EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 3, 2014, 312-334
ISSN 1307-5543 – www.ejpam.com

Fractional Helmholtz and Fractional Wave Equations with
Riesz-Feller and Generalized Riemann-Liouville Fractional
Derivatives

Ram K. Saxena1, Živorad Tomovski 2, Trifce Sandev3,∗

1 Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur - 342004, India
2 Faculty of Natural Sciences and Mathematics, Institute of Mathematics, Saints Cyril and Methodius
University, 1000 Skopje, Macedonia
3 Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia

Abstract. The objective of this paper is to derive analytical solutions of fractional order Laplace, Pois-
son and Helmholtz equations in two variables derived from the corresponding standard equations in
two dimensions by replacing the integer order partial derivatives with fractional Riesz-Feller deriva-
tive and generalized Riemann-Liouville fractional derivative recently defined by Hilfer. The Fourier-
Laplace transform method is employed to obtain the solutions in terms of Mittag-Leffler functions,
Fox H-function and an integral operator containing a Mittag-Leffler function in the kernel. Results for
fractional wave equation are presented as well. Some interesting special cases of these equations are
considered. Asymptotic behavior and series representation of solutions are analyzed in detail. Many
previously obtained results can be derived as special cases of those presented in this paper.

2010 Mathematics Subject Classifications: 26A33, 33E12, 33C60, 76R50, 44A10, 42A38.
Key Words and Phrases: Mittag-Leffler functions, Fox H-function, fractional Riesz-Feller derivative,
Hilfer-composite fractional derivative, Laplace-Fourier transform, asymptotic behavior

1. Introduction

Fractional differential equations have been used in different fields of science. To men-
tion a few examples: fractional relaxation equations have applications in the non-exponential
relaxation theory [11, 13, 23–25]; fractional diffusion [31, 32] and fractional Fokker-Planck
equations [30], as well as fractional master equations [10, 16, 33, 41], in the description of
anomalous diffusive processes; fractional wave equations has been used in the theory of vi-
brations of smart materials in media where the memory effects can not be neglected [20, 21];
etc.

∗Corresponding author.

Email addresses: ram.saxena@yahoo.com (R. Saxena), tomovski@pmf.ukim.mk (Ž. Tomovski),
trifce.sandev@drs.gov.mk (T. Sandev)

http://www.ejpam.com 312 c© 2014 EJPAM All rights reserved.



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 313

In the present paper, we introduce a new generalization of time-independent diffusion/wave
equations, i.e. fractional Laplace, fractional Poisson and fractional Helmholtz equations in
two variables in which both space variables x and y are of fractional orders. We use fractional
Riesz-Feller space derivative [6] for the first variable, and generalized Riemann-Liouville (R-L)
fractional derivative [11, 14, 49] for the second variable. Space-time fractional wave equation
with Riesz-Feller space derivative and generalized R-L fractional time derivative is consid-
ered as well. Similar time-dependent models are discussed earlier by many authors, such as
Haubold et al. [28], Saxena [41], Saxena et al. [42–44], Tomovski et al. [49, 51–53], etc.

Such generalized R-L time fractional derivative (or so-called Hilfer-composite fractional
time derivative in [7, 15, 26, 38, 50, 54]) was used by Hilfer [11, 12], Sandev et al. [38]
and Tomovski et al. [54] in the analysis of fractional diffusion equations, obtaining that such
models may be used in context of glass relaxation and aquifer problems. Hilfer-composite
time fractional derivative was also used by Saxena et al. [45] and Garg et al. [8] in the theory
of fractional reaction-diffusion equations, where the obtained results are presented through
Mittag-Leffler (M-L) and Fox H-functions. Furthermore, an operational method for solving
differential equations with the Hilfer-composite fractional derivative is presented in [14, 19].
From the other side, Riesz-Feller fractional derivative has been used in analysis of space-time
fractional diffusion equations by Mainardi, Pagnini and Saxena [27] and Tomovski et al. [54],
where they expressed the solutions in terms of Fox H-function. It is shown that space frac-
tional diffusion equation with fractional Riesz-Feller space derivative [3] gives same results
as those obtained from the continuous time random walk theory for Lévy flights [31, 32].
A numerical scheme for solving fractional diffusion equation with Hilfer-composite fractional
time derivative and Riesz-Feller space fractional derivative is elaborated in [54]. Furthermore,
the quantum fractional Riesz-Feller derivative has been used by Luchko et al. [22, 40] in the
Schrödinger equation for a free particle and a particle in an infinite potential well. Local frac-
tional derivative operators have been used as well [9] in Helmholtz and diffusion equations.

The paper is organized as following. In Section II we give an introduction to the frac-
tional derivatives and integrals used in the paper. Fractional form of the Laplace and Poisson
equations in two variables are considered in Section III. We give analytical results for different
forms of the boundary conditions and for the source term. Asymptotic behavior and series
representation of solutions are given. We also give remarks on the general space-time frac-
tional wave equation for a vibrating string with fractional Riesz-Feller space derivative and
Hilfer-composite fractional time derivative. In Section IV we analyze the fractional Helmholtz
equation for different forms of the boundary conditions and source term. The obtained results
are of general character and include those recently given by Thomas [48]. Conclusions are
given in Section V. At the end of the paper in an Appendix we give definitions, relations, and
some properties of M-L functions and Fox H-function.



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 314

2. Fractional Derivatives and Integrals

The Riesz-Feller fractional derivative of order α and skewness θ is defined by the following
Fourier transform formula [6]

F
�

x Dαθ f (x)
�

(κ) = −ψθα(κ)F
�

f (x)
�

(κ), (1)

where

F
�

f (x)
�

(κ) = f̂ (κ) =

∫ ∞

−∞
f (x)eıκxdx , and

F−1
�

f̂ (κ)
�

(x) =
1

2π

∫ ∞

−∞
f̂ (κ)e−ıκxdκ,

(2)

are Fourier transform and inverse Fourier transform, respectively, and ψθα(κ) is given by

ψθα(κ) = |κ|
α exp

�

ısign(κ)
θπ

2

�

, 0< α≤ 2, |θ | ≤min{α, 2−α}. (3)

Riesz-Feller fractional derivative is a pseudo-differential operator whose symbol −ψθα(κ) is the
logarithm of the characteristic function of a general Lévy strictly stable probability density with
stability index α and asymmetry parameter θ (for details, see Mainardi, Pagnini and Saxena

[27]). For θ = 0 one obtains Riesz fractional derivative x Dα0 = −
�

− d2

dx2

�α/2
, for which

F
�

x Dα0 f (x)
�

(κ) = −|κ|αF
�

f (x)
�

(κ). (4)

This special case has been used in the theory of Lévy flights [31, 32].
In this paper we also use the quantum fractional Riesz-Feller derivative x D∗,α

θ
of order α

and skewness θ , which is defined as a pseudo-differential operator with a symbolψθα(κ) given
by [22, 40]

F
�

x D∗,α
θ

f (x)
�

(κ) =ψθα(κ)F
�

f (x)
�

(κ). (5)

Note that the quantum fractional Riesz-Feller derivative is the Riesz-Feller fractional derivative
(1) multiplied by −1. Thus, the obtained solutions which correspond to the case of fractional
Riesz-Feller space derivative (5) can be easily transformed to those obtained in a case where
the quantum fractional Riesz-Feller space derivative (1) is applied.

The R-L fractional integral is defined by [11, 18, 36]

�

y Iµa+ f
�

(y) =
1
Γ(µ)

∫ y

a

f (y ′)
(y − y ′)1−µ

dy ′, y > a, ℜ(µ)> 0. (6)

For µ= 0, this is the identity operator,
�

y I0
a+ f

�

(y) = f (y). Similarly, R-L fractional derivative
is defined by [11, 18, 36]

�

y Dµa+ f
�

(y) =

�

d
dy

�n
�

y In−µ
a+ f

�

(y), ℜ(µ)> 0, n=
�

ℜ(µ)
�

+ 1, (7)



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 315

where [ℜ(µ)] denotes the integer part of the real number ℜ(µ). Hilfer generalized the frac-
tional derivative (7) by the following fractional derivative of order 0 < µ ≤ 1 and type
0≤ ν≤ 1 [11]:

�

y Dµ,ν
a+ f

�

(y) =

�

y Iν(1−µ)a+
d

dy

�

y I (1−ν)(1−µ)a+ f
�

�

(y). (8)

Note that when 0 < µ ≤ 1, ν = 0, a = 0, the generalized R-L fractional derivative (8) would
correspond to the classical R-L fractional derivative [11, 18, 36]

�

RL
y Dµ0+ f

�

(y) =
d

dy

�

y I (1−µ)0+ f
�

(y). (9)

Conversely, when 0 < µ ≤ 1, ν = 1, a = 0, it corresponds to the Caputo fractional derivative
[2]

�

C
y Dµ0+ f

�

(y) =

�

y I (1−µ)0+
d

dy
f

�

(y). (10)

The difference between fractional derivatives of different types becomes apparent when
we consider their Laplace transform. In Ref. [11] it is found for 0< µ < 1 that

L
�

y Dµ,ν
0+ f (y)

�

(s) = sµL
�

f (y)
�

(s)− sν(µ−1)
�

y I (1−ν)(1−µ)0+ f
�

(0+), (11)

where the initial-value term
�

y I (1−ν)(1−µ)0+ f
�

(0+) is evaluated in the limit y → 0+, in the space
of summable Lebesgue integrable functions

L(0,∞) =

¨

f : ‖ f ‖1 =
∫ ∞

0

| f (y)|dy <∞

«

. (12)

Hilfer, Luchko and Tomovski generalized Hilfer-composite derivative (8) to order
n− 1< µ≤ n (n ∈ N+) and type 0≤ ν≤ 1 in the following way [14]:

�

y Dµ,ν
a+ f

�

(y) =

�

y Iν(n−µ)a+
dn

dyn

�

y I (1−ν)(n−µ)a+ f
�

�

(y). (13)

Its Laplace transform is recently given by Tomovski [49]

L
�

y Dµ,ν
0+ f (y)

�

(s) = sµL
�

f (y)
�

(s)−
n−1
∑

k=0

sn−k−ν(n−µ)−1

�

dk

dyk

�

y I (1−ν)(n−µ)0+ f
�

�

(0+), (14)

where initial-value terms
�

dk

dyk

�

y I (1−ν)(n−µ)0+ f
��

(0+) are evaluated in the limit y → 0+.
Various operators for fractional integration were investigated by Srivastava and Saxena

[46]. Srivastava and Tomovski [47] introduced an integral operator (Eω;γ,κ
a+;α,βϕ)(y) of form

(yE
ω;γ,κ
a+;α,βϕ)(y) =

∫ y

a
(y − ξ)β−1Eγ,κ

α,β(ω(y − ξ)
α)ϕ(ξ)dξ, (15)



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 316

where Eγ,κ
α,β(z) is the four parameter M-L function (A5). In case when ω= 0 the integral oper-

ator (15) would correspond to the classical R-L integral operator. For κ = 1 integral operator
(15) becomes the Prabhakar integral operator

�

yE
ω;γ
a+;α,βϕ

�

(y) [35], which was extensively
investigated by Kilbas, Saigo and Saxena [17], and will be used here with γ= 1 for represen-
tation of solutions. These generalized integral operators was shown to appear in the expression
of solutions of fractional diffusion/wave equations with source terms [38, 39, 51–53].

3. Fractional Laplace and Fractional Poisson Equations

In this section we investigate generalized form of the Laplace equation for the field variable
N(x , y) in two dimensions

∂ 2

∂ x2
N(x , y) +

∂ 2

∂ y2
N(x , y) = 0, (16)

on the upper half plane y ≥ 0 and −∞< x <∞, with boundary conditions

N(x , 0+) = f (x),
d

dy
N(x , 0+) = g(x), (17a)

lim
x→±∞

N(x , y) = 0. (17b)

Since there is no dependence on time variable, Laplace equation gives the steady-state solu-
tion of, for example, diffusion/heat conduction and wave equations. Thus, initial conditions
are not required, only we use boundary conditions, which may be defined in a different ways.
Therefore, Laplace equation in two dimensions (16) may arise in analysis of two dimensional
steady-state diffusion/heat conduction, static deflection of a membrane, electrostatic poten-
tial, etc.

If in the Laplace equation in two dimensions (16) we add a source term Φ(x , y), then it
becomes Poisson equation

∂ 2

∂ x2
N(x , y) +

∂ 2

∂ y2
N(x , y) = Φ(x , y). (18)

This equation has applications in different field of science, such as gravitation theory, elec-
tromagnetism, elasticity, etc. For example, N(x , y) may be interpreted as a temperature field
variable subject to external force (source) Φ(x , y).

Before to formulate the corresponding fractional form of the Laplace equation (16) and
Poisson equation (18) we prove the following Lemmas.

Lemma 1. Let 1 < µ ≤ 2, 0 ≤ ν ≤ 1, ς ≥ 0 and r̂(κ) is a given function. Then the following
relation holds true

L −1

�

sς−ν(2−µ)

sµ ± r̂(κ)

�

(y) = y1−(1−ν)(2−µ)−ςEµ,2−(1−ν)(2−µ)−ς
�

∓r̂(κ)yµ
�

, (19)

where Eα,β(z) is the two parameter M-L function (A2).



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 317

Proof. From relation (A3), we directly prove Lemma 1.

Lemma 2. Let 1< µ≤ 2 and r̂(κ) and Φ̂(κ, y) are given functions. Then the following relation
holds true

L −1

�

1
sµ ± r̂(κ)

L
�

Φ̂(κ, y)
�

(κ, s)

�

(κ, y) =
�

E∓r̂(κ);1
0+;µ,µ Φ̂

�

(κ, y), (20)

where E∓r̂(κ);1
0+;µ,µ Φ̃ is the Prabhakar integral operator (see definition (15)) and Φ̂(κ, y) is a given

function.

Proof. From relation (A3) it follows that

1
sµ ± r̂(κ)

=L
�

yµ−1Eµ,µ

�

∓r̂(κ)yµ
��

(κ, s). (21)

Thus by applying the convolution theorem of the Laplace transform one obtains

L −1

�

1
sµ ± r̂(κ)

L
�

Φ̂(κ, t)
�

(κ, s)

�

(κ, t) =

∫ y

0

(y − ξ)µ−1E1
µ,µ

�

∓r̂(κ)(y − ξ)µ
�

Φ̂(κ,ξ)dξ,

(22)
from where we obtain the proof of Lemma 2.

Theorem 1. The solution of the following fractional Poisson equation

x Dαθ N(x , y) + y Dµ,ν
0+ N(x , y) = Φ(x , y), (23)

where x ∈ R, y ∈ R+, 1 < α ≤ 2, |θ | ≤ min{α, 2− α}, 1 < µ ≤ 2, 0 ≤ ν ≤ 1, with boundary
conditions

�

y I (1−ν)(2−µ)0+ N
�

(x , 0+) = f (x),

�

d
dy

�

y I (1−ν)(2−µ)0+ N
�

�

(x , 0+) = g(x), (24a)

lim
x→±∞

N(x , y) = 0, (24b)

is given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

yµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

yµψθα(κ)
�

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

∫ y

0

(y − ξ)µ−1Eµ,µ

�

(y − ξ)µψθα(κ)
�

Φ̂(κ,ξ)e−ıκxdξdκ

=
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

yµψθα(κ)
�

f̂ (κ)e−ıκxdκ



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 318

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

yµψθα(κ)
�

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

yE
ψθα(κ);1
0+;µ,µ Φ̂

�

(κ, y)e−ıκxdκ, (25)

where Φ̂(κ, y) =F
�

Φ(x , y)
�

(κ, y).

Proof. From the Laplace transform (14) to equation (23) and the boundary conditions
(24a), it follows

x Dθα Ñ(x , s) + sµÑ(x , s)− s1−ν(2−µ) f (x)− s−ν(2−µ)g(x) = Φ̃(x , s), (26)

where Ñ(x , s) =L
�

N(x , y)
�

(x , s). By applying Fourier transform (5) to relation (26) we find

ˆ̃N(κ, s) =
s1−ν(2−µ)

sµ −ψθα(κ)
f̂ (κ) +

s−ν(2−µ)

sµ −ψθα(κ)
ĝ(κ) +

1
sµ −ψθα(κ)

ˆ̃Φ(κ, s), (27)

where ˆ̃Φ(κ, s) = F
�

Φ̃(x , s)
�

(κ, s). Employing the results from Lemma 1 and Lemma 2, by
inverse Fourier transform we obtain solution (25). Thus, we finish with the proof of Theorem 1.

Remark 1. If in equation (23) instead of fractional Riesz-Feller derivative we use quantum frac-
tional Riesz-Feller derivative we obtain the following equation

x D∗,α
θ

N(x , y) + y Dµ,ν
0+ N(x , y) = Φ(x , y). (28)

For same boundary conditions as those used in Theorem 1, we obtain the solution in the following
form

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

−yµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

−yµψθα(κ)
�

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

yE
−ψθα(κ);1
0+;µ,µ Φ̂

�

(κ, y)e−ıκxdκ. (29)

Corollary 1. If we consider source term of form Φ(x , y) = δ(x) y−β

Γ(1−β) , the solutions of fractional
equations (23) and (28) are given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

±yµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

±yµψθα(κ)
�

ĝ(κ)e−ıκxdκ



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 319

+
yµ−β

2π

∫ ∞

−∞
Eµ,µ−β+1

�

±yµψθα(κ)
�

e−ıκxdκ

=
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

±yµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

±yµψθα(κ)
�

ĝ(κ)e−ıκxdκ

+
yµ−β

|x |
H2,1

3,3

�

∓
|x |α

yµeı θπ2

�

�

�

�

(1,1), (1+µ− β ,µ), (1, α2 )
(1,α), (1,1), (1, α2 )

�

, (30)

where the upper signs in the solution correspond to the case of fractional Riesz-Feller derivative
and lower signs to quantum fractional Riesz-Feller derivative.

Example 1. If we consider Φ(x , y) = δ(x)δ(y) and boundary conditions f (x) = δ(x),
g(x) = 0, for θ = 0, from relations (A8) and (A9), we obtain solutions (25) and (29) in terms
of Fox H-functions

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

±yµ|κ|α
�

e−ıκxdκ

+
yµ−1

2π

∫ ∞

−∞
Eµ,µ

�

±yµ|κ|α
�

e−ıκxdκ

=
y−(1−ν)(2−µ)

|x |
H2,1

3,3

�

∓
|x |α

yµ

�

�

�

�

(1,1), (1− (1− ν)(2−µ),µ), (1, α2 )
(1,α), (1, 1), (1, α2 )

�

+
yµ−1

|x |
H2,1

3,3

�

∓
|x |α

yµ

�

�

�

�

(1, 1), (µ,µ), (1, α2 )
(1,α), (1,1), (1, α2 )

�

, (31)

where the upper signs in the solution correspond to the case of fractional Riesz-Feller derivative
and lower signs to quantum fractional Riesz-Feller derivative. Moreover, for α= 2, solution (31)
in case of quantum fractional Riesz-Feller derivative becomes

N(x , y) =
y−(1−ν)(2−µ)

2|x |
H2,0

2,2

�

|x |
yµ/2

�

�

�

�

(1− (1− ν)(2−µ), µ2 ), (1, 1
2)

(1, 1), (1, 1
2)

�

+
yµ−1

2|x |
H2,0

2,2

�

|x |
yµ/2

�

�

�

�

(µ, µ2 ), (1, 1
2)

(1,1), (1, 1
2)

�

=
y−(1−ν)(2−µ)

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(1− (1− ν)(2−µ), µ2 )
(1, 1)

�

+
yµ−1

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(µ, µ2 )
(1,1)

�

. (32)

Remark 2. For the asymptotic behavior of solution (32) for |x |
yµ/2 � 1, we obtain

N(x , y)'

�µ
2

�1−2ν+ 1
2−µ

2
p

(2−µ)π
y−(1−ν)(2−µ)

|x |

�

|x |
yµ/2

�

1+2(1−ν)(2−µ)
2−µ



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 320

× exp



−
2−µ

2

�

µ

2

�

µ
2−µ
�

|x |
yµ/2

�
2

2−µ





+

�µ
2

�

1−2µ
2−µ

2
p

(2−µ)π
yµ−1

|x |

�

|x |
yµ/2

�

1−2µ
2−µ

exp



−
2−µ

2

�

µ

2

�

µ
2−µ
�

|x |
yµ/2

�
2

2−µ



 , (33)

where we employ relations (A10), (A11), (A12), (A13) and (A14).

Remark 3. From the series representation (A7) of Fox H-function, we obtain the following series
representation of solution (32)

N(x , y) =
y−(1−ν)(2−µ)−

µ
2

2

∞
∑

j=0

(−1) j

j!Γ
�

1− (1− ν)(2−µ)− µ
2 ( j + 1)

�

�

|x |
yµ/2

� j

+
y
µ
2−1

2

∞
∑

j=0

(−1) j

j!Γ
�

−µ j
�

�

|x |
yµ/2

� j

=
y−(1−ν)(2−µ)−

µ
2

2
φ

�

−
µ

2
, 1− (1− ν)(2−µ)−

µ

2
;−
|x |
yµ/2

�

+
y
µ
2−1

2
φ

�

−µ, 0;−
|x |
yµ/2

�

, (34)

from where by using the first few terms of the series (34) we can obtain the asymptotic behavior
for |x |

yµ/2 � 1. Here, φ(a, b; z) is the Wright function (A16).

Example 2. For Φ(x , y) = δ(x)δ(y) and boundary conditions f (x) = 0, g(x) = δ(x), for
θ = 0, from (A8) and (A9), we obtain the solutions (25) and (29) in the following form

N(x , y) =
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

±yµ|κ|α
�

e−ıκxdκ

+
yµ−1

2π

∫ ∞

−∞
Eµ,µ

�

±yµ|κ|α
�

e−ıκxdκ

=
y1−(1−ν)(2−µ)

|x |
H2,1

3,3

�

∓
|x |α

yµ

�

�

�

�

(1,1), (2− (1− ν)(2−µ),µ), (1, α2 )
(1,α), (1, 1), (1, α2 )

�

+
yµ−1

|x |
H2,1

3,3

�

∓
|x |α

yµ

�

�

�

�

(1, 1), (µ,µ), (1, α2 )
(1,α), (1,1), (1, α2 )

�

, (35)

where the upper signs in the solution correspond to the case of fractional Riesz-Feller derivative
and lower signs to quantum fractional Riesz-Feller derivative. For α = 2 solution (35) in case of
quantum fractional Riesz-Feller derivative becomes

N(x , y) =
y1−(1−ν)(2−µ)

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(2− (1− ν)(2−µ), µ2 )
(1, 1)

�



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 321

+
yµ−1

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(µ, µ2 )
(1,1)

�

. (36)

The asymptotic behavior and series representation of this solution can be found in a same way as
it was done in Remark 6 and Remark 7.

Corollary 2. The solution of the following fractional form of the Laplace equation

x Dαθ N(x , y) + y Dµ,ν
0+ N(x , y) = 0, (37)

where x ∈ R, y ∈ R+, 1 < α ≤ 2, |θ | ≤ min{α, 2− α}, 1 < µ ≤ 2, 0 ≤ ν ≤ 1, with boundary
conditions

�

y I (1−ν)(2−µ)0+ N
�

(x , 0+) = f (x),

�

d
dy

�

y I (1−ν)(2−µ)0+ N
�

�

(x , 0+) = g(x), (38a)

lim
x→±∞

N(x , y) = 0, (38b)

is given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

yµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

yµψθα(κ)
�

ĝ(κ)e−ıκxdκ, (39)

where f̂ (κ) =F
�

f (x)
�

(κ), ĝ(κ) =F
�

g(x)
�

(κ).

Proof. The proof of Corrolary 2 follows directly from Theorem 1 if we substitute
Φ(x , y) = 0.

Remark 4. If in equation (37) instead of fractional Riesz-Feller derivative we use quantum frac-
tional Riesz-Feller derivative we obtain the following equation

x D∗,α
θ

N(x , y) + y Dµ,ν
0+ N(x , y) = Φ(x , y). (40)

For same boundary conditions as those in Theorem 2, the solution of equation (40) is given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

−yµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

−yµψθα(κ)
�

ĝ(κ)e−ıκxdκ. (41)

Corollary 3. Solutions of equation (37) and (40) in case of Caputo fractional derivative (ν= 1),
become

N(x , y) =
1

2π

∫ ∞

−∞
Eµ
�

±yµψθα(κ)
�

f̂ (κ)e−ıκxdκ



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 322

+
y

2π

∫ ∞

−∞
Eµ,2

�

±yµψθα(κ)
�

ĝ(κ)e−ıκxdκ, (42)

and for R-L fractional derivative (ν= 0)

N(x , y) =
yµ−2

2π

∫ ∞

−∞
Eµ,µ−1

�

±yµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
yµ−1

2π

∫ ∞

−∞
Eµ,µ

�

±yµψθα(κ)
�

ĝ(κ)e−ıκxdκ, (43)

where the upper signs in the solution correspond to the case of fractional Riesz-Feller derivative
and lower signs to quantum fractional Riesz-Feller derivative.

Example 3. If we use the following boundary conditions f (x) = δ(x) and g(x) = 0, solutions
(39) and (41) are given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

±yµψθα(κ)
�

e−ıκxdκ, (44)

which for Caputo fractional derivative become

N(x , y) =
1

2π

∫ ∞

−∞
Eµ
�

±yµψθα(κ)
�

e−ıκxdκ, (45)

and for R-L fractional derivative

N(x , y) =
yµ−2

2π

∫ ∞

−∞
Eµ,µ−1

�

±yµψθα(κ)
�

e−ıκxdκ, (46)

where it is used thatF [δ(x)] = 1, and we use the upper signs in the solution in case of fractional
Riesz-Feller derivative and lower signs in case of quantum fractional Riesz-Feller derivative.

Remark 5. From relation between M-L and Fox H-function (A8), by using Mellin-cosine transform
formula (A9), for the solution (44) we find

N(x , y) =
y−(1−ν)(2−µ)

π

∫ ∞

0

cos(κx)H1,1
1,2

�

∓yµeı θπ2 |κ|α
�

�

�

�

(0,1)
(0,1), ((1− ν)(2−µ),µ)

�

dκ

=
y−(1−ν)(2−µ)

|x |
H2,1

3,3

�

∓
|x |α

yµeı θπ2

�

�

�

�

(1,1), (1− (1− ν)(2−µ),µ), (1, α2 )
(1,α), (1, 1), (1, α2 )

�

, (47)

which for θ = 0 becomes

N(x , y) =
y−(1−ν)(2−µ)

|x |
H2,1

3,3

�

∓
|x |α

yµ

�

�

�

�

(1, 1), (1− (1− ν)(2−µ),µ), (1, α2 )
(1,α), (1, 1), (1, α2 )

�

. (48)



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 323

Note that for µ = 2, ν = 1 solution (48) in case of quantum fractional Riesz-Feller derivative is
given by

N(x , y) =
1
α|x |

H2,1
3,3

�

|x |
y2/α

�

�

�

�

(1, 1
α), (1, 2

α), (1, 1
2)

(1,1), (1, 1
α), (1, 1

2)

�

. (49)

and for α= 2 by

N(x , y) =
y−(1−ν)(2−µ)

2|x |
H2,0

2,2

�

|x |
yµ/2

�

�

�

�

(1− (1− ν)(2−µ), µ2 ), (1, 1
2)

(1, 1), (1, 1
2)

�

=
y−(1−ν)(2−µ)

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(1− (1− ν)(2−µ), µ2 )
(1, 1)

�

, (50)

where we apply the definition (A6) and the known properties of H-function [29].

Remark 6. From the results in Remark 2, for the asymptotic behavior of solution (50) for
|x |

yµ/2 � 1, we obtain

N(x , y)'

�µ
2

�1−2ν+ 1
2−µ

2
p

(2−µ)π
y−(1−ν)(2−µ)

|x |

�

|x |
yµ/2

�

1+2(1−ν)(2−µ)
2−µ

× exp



−
2−µ

2

�

µ

2

�

µ
2−µ
�

|x |
yµ/2

�
2

2−µ



 . (51)

Remark 7. The series representation of solution (50) is given by

N(x , y) =
y−(1−ν)(2−µ)−

µ
2

2

∞
∑

j=0

(−1) j

j!Γ
�

1− (1− ν)(2−µ)− µ
2 ( j + 1)

�

�

|x |
yµ/2

� j

=
y−(1−ν)(2−µ)−

µ
2

2
φ

�

−
µ

2
, 1− (1− ν)(2−µ)−

µ

2
;−
|x |
yµ/2

�

. (52)

By using the first few terms of series (52), we can obtain the asymptotic behavior of solution (50)
for |x |

yµ/2 � 1.

Example 4. For the following boundary conditions f (x) = 0 and g(x) = δ(x), solutions (39)
and (41) are given by

N(x , y) =
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

±yµψθα(κ)
�

e−ıκxdκ

=
y1−(1−ν)(2−µ)

|x |
H2,1

3,3

�

∓
|x |α

yµeı θπ2

�

�

�

�

(1,1), (2− (1− ν)(2−µ),µ), (1, α2 )
(1,α), (1,1), (1, α2 )

�

. (53)

The case with θ = 0, α= 2, and quantum fractional Riesz-Feller derivative, yields

N(x , y) =
y1−(1−ν)(2−µ)

2|x |
H2,0

2,2

�

|x |
yµ/2

�

�

�

�

(2− (1− ν)(2−µ), µ2 ), (1, 1
2)

(1, 1), (1, 1
2)

�



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 324

=
y1−(1−ν)(2−µ)

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(2− (1− ν)(2−µ), µ2 )
(1, 1)

�

. (54)

From solution (54) in case of Caputo fractional derivative (ν= 1) one finds

N(x , y) =
y

2|x |
H2,0

2,2

�

|x |
yµ/2

�

�

�

�

(2, µ2 ), (1, 1
2)

(1,1), (1, 1
2)

�

=
y

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(2, µ2 )
(1, 1)

�

, (55)

and for R-L fractional derivative (ν= 0) [37]

N(x , y) =
yµ−1

2|x |
H2,0

2,2

�

|x |
yµ/2

�

�

�

�

(µ, µ2 ), (1, 1
2)

(1,1), (1, 1
2)

�

=
yµ−1

2|x |
H1,0

1,1

�

|x |
yµ/2

�

�

�

�

(µ, µ2 )
(1,1)

�

. (56)

Remark 8. Here we note that the considered equation (28) can be transformed to the following
general space-time fractional wave equation in presence of an external source Φ(x , t)

t D
µ,ν
0+ N(x , t) = x Dαθ N(x , t) +Φ(x , t), (57)

where we use fractional Riesz-Feller space derivative x Dα
θ
= −x D∗,α

θ
given by (1), x ∈ R, t ≥ 0,

1< α≤ 2, |θ | ≤min{α, 2−α}, 1< µ≤ 2, 0≤ ν≤ 1, with initial conditions

�

t I
(1−ν)(2−µ)
0+ N

�

(x , 0+) = f (x),
�

d
dt

�

t I
(1−ν)(2−µ)
0+ N

�

�

(x , 0+) = g(x), (58a)

and boundary conditions
lim

x→±∞
N(x , t) = 0. (58b)

So, its solution is given by

N(x , t) =
t−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

−tµψθα(κ)
�

f̂ (κ)e−ıκxdκ

+
t1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

−tµψθα(κ)
�

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

tE
−ψθα(κ);1
0+;µ,µ Φ̂

�

(κ, t)e−ıκxdκ, (59)

where Φ̂(κ, t) = F [Φ(x , t)] (κ, t). Thus, all the previously obtained results, series representa-
tions and asymptotic behaviors in case of quantum fractional Riesz-Feller derivative can be used
for this fractional wave equation with fractional Riesz-Feller space derivative and Hilfer-composite
fractional time derivative. From this solution many obtained results for fractional wave equations
with Caputo or R-L time fractional derivative can be recovered. For example, if Φ(x , t) = 0 we ob-
tain the general space-time fractional wave equation t D

µ,ν
0+ N(x , t) = x Dα

θ
N(x , t) which contains

a number of limiting cases.



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 325

4. Fractional Helmholtz Equation

The inhomogeneous Helmholtz equation in two variables is given by

∂ 2

∂ x2
N(x , y) +

∂ 2

∂ y2
N(x , y) + k2N(x , y) = Φ(x , y), (60)

where N(x , y) is the field variable, k is the wave number, and Φ(x , y) is a given function. For
Φ(x , y) = 0 it becomes homogeneous Helmholtz equation. Furthermore, if k = 0 it is related
with the Poisson and Laplace equations considered in previous section. For a given form of
Φ(x , y), equation (60) corresponds to the time-independent wave equation, which may be
used for modeling vibrating membrane.

Here we analyze fractional generalization of the inhomogeneous Helmholtz equation (60).

Theorem 2. The solution of the following inhomogeneous fractional Helmholtz equation

x Dαθ N(x , y) + y Dµ,ν
0+ N(x , y) + k2N(x , y) = Φ(x , y), (61)

where x ∈ R, y ≥ 0, 1 < α ≤ 2, |θ | ≤ min{α, 2 − α}, 1 < µ ≤ 2, 0 ≤ ν ≤ 1, with boundary
conditions

�

y I (1−ν)(2−µ)0+ N
�

(x , 0+) = f (x),

�

d
dy

�

y I (1−ν)(2−µ)0+ N
�

�

(x , 0+) = g(x), (62a)

lim
x→±∞

N(x , y) = 0, (62b)

is given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

yµ
�

ψθα(κ)− k2
��

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

yµ
�

ψθα(κ)− k2
��

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

yE
ψθα(κ)−k2;1
0+;µ,µ Φ̂

�

(κ, y)e−ıκxdκ, (63)

where Φ̂(κ, y) =F
�

Φ(x , y)
�

(κ, y).

Proof. In a same way as previously, by Laplace transform (14) to equation (61) and from
the boundary conditions (62a), we obtain

x Dθα Ñ(x , s) + sµÑ(x , s)− s1−ν(2−µ) f (x)− s−ν(2−µ)g(x) + k2Ñ(x , s) = Φ̃(x , s), (64)

where Ñ(x , s) =L [N(x , t)]. The Fourier transform (5) to (64) yields

ˆ̃N(κ, s) =
s1−ν(2−µ)

sµ −
�

ψθα(κ)− k2
� f̂ (κ) +

s−ν(2−µ)

sµ −
�

ψθα(κ)− k2
� ĝ(κ) +

1

sµ −
�

ψθα(κ)− k2
�

ˆ̃Φ(κ, s), (65)

where ˆ̃Φ(κ, s) = F
�

Φ̃(x , s)
�

. By application of Lemma (1) and Lemma 2, by inverse Fourier
transform we obtain solution (63).



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 326

Remark 9. If in equation (61) instead of fractional Riesz-Feller derivative we use quantum frac-
tional Riesz-Feller derivative we obtain the following equation

x D∗,α
θ

N(x , y) + y Dµ,ν
0+ N(x , y) + k2N(x , t) = Φ(x , y). (66)

If we use same boundary conditions as those used in Theorem (2), we obtain the following solution

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

−yµ
�

ψθα(κ) + k2
��

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

−yµ
�

ψθα(κ) + k2
��

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

yE
−(ψθα(κ)+k2);1
0+;µ,µ Φ̂

�

(κ, y)e−ıκxdκ. (67)

Corollary 4. For ν= 0 (R-L fractional derivative) one finds the following solutions [37]

N(x , y) =
yµ−2

2π

∫ ∞

−∞
Eµ,µ−1

�

yµ
�

±ψθα(κ)− k2
��

f̂ (κ)e−ıκxdκ

+
yµ−1

2π

∫ ∞

−∞
Eµ,µ

�

yµ
�

±ψθα(κ)− k2
��

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

yE
±ψθα(κ)−k2;1
0+;µ,µ Φ̂

�

(κ, y)e−ıκxdκ, (68)

and for ν= 1 - solutions

N(x , y) =
1

2π

∫ ∞

−∞
Eµ
�

yµ
�

±ψθα(κ)− k2
��

f̂ (κ)e−ıκxdκ

+
y

2π

∫ ∞

−∞
Eµ,2

�

yµ
�

±ψθα(κ)− k2
��

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

yE
±ψθα(κ)−k2;1
0+;µ,µ Φ̂

�

(κ, y)e−ıκxdκ, (69)

where the upper signs in the solution correspond to the case of fractional Riesz-Feller derivative
and lower signs to quantum fractional Riesz-Feller derivative.

Corollary 5. The solutions of equations (61) and (66) for a source term of form

Φ(x , y) = δ(x) y−β

Γ(1−β) are given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

yµ
�

±ψθα(κ)− k2
��

f̂ (κ)e−ıκxdκ

+
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

yµ
�

±ψθα(κ)− k2
��

ĝ(κ)e−ıκxdκ



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 327

+
yµ−β

2π

∫ ∞

−∞
Eµ,µ−β+1

�

yµ
�

±ψθα(κ)− k2
��

e−ıκxdκ, (70)

where the upper signs in the solution correspond to the case of fractional Riesz-Feller derivative
and lower signs to quantum fractional Riesz-Feller derivative.

Example 5. The solution of the following fractional Helmholtz equation

x Dαθ N(x , y) + y Dµ,ν
0+ N(x , y) + k2N(x , y) = δ(x)δ(y), (71)

where x ∈ R, x ≥ 0, 1 < α ≤ 2, |θ | ≤ min{α, 2 − α}, 1 < µ ≤ 2, 0 ≤ ν ≤ 1, with boundary
conditions

�

y I (1−ν)(2−µ)0+ N
�

(x , 0+) = δ(x),

�

d
dy

�

y I (1−ν)(2−µ)0+ N
�

�

(x , 0+) = 0, (72a)

lim
x→±∞

N(x , y) = 0, (72b)

is given by

N(x , y) =
y−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

yµ
�

ψθα(κ)− k2
��

e−ıκxdκ

+
yµ−1

2π

∫ ∞

−∞
Eµ,µ

�

yµ
�

ψθα(κ)− k2
��

e−ıκxdκ. (73)

Example 6. If the boundary conditions are given by f (x) = 0 and g(x) = δ(x), equation from
Example (5) has a solution of form

N(x , y) =
y1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

yµ
�

ψθα(κ)− k2
��

e−ıκxdκ

+
yµ−1

2π

∫ ∞

−∞
Eµ,µ

�

yµ
�

ψθα(κ)− k2
��

e−ıκxdκ. (74)

Remark 10. Note that equation (66) can be transformed to the following general space-time
fractional wave equation in presence of an external source Φ(x , t)

t D
µ,ν
0+ N(x , t) = x Dαθ N(x , t)− k2N(x , t) +Φ(x , t), (75)

where we use fractional Riesz-Feller space derivative x Dα
θ
= −x D∗,α

θ
given by (1), x ∈ R, t ≥ 0,

1< α≤ 2, |θ | ≤min{α, 2−α}, 1< µ≤ 2, 0≤ ν≤ 1, with initial conditions

�

t I
(1−ν)(2−µ)
0+ N

�

(x , 0+) = f (x),
�

d
dt

�

t I
(1−ν)(2−µ)
0+ N

�

�

(x , 0+) = g(x), (76a)

and boundary conditions
lim

x→±∞
N(x , t) = 0. (76b)



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 328

Thus, its solution is given by

N(x , t) =
t−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,1−(1−ν)(2−µ)

�

−tµ
�

ψθα(κ) + k2
��

f̂ (κ)e−ıκxdκ

+
t1−(1−ν)(2−µ)

2π

∫ ∞

−∞
Eµ,2−(1−ν)(2−µ)

�

−tµ
�

ψθα(κ) + k2
��

ĝ(κ)e−ıκxdκ

+
1

2π

∫ ∞

−∞

�

tE
−(ψθα(κ)+k2);1
0+;µ,µ Φ̂

�

(κ, t)e−ıκxdκ, (77)

where Φ̂(κ, t) =F [Φ(x , t)] (κ, t). This solution contains a number of limiting cases.

5. Conclusions

We consider fractional generalization of the Laplace equation, Poisson equation and Helm-
holtz equations in two variables. Since there is no dependence on the time variable, the solu-
tions of these equations can be considered as a steady-state solutions. The fractional deriva-
tives used in this paper are of Riesz-Feller and Hilfer-composite form. M-L type functions, Fox
H-functions, and the Prabhakar integral operator containing two parameter M-L function in
the kernel are used to express solutions analytically. Several special cases of these equations
are investigated. Asymptotic behavior of solutions is analyzed, and series expression of solu-
tions are provided. The general space-time fractional wave equation in presence of an external
source is considered as well.



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 329

Appendix: Mittag-Leffler and Fox H-functions

The standard (one parameter) M-L function, introduced by Mittag-Leffler, is defined by
[4, 5, 11, 18, 29, 34, 36]

Eα(z) =
∞
∑

k=0

zk

Γ(αk+ 1)
, (A1)

where (z ∈ C;ℜ(α)> 0). Later, two parameter M-L function which was introduced by Wiman,
and further analyzed by Agarwal and Humbert, is given by [4, 5, 11, 18, 29, 34, 36]

Eα,β(z) =
∞
∑

k=0

zk

Γ(αk+ β)
, (A2)

where (z,β ∈ C;ℜ(α) > 0). The M-L functions (A1) and (A2) are entire functions of order
ρ = 1/ℜ(α) and type 1. Note that Eα,1(z) = Eα(z). These functions are generalization of
the exponential, hyperbolic and trigonometric functions since E1,1(z) = ez , E2,1(z2) = cosh(z),
E2,1(−z2) = cos(z) and E2,2(−z2) = sin(z)/z.

The Laplace transform of the M-L function is given by [11, 18, 29, 34, 36]

L [tβ−1Eα,β(±atα)](s) =

∫ ∞

0

e−st tβ−1Eα,β(±atα)dt =
sα−β

sα ∓ a
, (A3)

where ℜ(s)> |a|1/α.
Prabhakar [35] introduced the following three parameter M-L function

Eγ
α,β(z) =

∞
∑

k=0

(γ)k
Γ(αk+ β)

zk

k!
, (A4)

where β ,γ, z ∈ C, ℜ(α) > 0, (γ)k is the Pochhammer symbol. It is an entire function of order
ρ = 1/ℜ(α) and type 1. Note that E1

α,β(z) = Eα,β(z). Later, in [47] it is used the following
four parameter generalized M-L function

Eγ,κ
α,β(z) =

∞
∑

n=0

(γ)κn

Γ(αn+ β)
·

zn

n!
, (A5)

where (z,β ,γ ∈ C;ℜ(α) > max{0,ℜ(κ) − 1};ℜ(κ) > 0) and (γ)κn is a notation of the
Pochhammer symbol, as a kernel of a generalized integral operator. It is an entire function of

order ρ = 1
ℜ(α−κ)+1 and type σ = 1

ρ

�

{ℜ(α)}ℜ(κ)

{ℜ(α)}ℜ(α)

�ρ

[47]. Note that Eγ,1
α,β(z) = Eγ

α,β(z).
The Fox H-function (or simply H-function) is defined by the following Mellin-Barnes inte-

gral [29]

Hm,n
p,q (z) = Hm,n

p,q

�

z

�

�

�

�

(a1, A1), . . . , (ap, Ap)
(b1, B1), . . . , (bq, Bq)

�

= Hm,n
p,q

�

z

�

�

�

�

(ap, Ap)
(bq, Bq)

�

=
1

2πı

∫

Ω
θ (s)zsds, (A6)



R. Saxena, Ž. Tomovski, T. Sandev / Eur. J. Pure Appl. Math, 7 (2014), 312-334 330

where θ (s) =
∏m

j=1 Γ(b j−B js)
∏n

j=1 Γ(1−a j+A js)
∏q

j=m+1 Γ(1−b j+B js)
∏p

j=n+1 Γ(a j−A js)
, 0 ≤ n ≤ p, 1 ≤ m ≤ q, ai , b j ∈ C, Ai , B j ∈ R+,

i = 1, . . . , p, j = 1, . . . , q. The contour Ω starting at c − ı∞ and ending at c + ı∞ separates
the poles of the function Γ(b j + B js), j = 1, . . . , m from those of the function Γ(1− ai − Ais),
i = 1, . . . , n. The expansion for the H-function (A6) is given by [29]

Hm,n
p,q

�

z

�

�

�

�

(a1, A1), . . . , (ap, Ap)
(b1, B1), . . . , (bq, Bq)

�

=
m
∑

h=1

∞
∑

k=0

∏m
j=1, j 6=h Γ

�

b j − B j
bh+k

Bh

�

∏n
j=1 Γ

�

1− a j + A j
bh+k

Bh

�

∏q
j=m+1 Γ

�

1− b j + B j
bh+k

Bh

�

∏p
j=n+1 Γ

�

a j − A j
bh+k

Bh

� ·
(−1)kz(bh+k)/Bh

k!Bh
. (A7)

From the Mellin-Barnes integral representation of two parameter M-L function, one can
find the following relation with the Fox H-function [29]

Eα,β(z) =
1

2πı

∫

Ω

Γ(s)Γ(1− s)
Γ(β −αs)

zsds = H1,1
1,2

�

−z

�

�

�

�

(0,1)
(0, 1), (1− β ,α)

�

, (A8)

where the contour Ω starts at c − ı∞, ends at c + ı∞, and separates the poles of function
Γ(s) from those of the function Γ(1− s). It is shown by Mathai, Saxena and Haubold that the
integral converges for all z [29].

The Mellin-cosine transform of the H-function is given by [29]
∫ ∞

0

kρ−1 cos(kx)Hm,n
p,q

�

akδ
�

�

�

�

(ap, Ap)
(bq, Bq)

�

dk

=
2ρ−1pπ

xρ
Hm,n+1

p+2,q

�

a
�

2
x

�δ
�

�

�

�

(2− ρ
2 , µ2 ), (ap, Ap), (

1−ρ
2 , δ2 )

(bq, Bq)

�

, (A9)

where ℜ
�

ρ
�

+δmin1≤ j≤mℜ
�

b j
B j

�

> 0, ρ = δmax1≤ j≤nℜ
�

a j−1
A j

�

< 0, |arg(a)|< πθ/2,

θ =
∑n

j=1 A j −
∑p

j=n+1 A j +
∑m

j=1 B j −
∑q

j=m+1 B j > 0.

The asymptotic expansion of the Fox H-function Hm,0
p,q (z) where q = m for large z is [1, 29]

Hm,0
p,q (z)∼ Bz(1−α)/m

∗
exp

�

−m∗C1/m∗z1/m∗
�

, (A10)

where

α=
p
∑

k=1

ak −
q
∑

k=1

bk +
1
2
(q− p+ 1), (A11)

m∗ =
q
∑

j=1

B j −
p
∑

j=1

A j > 0, (A12)

C =
p
∏

k=1

AAk
k

q
∏

k=1

B−Bk
k , (A13)



REFERENCES 331

B =(2π)(m−p−1)/2C (1−α)/m
∗
m∗−1/2

p
∏

k=1

A1/2−ak
k

m
∏

k=1

Bbk−1/2
k . (A14)

Closely related to the Fox H-function is the Fox-Wright function defined by [29]

pΨq(z) = pΨq

�

z

�

�

�

�

(a1, A1), . . . , (ap, Ap)
(b1, B1), . . . , (bq, Bq)

�

=
∞
∑

n=0

∏p
j=1 Γ

�

a j + nA j

�

∏q
j=1 Γ

�

b j + nB j

� ·
zn

n!
, (A15)

which as a special case gives the Wright function [29]

φ(a, b; z) = 0Ψ1(z) = 0Ψ1

�

z

�

�

�

� (b, a)

�

=
∞
∑

n=0

1
Γ(b+ na)

·
zn

n!
. (A16)

References

[1] B.L.J. Braaksma. Asymptotic expansions and analytic continuations for a class of Barnes-
integrals, Compositio Mathematica 15, pp. 239-341. 1964.

[2] M. Caputo. Elasticita Dissipacione (Bologna: Zanichelli). 1969.

[3] A. Compte. Stochastic foundations of fractional dynamics, Physical Review E 53, pp. 4191-
4193. 1996.

[4] M.M. Dzherbashyan. Harmonic Analysis and Boundary Value Problems in the Complex
Domain, Vol 65 ed I Gohberg (Basel: Birkhauser). 1993.

[5] A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi. Higher Transcedential Func-
tions 3 (New York, Toronto and London: McGraw-Hill Book Company. 1955.

[6] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New
York) 1968.

[7] K.M. Furati, M.D. Kassim, and N.e.-Tatar. Existence and uniqueness for a problem involving
Hilfer fractional derivative, Computers and Mathematics with Applications 64, pp. 1616-
1626. 2012.

[8] M. Garg, A. Sharma, and P. Manohar. Linear space-time fractional reaction-diffusion equa-
tion with composite fractional derivative in time, Journal of Fractional Calculus and Ap-
plications 5, pp. 114-121. 2014.

[9] Y.-J. Hao, H.M. Srivastava, H. Jafari, and X.-J. Yang. Helmholtz and Diffusion Equations
Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-
Type Cylindrical Coordinates, Advances in Mathematical Physics 2013, 754248. 2013.

[10] R. Hilfer. Fractional dynamics, irreversibility and ergodicity breaking, Chaos, Solitons and
Fractals 5, pp. 1475-1484. 1995.



REFERENCES 332

[11] R. Hilfer. Application of Fractional Calculus in Physics (Singapore: World Scientific Pub-
lishing Company). 2000.

[12] R. Hilfer. Experimental evidence for fractional time evolution in glass forming materials,
Chemical Physics 284, pp. 399-408. 2002.

[13] R. Hilfer. On fractional relaxation, Fractals 11, pp. 251-257. 2003.

[14] R. Hilfer, Y. Luchko, and Ž. Tomovski. Operational method for the solution of fractional
differential equations with generalized Riemann-Liouville fractional derivatives, Fractional
Calculus and Applied Analysis 12, pp. 299-318. 2009.

[15] S. Al-Homidan, R.A. Ghanam, and N.-e. Tatar. On a generalized diffusion equation arising
in petroleum engineering, Advances in Difference Equations 2013, 349. 2013.

[16] G. Jumarie. Non-standard analysis and Liouville-Riemann derivative, Chaos, Solitons and
Fractals 12, pp. 2577-2587. 2001.

[17] A.A. Kilbas, M. Saigo, and R.K. Saxena. Generalized Mittag-Leffler function and generalized
fractional calculus operators, Integral Transforms and Special Functions 15, pp. 31-49.
2004.

[18] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo. Theory and Applications of Fractional Dif-
ferential Equations (Amsterdam: Elsevier). 2006.

[19] M.-H. Kim, G.-C. Ri, and H.-C. O. Operational method for solving multi-term fractional
differential equations with the generalized fractional derivatives, Fractional Calculus and
Applied Analysis 17, pp. 79-95. 2014.

[20] J. Liang, W. Zhang, Y.Q. Chen, and I. Podlubny. Robustness of boundary control of frac-
tional wave equations with delayed boundary measurement using fractional order con-
troller and the Smith predictor, in: Proceedings of 2005 ASME Design Engineering Technical
Conferences, Long Beach, California, USA, 2005.

[21] J. Liang and Y.Q. Chen. Hybrid symbolic and numerical simulation studies of time-fractional
order wave-diffusion systems, International Journal of Control 79, pp. 1462-1470. 2006.

[22] Y. Luchko. Fractional Schrödinger equation for a particle moving in a potential well, Journal
of Mathematical Physics 54, 012111. 2013.

[23] F. Mainardi. Fractional relaxation-oscillation and fractional diffusion-wave phenomena,
Chaos, Solitons and Fractals 7, pp. 1461-1477. 1996.

[24] F. Mainardi. The fundamental solutions for the fractional diffusion-wave equation, Applied
Mathematics Letters 9, pp. 23-28. 1996.

[25] F. Mainardi and R. Gorenflo. On Mittag-Leffler-type functions in fractional evolution pro-
cesses, Journal of Computational and Applied Mathematics 118, pp. 283-299. 2000.



REFERENCES 333

[26] F. Mainardi and R. Gorenflo. Time-fractional derivatives in relaxation processes: a tutorial
survey, Fractional Calculus and Applied Analysis 10, pp. 269-308. 2007.

[27] F. Mainardi, G. Pagnini, and R.K. Saxena. Fox H functions in fractional diffusion, Journal
of Computational and Applied Mathematics 178, pp. 321-331. 2005.

[28] H.J. Haubold, A.M. Mathai, and R.K. Saxena. Solutions of fractional reaction-diffusion
equations in terms of the H-function, Bulletin of the Astronomical Society of India 35, pp.
681-689. 2007.

[29] A.M. Mathai, R.K. Saxena, and H.J. Haubold. The H-function: Theory and Applications
(New York: Springer). 2010.

[30] R. Metzler, E. Barkai, and J. Klafter. Anomalous diffusion and relaxation close to thermal
equilibrium: a fractional Fokker-Planck equation approach, Physical Review Letters 82,
pp. 3563-3567. 1999.

[31] R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional
dynamics approach, Physics Reports 339, pp. 1-77. 2000.

[32] R. Metzler and J. Klafter. The restaurant at the end of the random walk: recent develop-
ments in the description of anomalous transport by fractional dynamics, Journal of Physics
A: Mathematical and General 37, pp. R161-R208. 2004.

[33] G. Pagnini, A. Mura, and F. Mainardi. Generalized fractional master equation for self-
similar stochastic processes modelling anomalous diffusion, International Journal of
Stochastic Analysis 2012, 427383. 2012.

[34] I. Podlubny. Fractional Differential Equations, (San Diego: Acad. Press, 1999).

[35] T.R. Prabhakar. A singular integral equation with a generalized Mittag-Leffler function in
the kernel, Yokohama Mathematical Journal 19, pp. 7-15. 1971.

[36] S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional Integrals and Derivatives. Theory
and Applications (New York et al: Gordon and Breach). 1993.

[37] M.S. Samuel and A. Thomas. On fractional Helmholtz equations, Fractional Calculus and
Applied Analysis 13, pp. 295-308. 2010.

[38] T. Sandev, R. Metzler, and Ž. Tomovski. Fractional diffusion equation with a generalized
Riemann-Liouville time fractional derivative, Journal of Physics A: Mathematical and The-
oretical 44, pp. 255203. 2011

[39] T. Sandev and Ž. Tomovski. The general time fractional wave equation for a vibrating string,
Journal of Physics A: Mathematical and Theoretical 43, pp. 055204. 2010.

[40] B. Al-Saqabi, L. Boyadjiev, and Y. Luchko. Comments on employing the Riesz-Feller deriva-
tive in the Schrödinger equation, European Physical Journal 222, pp. 1779-1794. 2013.



REFERENCES 334

[41] R.K. Saxena. On a fractional master equation and a fractional diffusion equation, Mathe-
matics and Statistics 1, pp. 59-63. 2013.

[42] R.K. Saxena, A.M. Mathai, and H.J. Haubold. Unified fractional kinetic equation and a
fractional diffusion equation, Astrophysics and Space Science 290, pp. 299-310. 2004.

[43] R.K. Saxena, A.M. Mathai, and H.J. Haubold. Fractional reaction-diffusion equations, As-
trophysics and Space Science 305, pp. 289-296. 2006.

[44] R.K. Saxena, A.M. Mathai, and H.J. Haubold. Solutions of fractional reaction-diffusion
equations in terms of Mittag-Leffler functions, International Journal of Scientifc Research
17, pp. 1-17. 2008.

[45] R.K. Saxena, A.M. Mathai, and H.J. Haubold. Computational solutions of unified fractional
reaction-diffusion equations with composite fractional time derivative, arXiv:1210.1453v1.

[46] H.M. Srivastava and R.K. Saxena. Operators of fractional integration and their applica-
tions, Applied Mathematics and Computation 118. pp. 1-52. 2001.

[47] H.M. Srivastava and Ž. Tomovski. Fractional calculus with an integral operator containing
a generalized Mittag-Leffler function in the kernel, Applied Mathematics and Computation
211, pp. 198-210. 2009.

[48] A. Thomas. On a fractional master equation, International Journal of Differential Equa-
tions 2011, 346298. 2013.

[49] Ž. Tomovski. Generalized Cauchy type problems for nonlinear fractional differential equa-
tions with composite fractional derivative operator, Nonlinear Analysis: Theory, Methods
and Applications 75, pp. 3364-3384. 2012.

[50] Ž. Tomovski, R. Hilfer, and H.M. Srivastava. Fractional and operational calculus with gen-
eralized fractional derivative operators and Mittag-Leffler type functions, Integral Trans-
forms and Special Functions 21, pp. 797-814. 2010.

[51] Ž. Tomovski and T. Sandev. Effects of a fractional friction with power-law memory kernel
on string vibrations, Computers and Mathematics with Applications 62, pp. 1554-1561.
2011.

[52] Ž. Tomovski and T. Sandev. Fractional wave equation with a frictional memory kernel of
Mittag-Leffler type, Applied Mathematics and Computation 218, pp. 10022-10031. 2012.

[53] Ž. Tomovski and T. Sandev. Exact solutions for fractional diffusion equation in a bounded
domain with different boundary conditions, Nonlinear Dynamics 71, pp. 671-683. 2013.

[54] Ž. Tomovski, T. Sandev, R. Metzler, and J. Dubbeldam. Generalized space-time fractional
diffusion equation with composite fractional time derivative, Physica A 391, pp. 2527-2542.
2012.