arXiv:0704.1916v2  [math.CA]  7 Aug 2007
SOLUTIONS OF CERTAIN FRACTIONAL KINETIC
EQUATIONS AND A FRACTIONAL DIFFUSION EQUATION
R.K. SAXENA
Department of Mathematics and Statistics, Jai Narain Vyas University
Jodhpur-342 004, India
A.M. MATHAI
Department of Mathematics and Statistics, McGill University
Montreal, Canada H3A 2K6
Centre for Mathematical Sciences, Pala Campus, Pala-686 574, Kerala, India
H.J. HAUBOLD
Office for Outer Space Affairs, United Nations
P.O. Box 500, A-1400 Vienna, Austria
Centre for Mathematical Sciences, Pala Campus, Pala-686 574, Kerala, India
Abstract. In view of the usefulness and importance of kinetic equations in cer-
tain physical problems, the authors derive the explicit solution of a fractional
kinetic equation of general character, that unifies and extends earlier results.
Further, an alternative shorter method based on a result developed by the au-
thors is given to derive the solution of a fractional diffusion equation.
1 Introduction
Fractional reaction/diffusion equations involve fractional derivatives with re-
spect to time and space and are studied to describe anomalous reaction/diffusion
of dynamic systems with chaotic motion. Fractional kinetic equation for Hamil-
tonian chaos is discussed by Zaslavsky (1994). Solutions and applications of cer-
tain kinetic equations are studied by Saichev and Zaslavsky (1997). Solutions of
a fractional kinetic equation is investigated by Haubold and Mathai (2000) for
a simple production-destruction mechanism. This equation was generalized by
Saxena, Mathai, and Haubold (2002). In recent articles, Saxena, Mathai, and
Haubold (2002, 2004a, 2004b) discussed the solution of a number of generalized
fractional kinetic equations . In the present article, we investigate the solution
of a unified fractional kinetic equation, which provides unification and extension
of results on fractional kinetic equations given earlier by Haubold and Mathai
(2000) and Saxena, Mathai, and Haubold (2002, 2004a). We also present the
solution of a fractional integral equation discussed by Miller and Ross (1993).
Further, an alternative proof of the solution of a fractional diffusion equation
http://arxiv.org/abs/0704.1916v2
given earlier by Kochubei (1990) is investigated, which is based upon a result
given by Saxena, Mathai, and Haubold (2006). Most of the results are obtained
in terms of generalized Mittag-Leffler functions in elegant and compact forms,
which are suitable for numerical computation.
The paper is organized as follows. Section 2 contains the solution of a unified
fractional kinetic equation while Section 3 considers special cases of the equation.
A shorter alternative method for the solution of a diffusion equation discussed
earlier by Kochubei (1990) is presented in Section 4. A series representation
and asymptotic expansion of the solution are given in Section 5. Incidentally,
an H-function representation of a one-sided Lévy stable density is also obtained.
2 Unified fractional kinetic equations
In this Section, we present a method based on Laplace transform for deriving
the solution of the unified fractional kinetic equations.
Theorem 1. If Re(νj) > 0, aj > 0, j ∈ N, and f(t) be a given function, defined
on ℜ+, then the equation
N(t)−N0f(t) = −
aj 0D
t N(t), (1)
is solvable and its particular solution is given by
N(t) = N0
(−1)l
r1+...+rn−1=l
(r1)! . . . (rn−1)!
(aµ+1)
f(u)(t− u)
νµ+1−1
(l+1)
[−a1(t− u)
ν1 ]du, (2)
where the summation in (2) is taken over all nonnegative integers r1, . . . , rn
such that r1 + . . . + rn−1 = l, and provided that the series and integral in (2)
are convergent. Here 0D
t , j ∈ N are Riemann-Liouville fractional integrals,
defined by
t f(t) =
(t− u)ν−1f(u)du,Re(ν) > 0, (3)
with 0D
t f(t) = f(t) (Oldham and Spanier, 1974; Miller and Ross, 1993),
Eδβ,γ(z) is the generalized Mittag-Leffler function, defined by Prabhakar (1971)
in terms of series representation as
Eδβ,γ(z) =
(δ)τz
Γ(βτ + γ)(τ)!
(β, γ, δ ∈ C,Re(β) > 0, Re(γ) > 0). (4)
Proof. By the application of the convolution theorem of the Laplace transform
(Erdélyi et al., 1953, p. 259) to (3), we find that
t f(t); s
L(f(t)),
= s−νf∼(s), (5)
where f∼(s) =
e−stf(t)dt, s ∈ C,Re(s) > 0. Applying Laplace transform
to (1) and using (5), it gives
N∼(s) =
1 + a1s−ν1 + . . .+ ans−νn
= N0f
(−1)l
j=1 aj+1s
−νj+1
(1 + a1s−ν1)l+1
j=2 ajs
1 + a1s−ν1
If we employ the identity (Abramowitz and Stegun, 1968, p. 823)
(x1 + . . .+ xm)
r1+...+rn=l
(r1)! . . . (rn)!
xrµµ , (7)
where the summation is taken over all nonnegative integers, r1, . . . , rn, such
that r1 + . . .+ rn = l, then for |a1s
−ν1 | < 1, (7) transforms into the form
N∼(s) = N0f
(−1)l
r1+...+rn−1=l
r1>...rn−1>0
(r1)! . . . (rn−1)!
µ=1(aµ+1)
(1 + a1s−ν1)l+1
. (8)
Taking the inverse Laplace transform of (8) by making use of the formula (Kil-
bas, Saigo, and Saxena, 2004, eq. (12))
s−γ(1− as−β)−δ; t
= tγ−1Eδβ,γ(at
β), (9)
where Re(s) > |a|1/Re(γ), Re(γ) > 0, Re(s) > 0,
and applying the convolution theorem of the Laplace transform, the result (2)
is established.
Remark 1. The generalized Mittag-Leffler function defined by (4) is studied by
Prabhakar (1971) and Kilbas, Saigo, and Saxena (2004). Recently this function
is used in the theory of finite-size scaling of systems with strong anisotropy and
long-range interaction by Chamati and Tonshev (2006).
3 Special cases
Some special cases of Theorem 1 are of interest to be highlighted. If we set
νj = jν, aj = (
jν (j ∈ N), we obtain
Theorem 2. If Re(ν) > 0, c > 0 and f(x) ∈ ℜ+, then the equation
N(t)−N0f(t) = −
(nr )c
νrD−νrt N(t), (10)
is solvable and its solution has the form
N(t) = N0
f(u)Enν,1[−c
ν(t− u)ν ]du, (11)
where Enν,1(x) is the generalized Mittag-Leffler function defined by (4) and pro-
vided that the integral (11) is convergent.
When n = 1, we obtain the following result given by Hille and Tamarkin
(1930).
Corollary 2.1. Let Re(ν) > 0, c > 0 and let f(x) ∈ ℜ+, then for the solution
of the integral equation
N(t)−N0f(t) = −c
t N(t), (12)
holds the following formula
N(t) = N0
f(u)Eν [−c
ν(t− u)ν ]du, (13)
where Eν(z) is an entire function of order ρ =
and type σ = 1, defined by
Eν(z) =
Γ(µν + 1)
, (ν ∈ C,Re(ν) > 0). (14)
Note 1. The above result has also been given by the authors in a different form
(Saxena, Mathai, and Haubold, 2004a, 2004b).
If we set f(t) = tγ−1Eδν,γ [−(ct)
ν ], Theorem 2 yields
Corollary 2.2. Let Re(ν) > 0, Re(γ) > 0, c > 0, then for the solution of the
kinetic equation
N(t)−N0t
γ−1Eδν,γ [−(ct)
ν ] = −
(nr )c
t n ∈ N (15)
holds the relation
N(t) = N0t
γ−1Eδ+nν,γ [−(ct)
ν ], n ∈ N. (16)
For f(t) = tρ−1, Theorem 2 yields the following result
Corollary 2.3. If Re(ρ) > 0, Re(ν) > 0, c > 0, then for the solution of the
equation
N(t)−N0t
ρ−1 = −
(nr )c
t N(t), r ∈ N, (17)
holds the relation
N(t) = N0t
ρ−1Γ(ρ)Enν,ρ[−(ct)
ν ], r ∈ N. (18)
For n = 1, eq. (18) reduces to a result given by Saxena, Mathai, and Haubold
(2002, p. 283, eq. (15)). When aj = a
jsνj , for j = 1, . . . , n, we obtain
Theorem 3. Let Re(ν) > 0, a > 0, t > 0, n > 1, |an+1s−(n+1)ν | < 1, and f(x)
be a given function defined on ℜ+, then the equation
N(t)−N0f(t) = −
ar 0D
t N(t), (19)
is solvable and its solution is given by
N(t) = N0
f(u)E(n+1)ν,ν [a
n+1(t− u)(n+1)ν ]du
(t− u)ν−1E(n+1)ν,ν [a
n+1(t− u)(n+1)ν ]du
, (20)
where E(n+1)ν,ν(z) is the generalized Mittag-Leffler function Eα,β(z) defined as
Eα,β(z) =
Γ(αµ+ β)
, (α, β ∈ C,Re(α) > 0, Re(β) > 0) (21)
and provided that the integral in (20) is convergent.
If we take νj = jν, for j = 1, . . . n, then it is interesting to note that Theorem
1 yields the following result given by (Miller and Ross, 1993) in a different form:
Theorem 4. Let Re(ν) > 0, aj > 0, and f(x) be a given function defined on
ℜ+, |a1s
−ν | < 1, then the fractional kinetic equation
N(t)−N0f(t) = −
aj 0D
t N(t), (22)
is solvable and has the solution given by
N(t) = N0
(−1)l
r1+...+rn−1=l
(r1)! . . . (rn−1)!
(aµ+1)
f(u)(t− u)
ν(µ+1)rµ−1
(l+1)
ν(µ+1)rµ
[−a1(t− u)
ν1 ]du,
provided that the series and integral in (23) are convergent.
4 Fractional diffusion equation
In this Section we present an alternative shorter method for deriving the solution
of a diffusion equation discussed earlier by Kochubei (1990).
Theorem 5. Consider the Cauchy problem
t N(x, t) = −c
ν∆N(x, t), (0 < α < 1;x ∈ ℜn; 0 < t ≤ T ), (24)
N(x, t = 0) = δ(x), x ∈ ℜ, lim
N(x, t) = 0 (25)
t is the regularized Caputo (1969) partial fractional derivative with respect
to t, defined by
t N(x, t) =
Γ(1− α)
N(x, s)ds
(t− x)α
N(x, 0)
and ∆ is the Laplacian. The fundamental solution of the above Cauchy problem
is given by
N(x, t) = |x|−nπ−
|x|2t−α
(1,α)
(n/2,1),(1,1)
, (26)
where H
1,2 (.) is the H-function (Mathai and Saxena, 1978).
Proof. Applying the Laplace transform with respect to t, using the result
(Caputo, 1969)
L {0D
t N(x, t)} = s
αN∼(x, s)−
sα−r−1N (r)(x, 0),
m− 1 < α ≤ m, m ∈ N,
and Fourier transform with respect to x, gives
(k, s)− sα−1 = −cν |k|2N∼
(k, s),
where the symbol ”∼“ indicates the Laplace transform with respect to the time
variable t and the symbol ”*“ the Fourier transform with respect to the space
variable x.
Solving for N∼
, we have
(k, s) =
sα + cν |k|2
. (27)
By virtue of the following Fourier transform formula (Samko, Kilbas, andMarichev,
1990, p. 538, eq. (27.1))
|x|(2−n)/2K(n−2)/2(a|x|)
(τ) =
a2 + τ2
, (τ ∈ ℜn;n ∈ N, a > 0),
where the multidimensional Fourier transform with respect to x ∈ ℜn is defined
(FxN)(τ, t) =
N(x, t)eixτdx (τ ∈ ℜn; t > 0) (29)
and Kν(.) is the modified Bessel function of the second kind, yields
Ñ(x, s) = c−νsα−1(2π)−
|x|cν/2
Kn−2/2
2 |x|
. (30)
In order to invert the Laplace transform, we employ the following result
given by the authors (Saxena, Mathai, and Haubold, 2006)
s−ρKν(zs
σ); t
tρ−1H
z2t−2σ
(ρ,2σ)
,1)(− ν
, (31)
where Kν(x) is the modified Bessel function of the second kind, Re(z
0, Re(s) > 0. Thus we obtain the solution in the form
N(x, t) =
(2π)−
4 |x|1−
t−α|x|2
,1),(
. (32)
By virtue of a result in Mathai and Saxena (1978),
xσHm,np,q
(ap,ap)
(bq ,bq)
= Hm,np,q
(ap+σAp,Ap)
(bq+σBq ,Bq)
, (33)
the power of the expression [
t−ν |x|2
/4cν ] can be absorbed inside the H-
function and consequently we obtain
N(x, t) = |π
2x|−n H
t−α|x|2
(1,α)
,1),(1,1)
. (34)
Remark 1. If we employ the identity (Mathai and Saxena, 1978)
Hm,np,q
(ap,Ap)
(bq ,Bq)
Hm,np,q
(ap,Ap/λ)
(bq,Bq/λ)
, λ > 0 (35)
the solution given by (32) can be expressed in the form
N(x, t) =
2 x|−n H
4cνtα
(1,1)
),(1, 1
, (36)
where α > 0. We also note that the above form of the solution is due to
Schneider and Wyss (1989). There is one importance of our result (32) that it
includes the Lévy stable density in terms of the H-function as shown in (45).
Similarly, using the identity (35) we arrive at
N(c, t) =
2x|−nH
2 |x|
),(1, 1
, (37)
where n is not an even integer. This form of the H-function is useful in deter-
mining its expansion in powers of x. Due to importance of the solution, we also
discuss its series representation and behavior.
5 Series representation of the solution
Using the series expansion for the H-function given in Mathai and Saxena (1978),
it follows that
(1,1)
),(1, 1
)Γ(1 − s
Γ(1 − s)
xsds (38)
Γ(1− n
− l)(−1)lxα(
Γ(1− an
− αl)(l)!
− 1− l)(−1)lxα(1+l)
Γ(1− α− αl)(l!)
where n is not an even integer.
Thus for n = 1, we find that
N(x, t) =
(−1)l
Γ(1− α(l + 1)/2)(l!)
, (39)
where A = x
and the duplication formula for the gamma function is used.
For n = 2, the H-function of (37) is singular and in this case, the result is
explicitly given by Barkai (2001) in the form
N(x, t) ∼
πΓ(1− α)tα
]. (40)
For n = 3, the series expansion is given by
N(x, t) =
4πt3α/2A1/2
(−1)lAl/2
Γ[1− α(1 + l/2)]
. (41)
From above, it readily follows that for n = 3 and α 6= 1
N(x, t) ∼
, as x → ∞. (42)
It will not be out of place to mention that the one sided Lévy stable density
can be obtained from Laplace inversion formula (31) by virtue of the identity
(x) =
e−x, (43)
and can be conveniently expressed in terms of the Laplace transform
e−utΦρ(t)dt = e
−uρ , Re(u) > 0, Re(ρ) > 0. (44)
The result is
Φρ(t) =
(1,1)
, (ρ > 0). (45)
Note 2. This result is obtained earlier by Schneider and Wyss (1989) by fol-
lowing a different procedure. Asymptotic behavior of Φα(t) is also given by
Schneider (1986).
In conclusion, we mention that some of the results derived in this article
may find some applications in problems associated with models of long-memory
processes driven by Lévy noise and other related problems, see the article by
Anh, Heyde, and Leonenko (2002).
Acknowledgment. The authors would like to thank the Department of Science
and Technology, Government of India, New Delhi, for the financial assistance for
this work under project No. SR/S4/MS:287/05 which enabled this collaboration
possible.
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  In view of the usefulness and importance of the kinetic equation in certain
physical problems, the authors derive the explicit solution of a fractional
kinetic equation of general character, that unifies and extends earlier
results. Further, an alternative shorter method based on a result developed by
the authors is given to derive the solution of a fractional diffusion equation.

Introduction
Fractional reaction/diffusion equations involve fractional derivatives with re-
spect to time and space and are studied to describe anomalous reaction/diffusion
of dynamic systems with chaotic motion. Fractional kinetic equation for Hamil-
tonian chaos is discussed by Zaslavsky (1994). Solutions and applications of cer-
tain kinetic equations are studied by Saichev and Zaslavsky (1997). Solutions of
a fractional kinetic equation is investigated by Haubold and Mathai (2000) for
a simple production-destruction mechanism. This equation was generalized by
Saxena, Mathai, and Haubold (2002). In recent articles, Saxena, Mathai, and
Haubold (2002, 2004a, 2004b) discussed the solution of a number of generalized
fractional kinetic equations . In the present article, we investigate the solution
of a unified fractional kinetic equation, which provides unification and extension
of results on fractional kinetic equations given earlier by Haubold and Mathai
(2000) and Saxena, Mathai, and Haubold (2002, 2004a). We also present the
solution of a fractional integral equation discussed by Miller and Ross (1993).
Further, an alternative proof of the solution of a fractional diffusion equation
http://arxiv.org/abs/0704.1916v2
given earlier by Kochubei (1990) is investigated, which is based upon a result
given by Saxena, Mathai, and Haubold (2006). Most of the results are obtained
in terms of generalized Mittag-Leffler functions in elegant and compact forms,
which are suitable for numerical computation.
The paper is organized as follows. Section 2 contains the solution of a unified
fractional kinetic equation while Section 3 considers special cases of the equation.
A shorter alternative method for the solution of a diffusion equation discussed
earlier by Kochubei (1990) is presented in Section 4. A series representation
and asymptotic expansion of the solution are given in Section 5. Incidentally,
an H-function representation of a one-sided Lévy stable density is also obtained.
2 Unified fractional kinetic equations
In this Section, we present a method based on Laplace transform for deriving
the solution of the unified fractional kinetic equations.
Theorem 1. If Re(νj) > 0, aj > 0, j ∈ N, and f(t) be a given function, defined
on ℜ+, then the equation
N(t)−N0f(t) = −
aj 0D
t N(t), (1)
is solvable and its particular solution is given by
N(t) = N0
(−1)l
r1+...+rn−1=l
(r1)! . . . (rn−1)!
(aµ+1)
f(u)(t− u)
νµ+1−1
(l+1)
[−a1(t− u)
ν1 ]du, (2)
where the summation in (2) is taken over all nonnegative integers r1, . . . , rn
such that r1 + . . . + rn−1 = l, and provided that the series and integral in (2)
are convergent. Here 0D
t , j ∈ N are Riemann-Liouville fractional integrals,
defined by
t f(t) =
(t− u)ν−1f(u)du,Re(ν) > 0, (3)
with 0D
t f(t) = f(t) (Oldham and Spanier, 1974; Miller and Ross, 1993),
Eδβ,γ(z) is the generalized Mittag-Leffler function, defined by Prabhakar (1971)
in terms of series representation as
Eδβ,γ(z) =
(δ)τz
Γ(βτ + γ)(τ)!
(β, γ, δ ∈ C,Re(β) > 0, Re(γ) > 0). (4)
Proof. By the application of the convolution theorem of the Laplace transform
(Erdélyi et al., 1953, p. 259) to (3), we find that
t f(t); s
L(f(t)),
= s−νf∼(s), (5)
where f∼(s) =
e−stf(t)dt, s ∈ C,Re(s) > 0. Applying Laplace transform
to (1) and using (5), it gives
N∼(s) =
1 + a1s−ν1 + . . .+ ans−νn
= N0f
(−1)l
j=1 aj+1s
−νj+1
(1 + a1s−ν1)l+1
j=2 ajs
1 + a1s−ν1
If we employ the identity (Abramowitz and Stegun, 1968, p. 823)
(x1 + . . .+ xm)
r1+...+rn=l
(r1)! . . . (rn)!
xrµµ , (7)
where the summation is taken over all nonnegative integers, r1, . . . , rn, such
that r1 + . . .+ rn = l, then for |a1s
−ν1 | < 1, (7) transforms into the form
N∼(s) = N0f
(−1)l
r1+...+rn−1=l
r1>...rn−1>0
(r1)! . . . (rn−1)!
µ=1(aµ+1)
(1 + a1s−ν1)l+1
. (8)
Taking the inverse Laplace transform of (8) by making use of the formula (Kil-
bas, Saigo, and Saxena, 2004, eq. (12))
s−γ(1− as−β)−δ; t
= tγ−1Eδβ,γ(at
β), (9)
where Re(s) > |a|1/Re(γ), Re(γ) > 0, Re(s) > 0,
and applying the convolution theorem of the Laplace transform, the result (2)
is established.
Remark 1. The generalized Mittag-Leffler function defined by (4) is studied by
Prabhakar (1971) and Kilbas, Saigo, and Saxena (2004). Recently this function
is used in the theory of finite-size scaling of systems with strong anisotropy and
long-range interaction by Chamati and Tonshev (2006).
3 Special cases
Some special cases of Theorem 1 are of interest to be highlighted. If we set
νj = jν, aj = (
jν (j ∈ N), we obtain
Theorem 2. If Re(ν) > 0, c > 0 and f(x) ∈ ℜ+, then the equation
N(t)−N0f(t) = −
(nr )c
νrD−νrt N(t), (10)
is solvable and its solution has the form
N(t) = N0
f(u)Enν,1[−c
ν(t− u)ν ]du, (11)
where Enν,1(x) is the generalized Mittag-Leffler function defined by (4) and pro-
vided that the integral (11) is convergent.
When n = 1, we obtain the following result given by Hille and Tamarkin
(1930).
Corollary 2.1. Let Re(ν) > 0, c > 0 and let f(x) ∈ ℜ+, then for the solution
of the integral equation
N(t)−N0f(t) = −c
t N(t), (12)
holds the following formula
N(t) = N0
f(u)Eν [−c
ν(t− u)ν ]du, (13)
where Eν(z) is an entire function of order ρ =
and type σ = 1, defined by
Eν(z) =
Γ(µν + 1)
, (ν ∈ C,Re(ν) > 0). (14)
Note 1. The above result has also been given by the authors in a different form
(Saxena, Mathai, and Haubold, 2004a, 2004b).
If we set f(t) = tγ−1Eδν,γ [−(ct)
ν ], Theorem 2 yields
Corollary 2.2. Let Re(ν) > 0, Re(γ) > 0, c > 0, then for the solution of the
kinetic equation
N(t)−N0t
γ−1Eδν,γ [−(ct)
ν ] = −
(nr )c
t n ∈ N (15)
holds the relation
N(t) = N0t
γ−1Eδ+nν,γ [−(ct)
ν ], n ∈ N. (16)
For f(t) = tρ−1, Theorem 2 yields the following result
Corollary 2.3. If Re(ρ) > 0, Re(ν) > 0, c > 0, then for the solution of the
equation
N(t)−N0t
ρ−1 = −
(nr )c
t N(t), r ∈ N, (17)
holds the relation
N(t) = N0t
ρ−1Γ(ρ)Enν,ρ[−(ct)
ν ], r ∈ N. (18)
For n = 1, eq. (18) reduces to a result given by Saxena, Mathai, and Haubold
(2002, p. 283, eq. (15)). When aj = a
jsνj , for j = 1, . . . , n, we obtain
Theorem 3. Let Re(ν) > 0, a > 0, t > 0, n > 1, |an+1s−(n+1)ν | < 1, and f(x)
be a given function defined on ℜ+, then the equation
N(t)−N0f(t) = −
ar 0D
t N(t), (19)
is solvable and its solution is given by
N(t) = N0
f(u)E(n+1)ν,ν [a
n+1(t− u)(n+1)ν ]du
(t− u)ν−1E(n+1)ν,ν [a
n+1(t− u)(n+1)ν ]du
, (20)
where E(n+1)ν,ν(z) is the generalized Mittag-Leffler function Eα,β(z) defined as
Eα,β(z) =
Γ(αµ+ β)
, (α, β ∈ C,Re(α) > 0, Re(β) > 0) (21)
and provided that the integral in (20) is convergent.
If we take νj = jν, for j = 1, . . . n, then it is interesting to note that Theorem
1 yields the following result given by (Miller and Ross, 1993) in a different form:
Theorem 4. Let Re(ν) > 0, aj > 0, and f(x) be a given function defined on
ℜ+, |a1s
−ν | < 1, then the fractional kinetic equation
N(t)−N0f(t) = −
aj 0D
t N(t), (22)
is solvable and has the solution given by
N(t) = N0
(−1)l
r1+...+rn−1=l
(r1)! . . . (rn−1)!
(aµ+1)
f(u)(t− u)
ν(µ+1)rµ−1
(l+1)
ν(µ+1)rµ
[−a1(t− u)
ν1 ]du,
provided that the series and integral in (23) are convergent.
4 Fractional diffusion equation
In this Section we present an alternative shorter method for deriving the solution
of a diffusion equation discussed earlier by Kochubei (1990).
Theorem 5. Consider the Cauchy problem
t N(x, t) = −c
ν∆N(x, t), (0 < α < 1;x ∈ ℜn; 0 < t ≤ T ), (24)
N(x, t = 0) = δ(x), x ∈ ℜ, lim
N(x, t) = 0 (25)
t is the regularized Caputo (1969) partial fractional derivative with respect
to t, defined by
t N(x, t) =
Γ(1− α)
N(x, s)ds
(t− x)α
N(x, 0)
and ∆ is the Laplacian. The fundamental solution of the above Cauchy problem
is given by
N(x, t) = |x|−nπ−
|x|2t−α
(1,α)
(n/2,1),(1,1)
, (26)
where H
1,2 (.) is the H-function (Mathai and Saxena, 1978).
Proof. Applying the Laplace transform with respect to t, using the result
(Caputo, 1969)
L {0D
t N(x, t)} = s
αN∼(x, s)−
sα−r−1N (r)(x, 0),
m− 1 < α ≤ m, m ∈ N,
and Fourier transform with respect to x, gives
(k, s)− sα−1 = −cν |k|2N∼
(k, s),
where the symbol ”∼“ indicates the Laplace transform with respect to the time
variable t and the symbol ”*“ the Fourier transform with respect to the space
variable x.
Solving for N∼
, we have
(k, s) =
sα + cν |k|2
. (27)
By virtue of the following Fourier transform formula (Samko, Kilbas, andMarichev,
1990, p. 538, eq. (27.1))
|x|(2−n)/2K(n−2)/2(a|x|)
(τ) =
a2 + τ2
, (τ ∈ ℜn;n ∈ N, a > 0),
where the multidimensional Fourier transform with respect to x ∈ ℜn is defined
(FxN)(τ, t) =
N(x, t)eixτdx (τ ∈ ℜn; t > 0) (29)
and Kν(.) is the modified Bessel function of the second kind, yields
Ñ(x, s) = c−νsα−1(2π)−
|x|cν/2
Kn−2/2
2 |x|
. (30)
In order to invert the Laplace transform, we employ the following result
given by the authors (Saxena, Mathai, and Haubold, 2006)
s−ρKν(zs
σ); t
tρ−1H
z2t−2σ
(ρ,2σ)
,1)(− ν
, (31)
where Kν(x) is the modified Bessel function of the second kind, Re(z
0, Re(s) > 0. Thus we obtain the solution in the form
N(x, t) =
(2π)−
4 |x|1−
t−α|x|2
,1),(
. (32)
By virtue of a result in Mathai and Saxena (1978),
xσHm,np,q
(ap,ap)
(bq ,bq)
= Hm,np,q
(ap+σAp,Ap)
(bq+σBq ,Bq)
, (33)
the power of the expression [
t−ν |x|2
/4cν ] can be absorbed inside the H-
function and consequently we obtain
N(x, t) = |π
2x|−n H
t−α|x|2
(1,α)
,1),(1,1)
. (34)
Remark 1. If we employ the identity (Mathai and Saxena, 1978)
Hm,np,q
(ap,Ap)
(bq ,Bq)
Hm,np,q
(ap,Ap/λ)
(bq,Bq/λ)
, λ > 0 (35)
the solution given by (32) can be expressed in the form
N(x, t) =
2 x|−n H
4cνtα
(1,1)
),(1, 1
, (36)
where α > 0. We also note that the above form of the solution is due to
Schneider and Wyss (1989). There is one importance of our result (32) that it
includes the Lévy stable density in terms of the H-function as shown in (45).
Similarly, using the identity (35) we arrive at
N(c, t) =
2x|−nH
2 |x|
),(1, 1
, (37)
where n is not an even integer. This form of the H-function is useful in deter-
mining its expansion in powers of x. Due to importance of the solution, we also
discuss its series representation and behavior.
5 Series representation of the solution
Using the series expansion for the H-function given in Mathai and Saxena (1978),
it follows that
(1,1)
),(1, 1
)Γ(1 − s
Γ(1 − s)
xsds (38)
Γ(1− n
− l)(−1)lxα(
Γ(1− an
− αl)(l)!
− 1− l)(−1)lxα(1+l)
Γ(1− α− αl)(l!)
where n is not an even integer.
Thus for n = 1, we find that
N(x, t) =
(−1)l
Γ(1− α(l + 1)/2)(l!)
, (39)
where A = x
and the duplication formula for the gamma function is used.
For n = 2, the H-function of (37) is singular and in this case, the result is
explicitly given by Barkai (2001) in the form
N(x, t) ∼
πΓ(1− α)tα
]. (40)
For n = 3, the series expansion is given by
N(x, t) =
4πt3α/2A1/2
(−1)lAl/2
Γ[1− α(1 + l/2)]
. (41)
From above, it readily follows that for n = 3 and α 6= 1
N(x, t) ∼
, as x → ∞. (42)
It will not be out of place to mention that the one sided Lévy stable density
can be obtained from Laplace inversion formula (31) by virtue of the identity
(x) =
e−x, (43)
and can be conveniently expressed in terms of the Laplace transform
e−utΦρ(t)dt = e
−uρ , Re(u) > 0, Re(ρ) > 0. (44)
The result is
Φρ(t) =
(1,1)
, (ρ > 0). (45)
Note 2. This result is obtained earlier by Schneider and Wyss (1989) by fol-
lowing a different procedure. Asymptotic behavior of Φα(t) is also given by
Schneider (1986).
In conclusion, we mention that some of the results derived in this article
may find some applications in problems associated with models of long-memory
processes driven by Lévy noise and other related problems, see the article by
Anh, Heyde, and Leonenko (2002).
Acknowledgment. The authors would like to thank the Department of Science
and Technology, Government of India, New Delhi, for the financial assistance for
this work under project No. SR/S4/MS:287/05 which enabled this collaboration
possible.
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