Description
In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by
Fractional Integration and Fractional Differentiation of the Product of M-Series and H-Function
ABSTRACT In this paper, we have derived formulae for the Riemann-Liouville fractional integral and fractional derivative of the product of the Manoj Sharma's M-series and the Fox H-function. Also the fractional integrals defined by Saxena and Kumbhat of the M-series is found with the help of integral of H-function. The M- series is a particular case of the -function of Inayat-Hussain. Certain special cases of the formulae have also been discussed. Keywords and Phrases: Fractional calculus operators, H-function, M-series, Laplace transform. 1. INTRODUCTION The purpose of this paper is to establish theorems on the fractional integrals and fractional derivatives of the product of M-series and H-function. The theorems derived in this paper provide an extension of the work [6]. The Riemann-Liouville Fractional Integral of order [3] is defined and represented as
=
1
?
, >
1.1
where ?, > 0, ? , which is the Space of Lebesgue measurable function. The Riemann-Liouville Fractional differential of order [3] is defined and represented as !" = 1 $ % " ?
&
#?
?
&
, #='
( + 1;
>
1. 2
Various definitions of fractional integration have been given from time to time by many authors, viz. Kober Where
'
( means the integral part of ?0. (1940), Erdélyi (1950-51), Saxena (1967), Kalla (1969) and many others The fractional integral operator involving the H-function have been defined and denoted by Saxena and Khumbat [7] in the following manner:
For - = 1
(= .
.
?
1,2
34 $ % $1 ? % 6 9 9 ,/ =>5
.,
'
@
/ ,0 &
78 , : ;
7 , < ; ,0
9 9
1. 3
?
B B 1 ,2
A B, '
(=
?
34 C D C1 ? D 6
/ ,0
5
&
78 , : ;
9 9
,/
>
1.4
7 ,< ; =
9 9 ,0
The conditions of validity of these operators are as follows:
F
1 ? H , I
In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by
Fractional Integration and Fractional Differentiation of the Product of M-Series and H-Function
ABSTRACT In this paper, we have derived formulae for the Riemann-Liouville fractional integral and fractional derivative of the product of the Manoj Sharma's M-series and the Fox H-function. Also the fractional integrals defined by Saxena and Kumbhat of the M-series is found with the help of integral of H-function. The M- series is a particular case of the -function of Inayat-Hussain. Certain special cases of the formulae have also been discussed. Keywords and Phrases: Fractional calculus operators, H-function, M-series, Laplace transform. 1. INTRODUCTION The purpose of this paper is to establish theorems on the fractional integrals and fractional derivatives of the product of M-series and H-function. The theorems derived in this paper provide an extension of the work [6]. The Riemann-Liouville Fractional Integral of order [3] is defined and represented as
=
1
?
, >
1.1
where ?, > 0, ? , which is the Space of Lebesgue measurable function. The Riemann-Liouville Fractional differential of order [3] is defined and represented as !" = 1 $ % " ?
&
#?
?
&
, #='
( + 1;
>
1. 2
Various definitions of fractional integration have been given from time to time by many authors, viz. Kober Where
'
( means the integral part of ?0. (1940), Erdélyi (1950-51), Saxena (1967), Kalla (1969) and many others The fractional integral operator involving the H-function have been defined and denoted by Saxena and Khumbat [7] in the following manner:
For - = 1
(= .
.
?
1,2
34 $ % $1 ? % 6 9 9 ,/ =>5
.,
'
@
/ ,0 &
78 , : ;
7 , < ; ,0
9 9
1. 3
?
B B 1 ,2
A B, '
(=
?
34 C D C1 ? D 6
/ ,0
5
&
78 , : ;
9 9
,/
>
1.4
7 ,< ; =
9 9 ,0
The conditions of validity of these operators are as follows:
F
1 ? H , I