SOLUTION OF GENERALIZED FRACTIONAL REACTION-DIFFUSION

Proceeding of National Conference on Recent Trends in Mathematics Vol.1, pp. 7-19. ISBN: 978-81-932712-0-9.

SOLUTION OF GENERALIZED FRACTIONAL REACTION-DIFFUSION EQUATIONS IN TERMS OF MITTAG-LEFFLER FUNCTIONS BIRAJDAR G. A. AND DHAIGUDE D. B.

Abstract. This paper gives the solution of the generalized fractional reactiondiffusion equation using the joint transform method. For this we use generalized Riemann-Liouville fractional derivative operator. This gives the more general solution.

1. Introduction Fractional calculus provides a powerful tool for the description of memory and hereditary properties of different substances because of their non-locality property. Recently, fractional differential equations have played a key role in modeling particle transport, in a anomalous diffusion, in many diverse fields, including finance[37], semiconductor, biology[38], physics[4], electrical engineering and control theory [29]. Fractional diffusion equations accounts for typical anomalous features which are observed in many systems e.g. in the case of dispersive transport in amorphous semiconductors, porous medium, colloid, proteins, biosystems or even in ecosystems [3, 16, 17]. The interdisciplinary applications show the importance and necessity of fractional calculus. It motivates us to construct a variety of efficient methods for fractional differential equations such as integral transform method [29, 33], new iterative method [10, 14], numerical methods [5, 6, 7, 8] and Adomian decomposition method [1, 2, 9, 11, 12, 13]. Reaction-diffusion models have found numerous applications in pattern formation in biology, chemistry, and physics given in, Smoller[34], Grindrod[15] and Wilhelmsson and Lazzraro [36]. A general model for reaction-diffusion systems is discussed by Henry and Warne [18, 19] and Manne[24]. These systems show that diffusion can produce the spontaneous formation of spatio-temporal patterns. The fractional reaction-diffusion systems for nonlinear waves studied by Mathai et al.[25] and Saxena et al.[30, 31, 32]. The simplest reaction-diffusion models can be described by an equation ∂2N ∂N = D 2 + λF (N ), N = N (x, t) ∂t ∂x

(1)

2000 Mathematics Subject Classification. 35R11. Key words and phrases. Fractional reaction-diffusion equation, Integral transforms, Generalized Mittag-Leffler Function, Generalized Riemann-Liouville fractional derivative. 7

8

BIRAJDAR G. A. AND DHAIGUDE D. B.

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where D is the diffusion constant and F (N ) is a nonlinear function representing reaction kinetics. It is interesting to observe that for F (N ) = γN (1 − N ), it reduces to the Fisher-Kolmogorov equation and if F (N ) = γN (1 − N 2 ) reduces to the real Ginsburg-Landau equation. In this paper we use generalized Riemann-Liouville fractional derivative operator α,β Da± of order α and β for the time derivative. The generalized fractional derivative were introduced and investigated in several earlier works of Hilfer[20, 21], Hilfer et al.[22], Tomovski et al. [35]. In this article, we present a straightforward method for the systematic derivation of the solution of nonlinear reaction-diffusion equation. The result are derived by the application of Laplace and Fourier transforms, which are suitable for numerical computation. The plan of the paper as follows. In section 2, we define basic definitions. We find the inverse Laplace transform of some algebraic functions in section 3. Section 4 is devoted for find out the solution of generalized fractional reaction-diffusion equations under the suitable conditions, by using joint transform method. 2. Preliminaries and notations In this section we define useful definitions and properties as follows: The Mittag-Leffler function [23, 26, 27] is very useful in the study of fractional differential equations. Definition 2.1. The one parameter Mittag-Leffler function is defined as ∞ X zk , (α ∈ C, Re(α) > 0) Eα (z) = Γ(αk + 1)

(2)

k=0

Definition 2.2. A two parameter Mittag-Leffler function is defined as ∞ X zk Eα,β (z) = , (α, β ∈ C, Re(α) > 0, Re(β) > 0) Γ(αk + β)

(3)

k=0

Definition 2.3. A generalized Mittag-Leffler function is defined as ∞ X (γ)n z n γ Eα,β (z) = , (α, β, γ ∈ C, Re(α) > 0, Re(β) > 0, Re(γ) > 0) (n!)Γ(nα + β) n=0 (4) where (γ)n is the Pochhammer [28] symbol and is defined as (γ)0 = 1, (γ)r = γ(γ + 1)(γ + 2)...(γ + r + 1), r = 1, 2, ...; γ 6= 0 For γ = 1, this function coincides with (3), while for β = γ = 1 with (2), we have 1 1 Eα,β (z) = Eα,β (z), Eα,1 (z) = Eα (z)

The definitions of the well know Laplace and Fourier transform of a function N (x, t) and their inverses defined as follows. The Laplace transform of a function N (x, t) with respect to t is defined as Z ∞ ¯ (x, s) = e−st N (x, t)dt, (t > 0), (x ∈ R) (5) L{N (x, t)} = N 0

where Re(s) > 0, and its inverse transform with respect to s is given by Z γ+i∞ ¯ (x, s)} = N (x, t) = 1 ¯ (x, s)ds, γ is fixed real number L−1 {N est N 2πi γ−i∞

(6)

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SOLUTION OF GENERALIZED FRACTIONAL REACTION-DIFFUSION ...

9

Prabhakar [28] shows that γ L{tβ−1 e−st Eα,β (atα )} =

s−β (1 − as−α )γ

(7)

1

where Re(α) > 0, Re(β) > 0, Re(s) > 0 and s > | a | Re(α) . The Fourier transform of function N (x, t) with respect to x is defined as ⋆

F {N (x, t)} = F (k, t) =

Z

∞

eikx N (x, t)dx

(8)

−∞

The inverse Fourier transform with respect to k is given by F −1 {F ⋆ (x, t)} =

1 2π

Z

∞

e−ikx F ⋆ (k, t)dk

(9)

−∞

The Riemann-Liouville fractional integral of order α is defined [29] as α 0 It N (x, t)

1 = Γ(α)

Z

t

(t − u)α−1 N (x, u)du, Re(α) > 0.

(10)

0

The generalized Riemman-Liouville fractional derivatives of order α(0 < α < 1) and type β(0 ≤ β ≤ 1) and is denoted by Daα,β and is defined [20] with respect to x as Dxα,β N (x, t) = (I β(1−α)

d (1−α)(1−β) (I N ))(x, t) dx

(11)

whenever the second member of equation (11) exists. This generalization (11) yields the classical Riemann-Liouville fractional derivative operator when β = 0. Moreover for β = 1, it gives the Caputo fractional derivative. Several authors called the general operators in (11) the Hilfer fractional derivative operators. Applications of Daα,β are given in [21] The Laplace transform of generalized Riemann-Liouville fractional derivative Daα,β is defined [20] as L[Daα,β N (x, t)](s) = sα L[N (x, t)](s) − sβ(α−1) (I (1−β)(1−α)N )(x, 0)

3. Inverse Laplace Transform of Some Algebraic Function In this section, we find the inverse Laplace transform of certain algebraic functions which are directly applicable to obtaining solution. A)L−1

sα1

X ∞ ∞ X sα2 (α1 −1) r l r−l r × = (−1) a b 0 0 l + a0 sβ 1 + b 0 sγ1 + c (12) r=0 l=0

tα2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 −1 Eαr+1 (−ctα1 ) 1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1

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Proof: We have sα1 =

= =

sα2 (α1 −1) sα2 (α1 −1) = + a0 sβ 1 + b 0 sγ1 + c sβ1 (sα1 −β1 + a0 + b0 sγ1 −β1 + csβ1 ) sα2 (α1 −1)−β1

a0 +b0 sγ1 −β1 ) β (sα1 −β1 +cs 1 ) ( ∞ r X X r s(γ1 −β1 ) r−l) sα2 (α1 −1)−α1 (−1)r al0 br−l 0 l (sα1 −β1 )r+1 (1 + cs−α1 )r+1 r=0 l=0 α2 (α1 −1)+(γ1 −α1 )r+(β1 −γ1 )l−α1 ∞ r X X r l r−l r s (−1) a0 b 0 (1 + cs−α1 )r+1 l r=0 l=0 β

(sα1 −β1 +cs 1 )(1 +

Applying Laplace inverse on both sides, we get L−1

sα1

X ∞ ∞ X sα2 (α1 −1) r l r−l r = (−1) L−1 × a b 0 0 + a0 sβ 1 + b 0 sγ1 + c l r=0 l=0 α2 (α1 −1)+(γ1 −α1 )r+(β1 −γ1 )l−α1 s (1 + s−α1 )r+1

we get, =

∞ X

(−1)r

r=0

∞ X l=0

al0 br−l 0

r α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 −1 t × l

Eαr+1 (−ctα1 ) 1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1

B)L

−1

sα1

X ∞ r X sβ2 (β1 −1) r l r−l r × = (−1) a0 b 0 l + a0 sβ 1 + b 0 sγ1 + c r=0 l=0

tβ2 (1−β1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 −1 Eαr+1 (−ctα1 ) 1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 (13) α1 β1 where Re(α1 ) > 0, Re(β1 > 0), Re(γ1 ) > 0, | as sα+bs |< 1 1 +c Proof:We have sα1

sβ2 (β1 −1) sβ2 (β1 −1) = β1 α1 −β1 γ β 1 1 + a0 + b0 sγ1 −β1 + cs−β1 ) + a0 s + b 0 s + c s (s = =

(sα1 −β1

sβ2 (β1 −1)−β1 + a0 + b0 sγ1 −β1 + cs−β1 ) sβ2 (β1 −1)−β1

(sα1 −β1 + cs−β1 )(1 +

= sβ2 (β1 −1)−β1

∞ X r=0

(−1)r

a0 +b0 sγ1 −β1 ) sα1 −β1 +cs−β

(a0 + b0 sγ1 −β1 )r (sα1 −β1 + cs−β1 )r+1

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=

∞ X

r

(−1)

r=0

=

∞ X

(−1)r

r=0

r X l=0

r X

al0 br−l 0

l=0

11

β2 (β1 −1)−β1 +(γ1 −β1 )(r−l) r s l (sα1 −β1 )r+1 (1 + cs−α1 )r+1

β2 (β1 −1)−β1 +(γ1 −β1 )(r−l)−(α1 −β1 )(r+1) l r−l r s a0 b 0 (1 + cs−α1 )r+1 l

Applying the inverse Laplace transform on both sides, we get ∞ r X X sβ2 (β1 −1) r l r−l r × ] = (−1) a b L−1 [ α1 0 0 s + a0 sβ 1 + b 0 sγ1 + c l r=0 l=0 β2 (β1 −1)−β1 +(γ1 −β1 )(r−l)−(α1 −β1 )(r+1) s L−1 (1 + cs−α1 )r+1 =

∞ X

r

(−1)

r X

al0 br−l 0

r β2 (1−β1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 −1 t × l

r=0 l=0 Eαr+1 (−ctα1 ) 1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1

C)L−1

sα1

X ∞ r X 1 r l r−l r t(α1 −γ1 )r+(γ1 −β1 )l+α1 −1 × = (−1) a b 0 0 l + a0 sβ 1 + b 0 sγ1 + c r=0 l=0 r+1 Eα1 ,(α1 −γ1 )r+(γ1 −β1 )l+α1

(14) [D]L−1 {

∞ X

sα2 (α1 −1) } =tα1 −α2 (α1 −1)−1 t(α1 −β1 )r−1 + asβ1 + b r=0

sα1

Eαr+1 (−btα1 ) 1 ,(α1 −β1 )r+α1 −α2 (α1 −1) β1

r+1 where Re(α1 > 0), Re(β1 > 0), Re(s > 0), | sas α1 +b |< 1 and Eα,β (t) is the generalized Mittag-Leffler function. Proof: sα2 (α1 −1) sα2 (α1 −1) = sα1 + asβ1 + b sβ1 (a + sα1 −β1 + bs−β1 )

=

sα2 (α1 −1)−β1 = (a + sα1 −β1 + bs−β1 ) (1 +

= sα2 (α1 −1)−β1

∞ X r=0

sα2 (α1 −1)−β1 a )(sα1 −β1 + bs−β1 ) sα1 −β1 +bs−β1

(−a)r (sα1 −β1 + bs−β1 )r+1

∞ X sα2 (α1 −1)−β1 +β1 (r+1) = (−a)r (sα1 + b)r+1 r=0

=

∞ X sα2 (α1 −1)+β1 r (−a)r α1 (s + b)r+1 r=0

Applying inverse Laplace transform on both sides, we get L−1 {

∞ α2 (α1 −1)+β1 r X sα2 (α1 −1) −1 rs } = L { (−a) } sα1 + asβ1 + b (sα1 + b)r+1 r=0

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we have L

−1

∞ X sα2 (α1 −1) α1 −α2 (α1 −1)−1 }=t t(α1 −β1 )r−1 Eαr+1 (−btα1 ) { α1 1 ,(α1 −β1 )r+α1 −α2 (α1 −1) s + asβ1 + b r=0

∞ X sβ2 (β1 −1) α1 −β1 β2 [E]L { α1 }=t (−a)r t(α1 −β1 )r Eαr+1 (−btα1 ) 1 ,(α1 −β1 )r−β2 (β1 −1)+α s + asβ1 + b r=0 (15) β1 r+1 |< 1 and E (t) is the generalwhere Re(α1 > 0), Re(β1 > 0), Re(s > 0), | sas α1 +b α,β ized Mittag-Leffler function. Proof: −1

sβ2 (β1 −1) sβ2 (β1 −1) = sα1 + asβ1 + b sβ1 (a + sα1 −β1 + bs−β1 ) =

sβ2 (β1 −1)−β1 sβ2 (β1 −1)−β1 = (a + sα1 −β1 + bs−β1 ) (1 + sα1 −β1a+bs−β1 )

= sβ2 (β1 −1)−β1

∞ X r=0

= =

∞ X

r=0 ∞ X r=0

(−a)r (−a)r

s

(−a)r (sα1 −β1 + bs−β1 )r+1

β2 (β1 −1)−β1 +β1 (r+1)

(sα1 + b)r+1 sβ2 (β1 −1)+β1 r (sα1 + b)r+1

Applying inverse Laplace transform on both side,we obtain L−1 {

L−1 {

∞ β2 (β1 −1)+β1 r X sβ2 (β1 −1) −1 rs } } = L { (−a) sα1 + asβ1 + b (sα1 + b)r+1 r=0

∞ X sβ2 (β1 −1) α1 −β1 β2 } = t (−a)r t(α1 −β1 )r Eαr+1 (−btα1 ) 1 ,(α1 −β1 )r−β2 (β1 −1)+α sα1 + asβ1 + b r=0

[F ]L−1 {

sα1

∞ X 1 α1 −1 } = t (−a)r t(α1 −β1 )r Eαr+1 (−btα1 ) 1 ,α1 +(α1 −β1 )r + asβ1 + b r=0

here Re(α1 > 0), Re(β1 > 0), Re(s > 0), | Mittag-Leffler function.

asβ1 sα1 +b

r+1 |< 1 and Eα,β (t) is the generalized

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Proof: 1 1 = β1 sα1 + asβ1 + b s (a + sα1 −β1 + bs−β1 ) =

s−β1 = (a + sα1 −β1 + bs−β1 ) (1 +

= s−β1

∞ X r=0

a sα1 −β1 +bs−β1

)

r

(−a) + bs−β1 )r+1

(sα1 −β1

=

∞ X

(−a)r

=

∞ X

(−a)r

r=0

r=0

s−β1

s−β1 +β1 (r+1) (sα1 + b)r+1 (sα1

sβ 1 r + b)r+1

Applying inverse Laplace transform on both side,we obtain ∞ X 1 α1 −1 } = t (−a)r t(α1 −β1 )r Eαr+1 (−btα1 ) L−1 { α1 1 ,α1 +(α1 −β1 )r s + asβ1 + b r=0 4. Solution of Fractional Reaction-Diffusion Equations In this section we find the solution of the fractional-diffusion system. Theorem 4.1. Consider the fractional reaction-diffusion equation 2 Dxγ N (x, t) + ξ 2 N (x, t) + φ(x, t) Dα1 ,α2 N (x, t) + a0 Dβ1 ,β2 N (x, t) = v−∞

(16)

x ∈ ℜ, 0 < α1 < 1, 0 < β1 < 1, η > 0, 0 < α1 ≤ 1, 0 < β1 ≤ 1, γ > 0 with initial condition I (1−α1 )(1−α2 ) N (x, 0) = f (x) = I (1−β1 )(1−β2 ) N (x, 0), f or(x ∈ ℜ)

(17)

2

where v is a diffusion coefficient φ is a constant which describes the nonlinearity in the system, and φ(x, t) is a nonlinear function for reaction kinetics, then there holds the following formula for the solution of (16) Z ∞ X (−a)r ∞ (α1 −β1 )r ∗ N (x, t) = t f (k)[tα1 −α2 (α1 +1)−1 Eαr+1 (−ctα1 ) 1 ,(α1 −β1 )r+α1 −α2 (α1 −1) 2π −∞ r=0 r+1 + tα1 −β1 β2 E(α (−ctα1 )]eikx dk 1 −β1 )r−β2 (β1 −1)+α1 Z Z ∞ X (−a)r t α1 +(α1 −β1 )r−1 ∞ ikx ∗ ξ e φ (k, t − ξ)Eαr+1 (−cξ α1 )dkdξ + 1 ,(α1 −β1 )r 2π 0 −∞ r=0 (18)

Proof: Given fractional-diffusion equation is 2 Dxγ N (x, t) + ξ 2 N (x, t) + φ(x, t) Dα1 ,α2 N (x, t) + Dβ1 ,β2 N (x, t) = v−∞

(19)

Applying the Laplace transform to equation (19) with respect to the time variable t and using the boundary conditions, we get 2 ¯ s) ¯ (x, s)− sα2 (α1 −1) f (x)+ a0 sβ2 (β1 −1) f (x) = v−∞ ¯ (x, s)+ ξ 2 N ¯ (x, s)+ φ(x, Dxγ N sα1 N (20)

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Applying the Fourier transform to equation (20) with respect to the space variable x, we have ¯ ∗ (k, s)−sα2 (α1 −1) f ∗ (k)+a0 sβ2 (β1 −1) f ∗ (k) = v 2 Dγ N ¯ ∗ (k, s)+ξ 2 N ¯ ∗ (k, s)+φ¯∗ (k, s) sα1 N −∞ x (21) ¯ ∗ (k, s), we solve the above equation for N α2 (α1 −1) + a0 sβ2 (β1 −1) )f ∗ (k) + φ¯∗ (k, s) ¯ ∗ (k, s) = (s N sα1 + a0 sβ1 + c

(22)

Applying inverse Laplace transform to above equation on both side, we obtain N ∗ (k, t) =

∞ X (−a)r t(α1 −β1 )r f ∗ (k)[tα1 −α2 (α1 −1) Eαr+1 (−ctα1 ) 1 ,(α1 −β1 )r+α1 −α2 (α1 −1) r=0

+ a0 tα1 −β1 β2 Eαr+1 (−ctα1 )] 1 ,(α1 −β1 )r−β2 (β1 −1)+α1 Z t ∞ X r (−a) + φ∗ (k, t − ξ)ξ α1 +(α1 −β1 )r−1 Eαr+1 (−cξ α1 )dξ 1 ,(α1 −β1 )r 0

r=0

(23)

Applying inverse Fourier transform to equation (23), we get

N (x, t) =

Z ∞ X (−a)r r=0

2π

∞

−∞

t(α1 −β1 )r f ∗ (k)[tα1 −α2 (α1 −1) Eαr+1 (−ctα1 ) 1 ,(α1 −β1 )r+α1 −α2 (α1 −1)

+ a0 tα1 −β1 β2 Eαr+1 (−ctα1 )]eikx dk 1 ,(α1 −β1 )r−β2 (β1 −1)+α1 Z Z ∞ X (−a)r t α1 +(α1 −β1 )r−1 ∞ ∗ ξ φ (k, t − ξ)eikx Eαr+1 (−cξ α1 )dkdξ + 1 ,(α1 −β1 )r 2π 0 −∞ r=0 (24) This completes the proof. Special Case: When f (x) = δ(x), where δ(x) is Dirac delta function, the above theorem reduces to the following corollary.

Corollary 4.1. Consider the fractional reaction-diffusion system 2 Dxγ N (x, t) + ξ 2 N (x, t) + φ(x, t) Dα1 ,α2 N (x, t) + a0 Dβ1 ,β2 N (x, t) = v−∞

(25)

(0 < α1 < 1, 0 < β1 < 1, 0 < α1 ≤ 1, 0 < β1 ≤ 1) subject to the initial condition I (1−α1 )(1−α2 ) N (x, 0) = δ(x) = I (1−β1 )(1−β2 ) N (x, 0), f or(x ∈ ℜ)

(26)

where δ(x) is the Dirac delta function. Here ξ is a constant that describes the nonlinearity in the system and φ(x, t) is nonlinear function which belongs to the reaction kinetics. The following is solution for equation (25) together with initial

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15

condition (26) N (x, t) =

Z ∞ X (−a)r r=0

2π

∞

−∞

t(α1 −β1 )r [tα1 −α2 (α1 −1) Eαr+1 (−ctα1 ) 1 ,(α1 −β1 )r+α1 −α2 (α1 −1)

+ a0 tα1 −β1 β2 Eαr+1 (−ctα1 )]eikx dk 1 ,(α1 −β1 )r−β2 (β1 −1)+α1 Z Z ∞ X (−a)r t α1 +(α1 −β1 )r−1 ∞ ∗ ξ φ (k, t − ξ)eikx Eαr+1 (−cξ α1 )dkdξ + 1 ,(α1 −β1 )r 2π 0 −∞ r=0 (27) Corollary 4.2. Consider the fractional reaction-diffusion system 1 1

1 1

1

2 D 2 , 2 N (x, t) + a0 D 2 , 2 N (x, t) = v−∞ Dx2 N (x, t) + ξ 2 N (x, t) + φ(x, t)

(28)

subject to the initial condition 1 1

1 1

I 2 , 2 N (x, 0) = δ(x) = I 2 , 2 N (x, 0), f or(x ∈ ℜ)

(29)

where v 2 is a diffusion constant, ξ is constant which describes the nonlinearity in the system, and φ(x, t) is nonlinear function. The solution of above equation (??) together with initial condition (29) is given by Z ∞ X −1 1 1 (−a)r ∞ ∗ 2 4 + a t 4 ]dk (30) f (k)E r+1 N (x, t) = 0 1 3 (−ct )[t 2,4 2π −∞ r=0 Z Z ∞ X 1 (−a)r t ∞ −1 ξ 2 e−ikx φ∗ (k, t − ξ)E r+1 −cξ 2 dkdξ + (31) 1 2 2π 0 −∞ r=0 Theorem 4.2. Consider the fractional reaction-diffusion equation 2 Dxη N (x, t)+ξ 2 N (x, t)+φ(x, t) Dα1 ,α2 N (x, t)+a0 Dβ1 ,β2 N (x, t)+b0 Dγ1 ,γ2 N (x, t) = v−∞ (32) x ∈ ℜ, 0 < α1 < 1, 0 < β1 < 1, 0 < γ1 < 1, η > 0, 0 < α1 ≤ 1, 0 < β1 ≤ 1, 0 < γ1 ≤ 1 with initial condition

I (1−α1 )(1−α2 ) N (x, 0) = f (x) = I (1−β1 )(1−β2 ) N (x, 0) = I (1−γ1 )(1−γ2 ) N (x, 0), f or(x ∈ ℜ) (33) and limN (x, t) = 0, t > 0 where v 2 is a diffusion coefficient ξ is a constant which describes the nonlinearity in the system, and φ(x, t) is a nonlinear function which belongs to the area of reaction-diffusion; then there holds the following formula for the solution of the equation (36) Z ∞ r X (−a)r ∞ −kx X r r r−l (α1 −γ1 )r+(γ1 −β1 )l+α1 −1 ∗ N (x, t) = e a b t f (k) 2π l 0 0 −∞ r=0 [t

α2 (1−α1 )

l r+1 Eα1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 (−ctα1 ) + a0 tβ2 (1−β2 ) Eαr+1 ] (−ctα1 ) + b0 (−ctα1 )Eαr+1 1 ,(α1 −β1 )r+(γ1 −β1 )l+α1 1 ,β2 (1−β2 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 Z r ∞ X (−1)r X r r r−l t (α1 −γ1 )r+(γ1 −β1 )l+α1 −1 a b ξ dξ + 2pi l 0 0 0 r=0 l=0 Z ∞ r+1 e−ikx φ∗ (k, t − ξ)E(α dk 1 −γ1 )r+(γ1 −β1 )l+α1 −∞

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Proof:Given fractional reaction-diffusion equation is D

α1 ,α2

2 Dxη N (x, t)+ξ 2 N (x, t)+φ(x, t) N (x, t)+a0 Dβ1 ,β2 N (x, t)+b0 Dγ1 ,γ2 N (x, t) = v−∞

Applying Laplace transform to above equation with respect to t using initial conditions, we get 2 L[Dxη N (x, t)]+ L[Dα1 ,α2 N (x, t)] + a0 L[Dβ1 ,β2 N (x, t)] + b0 L[Dγ1 ,γ2 N (x, t)] = v−∞

L[ξ 2 N (x, t) + φ(x, t)] We have ¯ (x, s) − sγ2 (γ1 −1) f (x) ¯ (x, s) − sα2 (α1 −1) f (x) + a0 sβ2 (β1 −1) f (x) + sγ1 N sα1 N 2 ¯ s) ¯ (x, s) + ξ 2 N ¯ (x, s) + φ(x, = v−∞ Dxγ N

(34)

Applying the Fourier transform to equation (34) with respect to the space variable x, we have ¯ ∗ (k, s)− ¯ ∗ (k, s) + a0 sβ2 (β1 −1) f ∗ (k) + b0 sγ1 N ¯ ∗ (k, s)−sα2 (α1 −1) f ∗ (k) + a0 sβ2 N sα1 N ¯ ∗ (k, s) + φ¯∗ (k, s) ¯ ∗ (k, s) + ξ 2 N b0 sγ2 (γ1 −1) f ∗ (k) = v 2 | k |η N

(35)

¯ ∗ (k, s), we solve the above equation for N α2 (α1 −1) + a0 sβ2 (β1 −1) + b0 sγ2 (γ1 −1) )f ∗ (k) + φ¯∗ (k, s) ¯ ∗ (k, s) = (s N sα1 + a0 sβ1 + b0 sγ1 + c Applying inverse Laplace transform to above equation on both side, we obtain α2 (α1 −1) + a0 sβ2 (β1 −1) + b0 sγ2 (γ1 −1) )f ∗ (k) + φ¯∗ (k, s) −1 ¯ ∗ −1 (s L [N (k, s)] = L sα1 + a0 sβ1 + b0 sγ1 + c ∗

N (k, t) =

∞ X

r

(−a)

r X

al0 br−l 0

r r r−l (α1 −γ1 )r+(γ1 −β1 )l+α1 −1 ∗ a b t f (k) l 0 0

r=0 l=0 [tα2 (α1 −1) Eαr+1 (−ctα1 ) + a0 tβ2 (1−β1 ) 1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 Eαr+1 (−ctα1 ) + b0 Eαr+1 (−ctα1 )] 1 ,β2 (1−β1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 1 ,(α1 −γ1 )r+(γ1 −β1 )l+α1 ∞ r X X r r r−l a b + (−a)r al0 br−l 0 l 0 0 r=0 l=0

Finally, taking inverse Fourier transform on both sides, we have Z ∞ r X (−a)r ∞ −kx X r r r−l (α1 −γ1 )r+(γ1 −β1 )l+α1 −1 ∗ N (x, t) = e a b t f (k) 2π l 0 0 −∞ r=0 l=0

[tα2 (1−α1 ) Eαr+1 (−ctα1 )+ 1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 a0 tβ2 (1−β2 ) Eαr+1 (−ctα1 )+ 1 ,β2 (1−β2 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 b0 (−ctα1 )Eαr+1 ] 1 ,(α1 −β1 )r+(γ1 −β1 )l+α1 Z ∞ X (−1)r r r r−l t (α1 −γ1 )r+(γ1 −β1 )l+α1 −1 a b ξ dξ + 2p l 0 0 0 r=0 Z ∞ e−ikx φ∗ (k, t − ξ)Eαr+1 (−cξ α1 )dk 1 ,(α1 −γ1 )r+(γ1 −β1 )l+α1 −∞

Special Case:

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Some special cases of the above theorem given below. If we set α1 = α2 = β1 = β2 = γ1 = γ2 = η = 21 Corollary 4.3. Consider the fractional reaction-diffusion equation 1 1

1 1

1 1

1

2 Dx2 N (x, t)+ξ 2 N (x, t)+φ(x, t) D 2 , 2 N (x, t)+a0 D 2 , 2 N (x, t)+b0 D 2 , 2 N (x, t) = v−∞ (36) with initial condition 1

I04 N (x, t) = f (x), x ∈ ℜ, limx→∞ N (x, t) = 0, t > 0

(37)

2

where v is diffusion constant, ξ is a constant which describes the nonlinearity in the system, and φ(x, t) is nonlinear function which belongs to the area of reactiondiffusion, then there holds the following formula Z ∞ r X (−a)r ∞ −kx X r r r−l −1 ∗ 1 1 1 2 N (x, t) = e a0 b0 t 2 f (k)[(t 4 + t 4 a0 + b0 )E r+1 1 3 (−ct )] , 2 4 2π l −∞ r=0 l=0 Z ∞ Z ∞ r r X X 1 (−1) r r r−l t −1 2 e−ikx φ∗ (k, t − ξ)E r+1 + a0 b 0 ξ 2 dξ 1 1 (−cξ ) , 2 2 2π l −∞ 0 r=0 l=0

Corollary 4.4. Consider the fractional reaction-diffusion equation 2 Dxη N (x, t)+ξ 2 N (x, t)+φ(x, t) Dα1 ,α2 N (x, t)+a0 Dβ1 ,β2 N (x, t)+b0 Dγ1 ,γ2 N (x, t) = v−∞ (38) x ∈ ℜ, 0 < α1 < 1, 0 < β1 < 1, 0 < γ1 < 1, η > 0, 0 < α1 ≤ 1, 0 < β1 ≤ 1, 0 < γ1 ≤ 1 with initial condition

I (1−α1 )(1−α2 ) N (x, 0) = δ(x) = I (1−β1 )(1−β2 ) N (x, 0) = I (1−γ1 )(1−γ2 ) N (x, 0), f or(x ∈ ℜ) (39) and limN (x, t) = 0, t > 0 where δ(x) is the Dirac delta function. Here ξ is a constant that describes nonlinearity in the system, and φ(x, t) is a nonlinear function which belongs to the area of reaction-diffusion; then there exists the following formula for the solution of the equation (38) subject to the initial condition (39). Z ∞ r X (−a)r ∞ −kx X r r r−l (α1 −γ1 )r+(γ1 −β1 )l+α1 −1 N (x, t) = e a b t × 2π l 0 0 −∞ r=0 [t

α2 (1−α1 )

l r+1 Eα1 ,α2 (1−α1 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 (−ctα1 )+

a0 tβ2 (1−β2 ) Eαr+1 (−ctα1 )+ 1 ,β2 (1−β2 )+(α1 −γ1 )r+(γ1 −β1 )l+α1 b0 (−ctα1 )Eαr+1 ]+ 1 ,(α1 −β1 )r+(γ1 −β1 )l+α1 Z r ∞ t X (−1)r X r ar0 br−l ξ (α1 −γ1 )r+(γ1 −β1 )l+α1 −1 dξ 0 2pi l 0 r=0 l=0 Z ∞ r+1 e−ikx φ∗ (k, t − ξ)E(α dk 1 −γ1 )r+(γ1 −β1 )l+α1 −∞

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[32] R.K.Saxena, A.M.Mathai and H.J.Haubold, Solution of fractional reaction-diffusion equations in termes of Mittag-Leffler functions, Int.J. Sci.Res.15(2006)1-17. [33] S.G. Samoke, A.A. Kilbas and O.I. Marichev,Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach Sciences Publishers, Amsterdam, 1993. [34] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New YorkHeidelberg-Berlin, 1983. [35] Z.Tomovski, R.Hilfer and H.M. Srivastava, Fractional and Operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integr. Transf. Spec. Funct. 21 (2010), 797-814. [36] H.Wilhelmsson and E.Lazzaro, Reaction-diffusion Problems in the Physics of Hot Plasmas, Institute of Physics Publishing, Bristol and Philadelphia 2001. [37] W.Wyss,The fractional Black-Scholes equation. Fract. Calc. Appl. Anal.(3)(2000), 51-61. [38] S.B.Yuste and K.Lindenberg,Subdiffusion-limited A+A reactions, Phys. Rev. Lett. 87 (11) (2001) 118301. Gunvant A. Birajdar Tata Institute of Social Sciences, Tuljapur Campus Tuljapur, Dist: Osmanabad (MS)-413 601 India. E-mail address: [email protected] D. B. Dhaigude Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada, University, Aurangabad. 431 004, (M.S) India. E-mail address: [email protected]