. We know that

T

2 2

=

∑ N = −∞

| ξ m ( N ) |2 .

(31)


69

An Efficient Class of FDM Based on Hermite Formula for Solving Fractional Reaction-Subdiffusion Equations

THEOREM 1 The difference scheme (17) is stable under the condition (23). PROOF

From Proposition 3, (31) and the formula T

m 2

≤ T 0 , m = 1,2,.., M , which means that the difference scheme is 2

stable. By the Lax equivalence theorem (Richtmyer et al. (1967)) we can show that the numerical solution converges to the exact solution as h,τ → 0 .

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

∂ ∂2 u ( x, t ) = Dt1−α [ 2 u ( x, t ) − u ( x, t )] + (1 + α )e x t α , 0 < x < 1, 0 ≤ t ≤ T , ∂t ∂x α +1 α +1 with the following boundary conditions u (0, t ) = t , u (1, t ) = et , 0 ≤ t ≤ T ,

(32)

0 ≤ x ≤ 1.

and the initial condition u ( x,0) = 0,

x α +1

The exact solution of this problem is u ( x, t ) = e t

.

The behavior of the analytical solution and the numerical solution of the proposed fractional reaction-subdiffusion equation (33) by means of the proposed finite difference scheme with different values of

α , h , τ , and final time T

are presented

in figures 1-5.

Figure 1: The Behavior of the Exact Solution and the Numerical Solution of (32) by Means of the Proposed Method at

α = 0.8 , h =

1 1 ,τ = and T = 0.5 40 20


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

α = 0.3 , h =

1 1 ,τ = and T = 1 . 60 70

Figure 3: The Behavior of the Exact Solution and the Numerical Solution of (32) by Means of the Proposed Method at

α = 0.5 , h =

1 1 ,τ = , and T = 2 . 60 20

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

h=

1 1 ,τ = , and T = 0.5 , with different values of α 20 20


71

An Efficient Class of FDM Based on Hermite Formula for Solving Fractional Reaction-Subdiffusion Equations

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

h=

1 1 ,τ = , α = 0.7 , with different values of T 40 40

EXAMPLE 2

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

∂ ∂2 u ( x, t ) = Dt1−α [ 2 u ( x, t ) − u ( x, t )] + f ( x, t ), 0 < α ≤ 1, ∂t ∂x on a finite domain 0 < x < 1 , with 0 ≤ t ≤ T and the following source term

(33)

 tΓ(2 + α ) + (π 2 + 1)t α +1   sin (πx) . f ( x, t ) = 2 Γ(2 + α )   Under the boundary conditions u (0, t ) = u (1, t ) = 0 , and the initial condition u ( x,0) = 0 . 2

The exact solution of Eq.(34) in this case is u ( x, t ) = t sin (πx ) . The behavior of the exact solution and the numerical solution of the proposed fractional reaction-subdiffusion Eq.(34) by means of the proposed finite difference scheme with different values of

α , h, τ ,

and final time T are

presented in figures 6-8.

Figure 6: The Behavior of the Exact Solution and the Numerical Solution of (33) by Means of the Proposed Method at

α = 0.5 , h =

1 1 ,τ = , and T = 2 70 60


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

α = 0.2 , h =

1 1 ,τ = , and T = 0.2 60 20

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

α = 0.9 , h =

1 1 ,τ = and T = 0.01 50 150

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

α = 0.4 , h =

1 1 ,τ = , and T = 1 40 100


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An Efficient Class of FDM Based on Hermite Formula for Solving Fractional Reaction-Subdiffusion Equations

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 of

Table 1: The Maximum Error with Different Values of

h

τ

1 50 1 70 1 120 1 140 1 160

1 50 1 80 1 100 1 150 1 180

τ

1 10 1 10 1 20 1 20 1 30

1 10 1 20 1 20 1 20 1 30

h , τ at α = 0.5 , and T = 0.2

Maximum Error 0.01602

0.01303 0.01216 0.01118 0.01091

Table 2: The Maximum Error with Different Values of

h

h , τ and the final time T .

h , τ at α = 0.2 , and T = 0.1

Maximum Error 0.01013

0.00290 0.00283 0.00281 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.


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