International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 8, August 2013
ISSN 2319 - 4847
On the Fractional Calculus Involving Mainardi
Function for exponential Function
Mohd. Farman Ali1, Manoj Sharma2, Renu Jain3
1, 3
School of Mathematics and Allied Sciences, Jiwaji University, Gwalior, Address
2
Department of Mathematics RJIT, BSF Academy, Tekanpur, Address
ABSTRACT
This paper is devoted to study of fractional calculus of Mainardi function. The Mainardi function is a special case of Wright
function given by British mathematician (E. Maitland Wright), starting from 1933. The author establishes the some results with
Mainardi function for exponential function.
Mathematics Subject Classification— 26A33, 33C60, 44A15.
Key Words and Phrases—Riemann-Liouville fractional integral and derivative operators, Special functions and
Mainardi Function for exponential.
1. INTRODUCTION
The Mainardi function is a particular case of wright function. The application of the functions are useful in certain areas
of physical and applied science. The Mainardi function is
with C being the set of complex number.
2. THE M AINARDI FUNCTION FOR EXPONENTIAL FUNCTION
We first give the definition of Mainardi function for exponential function
3. Some Special Cases
3.1. When
3.2. When
,
, we get the Mainardi function (1).
, the function reduces to exponential series[8].
4. RELATION WITH MAINARDI FUNCTION FOR EXPONENTIAL
FRACTIONAL INTEGRAL OPERATOR.
FUNCTION AND
In this section, we establish the relation between Riemann-Liouville Fractional calculus operators and Mainardi function
for exponential function.
Theorem 5.1. Let
relation
and
be the Riemann-Liouville fractional derivative operator then there holds the
Proof – By virtue of the definition (3.1) and fractional derivative operator, we have
We use the modified Beta-function:
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 8, August 2013
ISSN 2319 - 4847
On the differentiating n times the term
w.r.t. z, representation (7) reduces to
This is complete proof of theorem (5.1).
5. RELATION BETWEEN M AINARDI FUNCTION FOR EXPONENTIAL FUNCTION AND
FRACTIONAL INTEGRAL OPERATOR.
In this section, integral associated with Mainardi function for exponential function are presented, using fractional
calculus integral operator and beta function.
Theorem-6.1. Let
and Riemann-Liouville fractional integral is
, then
Proof In this section, the fractional Riemann-Liouville (R-L) integral operator [6] (for lower limit
variable z) of the function (3.1).
with respect to
Using the modified Beta function, we have
This theorem is proved.
6. RECURRENCE RELATIONS
In this section, the relation is obtained by virtue of the definition (2).
Theorem 7.1. Let
and
then there holds the relation
Proof- we start from the side of (7.1),
Where
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 8, August 2013
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Now, we obtain the following results, on putting
Corollary 7.1 – If
then the relation turned into
Corollary 7.2 – If
then the relation transform into
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