Solutions of fractional reaction-diffusion equations in terms of the
  H-function
arXiv:0704.0329v2  [math.PR]  7 Aug 2007
SOLUTIONS OF FRACTIONAL REACTION-DIFFUSION
EQUATIONS IN TERMS OF THE H-FUNCTION
H.J. HAUBOLD
Office for Outer Space Affairs, United Nations, Vienna International Centre
P.O. Box 500, A-1400, Vienna, Austria and
Centre for Mathematical Sciences, Pala Campus
Arunapuram P.O., Pala-686 574, Kerala, India
A.M .MATHAI
Department of Mathematics and Statistics, McGill University
Montreal, Canada H3A 2K6 and
Centre for Mathematical Sciences, Pala Campus
Arunapuram P.O., Pala-686 574, Kerala, India
R.K. SAXENA
Department of Mathematics and Statistics, Jai Narain Vyas University
Jodhpur-342004, India
Abstract. This paper deals with the investigation of the solution of an unified
fractional reaction-diffusion equation associated with the Caputo derivative as
the time-derivative and Riesz-Feller fractional derivative as the space-derivative.
The solution is derived by the application of the Laplace and Fourier transforms
in closed form in terms of the H-function. The results derived are of general
nature and include the results investigated earlier by many authors, notably
by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time
fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction-
diffusion equations. The advantage of using Riesz-Feller derivative lies in the
fact that the solution of the fractional reaction-diffusion equation containing this
derivative includes the fundamental solution for space-time fractional diffusion,
which itself is a generalization of neutral fractional diffusion, space-fractional
diffusion, and time-fractional diffusion. These specialized types of diffusion can
be interpreted as spatial probability density functions evolving in time and are
expressible in terms of the H-functions in compact form.
1 Introduction
The review of the theory and applications of reaction-diffusion systems is con-
tained in many books and articles. In recent work authors have demonstrated
the depth of mathematics and related physical issues of reaction-diffusion equa-
tions such as nonlinear phenomena, stationary and spatio-temporal dissipative
pattern formation, oscillations, waves etc. (Frank, 2005; Grafiychuk, Datsko,
http://arxiv.org/abs/0704.0329v2
and Meleshko, 2006, 20076). In recent time, interest in fractional reaction-
diffusion equations has increased because the equation exhibits self-organization
phenomena and introduces a new parameter, the fractional index, into the equa-
tion. Additionally, the analysis of fractional reaction-diffusion equations is of
great interest from the analytical and numerical point of view.
The objective of this paper is to derive the solution of an unified model of
reaction-diffusion system (14), associated with the Caputo derivative and the
Riesz-Feller derivative. This new model provides the extension of the models
discussed earlier by Mainardi, Luchko, and Pagnini (2001), Mainardi, Pagnini,
and Saxena (2005), and Saxena, Mathai, and Haubold (2006a). The present
study is in continuation of our earlier work, Haubold and Mathai (1995, 2000)
and Saxena, Mathai, and Haubold (2006a, 2006b).
2 Results Required in the Sequel
In view of the results
J−1/2(x) =
cosx. (1)
and (Mathai and Saxena, 1978, p. 49), the cosine transform of the H-function
is given by
tρ−1cos(kt)Hm,np,q
(ap,Ap)
(bq,Bq)
dt (2)
n+1,m
q+1,p+2
(1−bq,Bq),(
(ρ,µ),(1−ap,ap),(
, (3)
where Re[ρ + µmin1≤j≤m(
)] > 0, Re[ρ+ µmax1≤j≤n
] < 0, |argα| < 1
πΩ, Ω >
k > 0 and Ω =
j=1 Bj −
j=m+1 Bj +
j=1 Aj −
j=n+1 Aj .
The Riemann-Liouville fractional integral of order ν is defined by (Miller and
Ross, 1993, p. 45; Kilbas et al., 2006)
t N(x, t) =
(t − u)ν−1N(x, u)du, (4)
where Re(ν) > 0.
The following fractional derivative of order α > 0 is introduced by Caputo
(1969; see also Kilbas et al., 2006) in the form
t f(x, t) =
Γ(m − α)
f (m)(x, τ)dτ
(t − τ)α+1−m
, m − 1 < α ≤ m, Re(α) > 0, m ∈ N.
∂mf(x, t)
, if α = m. (5)
where ∂
f(x, t) is the mth partial derivative of f(x,t) with respect to t.
The Laplace transform of the Caputo derivative is given by Caputo (1969;
see also Kilbas et al., 2006) in the form
L {0D
t f(x, t); s} = s
αF (x, s)−
sα−r−1f (r)(x, 0+), (m− 1 < α ≤ m). (6)
Following Feller (1952, 1971), it is conventional to define the Riesz-Feller
space-fractional derivative of order α and skewness θ in terms of its Fourier
transform as
F {xD
θ f(x); k} = −Ψ
α(k)f
∗(k), (7)
where
Ψθα(k) = |k|
αexp[i(signk)
], 0 < α ≤ 2, |θ| ≤ min {α, 2 − α} . (8)
When θ = 0, then (8) reduces to
F {xD
0 f(x); k} = −|k|
α, (9)
which is the Fourier transform of the Weyl fractional operator, defined by
xf(t) =
Γ(n − µ)
f(u)du
(t − u)µ−n+1
. (10)
This shows that the Riesz-Feller operator may be regarded as a generalization
of the Weyl operator.
Further, when θ = 0, we have a symmetric operator with respect to x that
can be interpreted as
0 = −
This can be formally deduced by writing −(k)α = −(k2)α/2. For 0 < α < 2 and
|θ| ≤ min {α, 2 − α}, the Riesz-Feller derivative can be shown to possess the
following integral representation in the x domain:
θ f(x) =
Γ(1 + α)
sin[(α + θ)π/2]
f(x + ξ) − f(x)
+ sin[(α − θ)π/2]
f(x − ξ) − f(x)
. (12)
Finally, we need the following property of the H-function (Mathai and Sax-
ena, 1978)
Hm,np,q
(ap,ap)
(bq ,Bq)
Hm,np,q
(ap,Ap/δ)
(bq,Bq/δ
, (δ > 0). (13)
3 Unified Fractional Reaction-Diffusion Equa-
In this section, we will investigate the solution of the reaction-diffusion equation
(14) under the initial conditions (15). The result is given in the form of the
following
Theorem. Consider the unified fractional reaction-diffusion model
t N(x, t) = ηxD
θ N(x, t) + Φ(x, t), (14)
where η, t > 0, x ∈ r; α, θ, β are real parameters with the constraints
0 < α ≤ 2, |θ| ≤ min(α, 2 − α), 0 < β ≤ 2, and the initial conditions
N(x, 0) = f(x), Nt(x, 0) = g(x) ); for x ∈ R,
|x|→±∞ N(x, t) = 0, t > 0. (15)
Here Nt(x, 0) means the first partial derivative of N(x, t) with respect to t
evaluated at t = 0, η is a diffusion constant and Φ(x, t) is a nonlinear function
belonging to the area of reaction-diffusion. Further xD
θ is the Riesz-Feller
space-fractional derivative of order α and asymmetry θ. 0D
t is the Caputo
time-fractional derivative of order β. Then for the solution of (14), subject to
the above constraints, there holds the formula
N(x, t) =
f∗(k)Eβ,1(−ηt
βΨθα(k))exp(−ikx)dk (16)
tg∗(k)Eβ,2(−ηk
αtβΨθα(k))exp(−ikx)dk
ξβ−1dξ
Φ∗(k, t − ξ)Eβ,β(−ηk
αtβΨθα(k))exp(−ikx)dk.
In equation (16) and the following, Eα,β(z) denotes the generalized Mittag-
Leffler function (Saxena, Mathai, and Haubold, 2004; Berberan-Santos, 2005;
Chamati and Tonchev, 2006).
Proof. If we apply the Laplace transform with respect to the time variable t,
Fourier transform with respect to space variable x, and use the initial conditions
(15) and the formula (7), then the given equation transforms into the form
∼(k, s) − sβ−1f∗(k) − sβ−2g∗(k) = −ηΨθα(k)N
∼(k, s) + Φ
∼(k, s),
where according to the conventions followed , the symbol ∼ will stand for the
Laplace transform with respect to time variable t and * represents the Fourier
transform with respect to space variable x.
Solving for N
∼ , it yields
∼(k, s) =
f∗(k)sβ−1
sβ + ηΨθα(k)
g∗(k)sβ−2
sβ + ηΨθα(k)
sβ + ηΨθα(k)
. (17)
On taking the inverse Laplace transform of (17) and applying the formula
a + sα
= tα−βEα,α−β+1(−at
α), (18)
where Re(s) > 0, Re(α) > 0, Re(α − β) > −1; it is seen that
N∗(k, t) = f∗(k)Eβ,1(−ηt
βΨθα(k)) + g
∗(k)tEβ,2(−ηt
βΨθα(k))
Φ∗(k, t − ξ)ξβ−1Eβ,β(−ηΨ
α(k)ξ
β)dξ. (19)
The required solution (16) is now obtained by taking the inverse Fourier trans-
form of (19). This completes the proof of the theorem.
4 Special Cases
When g(x) = 0, then by the application of the convolution theorem of the
Fourier transform to the solution (16) of the theorem, it readily yields
Corollary 1. The solution of the fractional reaction-diffusion equation
N(x, t) − η
N(x, t) = Φ(x, t), x ∈ R, t > 0, η > 0, (20)
with initial conditions
N(x, 0) = f(x), Nt(x, 0) = 0 for x ∈ R, 1 < β ≤ 2,
x→±∞ N(x, t) = 0, (21)
where η is a diffusion constant and Φ(x, t) is a nonlinear function belonging to
the area of reaction-diffusion, is given by
N(x, t) =
G1(x − τ, t)f(τ)dτ
(t − ξ)β−1dξ
G2(x − τ, t − ξ)Φ(τ, ξ)dτ, (22)
where
α − θ
G1(x, t) =
exp(−ikx)Eβ,1(−η|t
β |Ψθα(k))dk (23)
η1/αtβ/α
(1,1/α),(β,β/α),(1,ρ)
(1,1/α),(1,1),(1,ρ)
, (α > 0)
G2(x, t) =
exp(−ikx)Eβ,β(−ηt
βΨθα(k))dk
η1/αtβ/α
(1,1/α),(β,β/α),(1,ρ)
(1,1/α),(1,1),(1,ρ)
, (α > 0). (24)
In deriving the above results, we have used the inverse Fourier transform formula
F−1[Eβ,γ(−ηt
βΨαθ (k)); x] =
3,3 [
η1αtβ/α
(1,1/α),(γ,β/α),(1,ρ)
(1,1/α),(1,1),(1,ρ)
], (25)
where Re(β) > 0, Re(γ) > 0, which can be established by following a procedure
similar to that employed by Mainardi, Luchko, and Pagnini (2001). Next , if
we set f(x) = δ(x), Φ = 0, g(x) = 0, where δ(x) is the Dirac delta-function,
then we arrive at the following interesting result given by Mainardi, Pagnini,
and Saxena (2005).
Corollary 2. Consider the following space-time fractional diffusion model
∂βN(x, t)
= η xD
θ N(x, t), η > 0, x ∈ R, 0 < β ≤ 2, (26)
with the initial conditions N(x, t = 0) = δ(x), Nt(x, 0) = 0,
x→±∞ N(x, t) = 0
where η is a diffusion constant and δ(x) is the Dirac delta-function. Then for
the fundamental solution of (26) with initial conditions, there holds the formula
N(x, t) =
3.3 [
(ηtβ)1/α
(1,1/α),(1,β/α),(1,ρ)
(1,1/α),(1,1),(1,ρ)
], (27)
where ρ = α−θ
Some interesting special cases of (26) are enumerated below.
(i) We note that for α = β, Mainardi, Pagnini, and Saxena (2005) have
shown that the corresponding solution of (26), denoted by Nθα, which we call as
the neutral fractional diffusion, can be expressed in terms of elementary function
and can be defined for x > 0 as
Neutral fractional diffusion: 0 < α = β < 2; θ ≤ min {α, 2 − α} ,
Nθα(x) =
xα−1sin[(π/2)(α − θ)]
1 + 2xαcos[(π/2)(α − θ)] + x2α
. (28)
The neutral fractional diffusion is not studied at length in the literature.
Next we derive some stable densities in terms of the H-functions as special
cases of the solution of the equation (26)
(ii) If we set β = 1, 0 < α < 2; θ ≤ min {α, 2 − α}then (26) reduces to space
fractional diffusion equation, which we denote by Lθα(x) is the fundamental
solution of the following space-time fractional diffusion model:
∂N(x, t)
= η xD
θ N(x, t), η > 0, x ∈ R, (29)
with the initial conditions N(x, t = 0) = δ(x), limx→±∞N(x, t) = 0,, where η is a
diffusion constant and δ(x) is the Dirac-delta function. Hence for the solution
of (29) there holds the formula
Lθα(x) =
α(ηt)1/α
(ηt)1/α
(1,1),(ρ,ρ)
),(ρ,ρ)
, 0 < α < 1, |θ| ≤ α, (30)
where ρ = α−θ
. The density represented by the above expression is known as
α-stable Lévy density. Another form of this density is given by
Lθα(x) =
α(ηt)1/α
(ηt)1/α
(1− 1
),(1−ρ,ρ)
(0,1),(1−ρ,ρ)
, 1 < α < 2, |θ| ≤ 2 − α,
(iii) Next, if we take α = 2, 0 < β < 2, θ = 0, then we obtain the time
fractional diffusion, which is governed by the following time fractional diffusion
model:
∂βN(x, t)
N(x, t), η > 0, x ∈ R, 0 < β ≤ 2, (32)
with the initial conditions N(x, t = 0) = δ(x), Nt(x, 0) = 0,
x→±∞ N(x, t) = 0
where η is a diffusion constant and δ(x) is the Dirac delta-function, whose
fundamental solution is given by the equation
N(x, t) =
(ηtβ)1/2
(1,β/2)
(1,1)
. (33)
(iv) Further, if we set α = 2, β = 1 and θ → 0 then for the fundamental
solution of the standard diffusion equation
N(x, t) = η
N(x, t), (34)
with initial condition
N(x, t = 0) = δ(x), limx→±∞N(x, t) = 0, (35)
there holds the formula
N(x, t) =
η1/2t1/2
(1,1/2)
(1,1)
= (4πηt)−1/2exp[−
], (36)
which is the classical Gaussian density. For further details of these special cases
based on the Green function, one can refer to the paper by Mainardi, Luchko,
and Pagnini (2001) and Mainardi, Pagnini, and Saxena (2005).
Remark. Fractional order moments and the asymptotic expansion of the solu-
tion (27) are discussed by Mainardi, Luchko, and Pagnini (2001).
Finally, for β = 1/2 in (14), we arrive at
Corollary 3. Consider the following fractional reaction-diffusion model
t N(x, t) = ηxD
θ N(x, t) + Φ(x, t), (37)
where η, t > 0, x ∈ R; α, θ are real parameters with the constraints
0 < α ≤ 2, |θ| ≤ min(α, 2 − α), and the initial conditions
N(x, 0) = f(x), for x ∈ R, limx→±∞N(x, t) = 0. (38)
Here η is a diffusion constant and Φ(x, t) is a nonlinear function belonging to
the area of reaction-diffusion. Further xD
θ is the Riesz-Feller space fractional
derivative of order α and asymmetry θ and D
t is the Caputo time-fractional
derivative of order 1/2. Then for the solution of (37), subject to the above
constraints, there holds the formula
N(x, t) =
f∗(k)E1/2,1(−ηt
βΨθα(k))exp(−ikx)dk (39)
ξ−1/2dξ
Φ∗(kct − ξ)E 1
(−ηkαt1.2Ψθα(k))exp(−ikx)dk.
If we set θ = 0 in (39), then it reduces to the result recently obtained by the
authors (2006a) for the fractional reaction-diffusion equation.
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