Article
KYUNGPOOK Math. J. 2019; 59(3): 433-443
Published online September 23, 2019
Copyright © Kyungpook Mathematical Journal.
A Study of Marichev-Saigo-Maeda Fractional Integral Operators Associated with the S-Generalized Gauss Hypergeometric Function
Manish Kumar Bansal, Devendra Kumar∗, Rashmi Jain
Department of Applied Science, Govt. Engineering College, Banswara 327001, Rajasthan, India
e-mail : bansalmanish443@gmail.com
Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India
e-mail : devendra.maths@gmail.com
Department of Mathematics, Malaviya National Institute of Technology, Jaipur 302017, Rajasthan, India
e-mail : rashmiramessh1@gmail.com
Received: July 30, 2017; Revised: December 17, 2018; Accepted: December 20, 2018
In this work, we evaluate the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator with Appell’s function
Keywords: S-Generalized Gauss hypergeometric function, Marichev-SaigoMaeda fractional integral operators, Appell’s function, Mellin transform.
Introduction and Definitions
Fractional calculus is an interesting and useful branch of mathematical analysis that studies differential and integral operators of arbitrary order. In recent years, several schemes of fractional calculus– namely Riemann-Liouville, Weyl, Gruunwald-Letnikow and Caputo–have been studied, and several researchers, such as Miller and Ross [9], Podlubny [10], Kilbas et al. [3], Yang et al.[22], Kumar [4, 5, 6], Srivastava et al. [18], Singh et al. [12, 13] and Choudhary et al. [1], have made valuable contributions in the area. Srivastava and Saigo [19] studied multiplication, and its applictions, on certain classes of operators of fractional calculus which include the Gaussian hypergeometric function. Then in [14, 21], Srivastava et al. made basic contributions to the development of Marichev-Saigo-Maeda fractional integral operators, which generalise both Riemann-Liouville and Erd
The generalized integral operators of fractional order [7, 11] involving Appell’s hypergeometric function
The S-generalized Gauss hypergeometric function (S-GGHF)
Special cases of the S-GGHF and S-generalized beta function were given by Srivastava et al. [16].
Mellin Transform
In the present section, we obtain the Mellin transform of Marichev-Saigo-Maeda fractional integral operator its particular cases.
Mellin Transform of Fractional Integral OperatorThe Mellin transform of a function
To prove the result
Theorem 2.2
To prove the result
On evaluating the resulting z-integral, we get
Now, we present the following special cases of Theorems 2.1 and 2.2. On setting
Corollary 2.1
On setting
Corollary 2.2
Further, on setting
Corollary 2.3
Corollary 2.4
Corollary 2.5
Corollary 2.6
Remark 2.1
It can be noticed that the above six corollaries are also quite general in nature and reduce to several interesting results. Thus, the six interesting results recorded in the text by Mathai et al. [8] follow as simple special cases of these corollaries.
In this section, we establish some image formulas for the S-GGHF under the Marichev-Saigo-Maeda fractional integral operator
In order to prove the result
Theorem 3.2
To prove the result
Now, we give six interesting special cases of the Theorems 3.1 and 3.2. On setting
Corollary 3.1
On setting
Corollary 3.2
Further, on setting
Corollary 3.3
Corollary 3.4
Corollary 3.5
Corollary 3.6
In this article, we evaluated the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator, the kernel of which is the Appell’s function
- A. Choudhary, D. Kumar, and J. Singh.
Numerical simulation of a fractional model of temperature distribution and heat flux in the semi infinite solid . Alexandria Eng. J..,55 (1)(2016), 87-91. - L. Debnath, and D. Bhatta. Integral transforms and their applications,
, Chapman & Hall/CRC, Boca Raton, London, New York, 2007. - AA. Kilbas, HM. Srivastava, and JJ. Trujillo. Theory and applications of fractional differential equations,
204 , North-Holland, Mathematical Studies, 2006. Elsevier Science Publishers, Amsterdam, London and New York. - D. Kumar, J. Singh, and D. Baleanu.
A hybrid computational approach for Klein-Gordon equations on Cantor sets . Nonlinear Dynam..,87 (2017), 511-517. - D. Kumar, J. Singh, and D. Baleanu.
A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves . Math. Methods Appl. Sci..,40 (15)(2017), 5642-5653. - D. Kumar, J. Singh, and D. Baleanu.
Modified Kawahara equation within a fractional derivative with non-singular kernel . Thermal Science.,22 (2018), 789-796. - OI. Marichev.
Volterra equation of Mellin convolution type with a Horn function in the kernel . Izvestiya Akademii Nauk BSSR. Seriya Fiziko-Matematicheskikh Nauk.,1 (1974), 128-129. - AM. Mathai, RK. Saxena, and HJ. Haubold. The H-function: theory and applications,
, Springer, New York, 2010. - KS. Miller, and B. Ross. An introduction to the fractional calculus and fractional differential equations,
, John Willey & Sons, New York, NY, 1993. - I. Podlubny. Fractional differential equations,
, Academic Press, San Diego, CA:1999. - M. Saigo, and N. Maeda. More generalization of fractional calculus,
, 1998. Transform Methods & Special Functions, Varna ’96, 386?400, Bulgarian Acad. Sci., Sofia. - J. Singh, D. Kumar, and JJ. Nieto.
A reliable algorithm for local fractional Tricomi equation arising in fractal transonic flow . Entropy.,18 (6)(2016) Paper 206, 8. - J. Singh, D. Kumar, MA. Qurashi, and D. Baleanu.
Analysis of a new fractional model for damped Burgers’ equation . Open Physics.,15 (2017), 35-41. - HM. Srivastava, and P. Agarwal.
Certain fractional integral operators and the generalized incomplete hypergeometric functions . Appl. Appl. Math..,8 (2)(2013), 333-345. - HM. Srivastava, P. Agarwal, and S. Jain.
Generating functions for the generalized Gauss hypergeometric functions . Appl. Math Comput.,247 (2014), 348-352. - HM. Srivastava, R. Jain, and MK. Bansal.
A study of the S-generalized Gauss hypergeometric function and its associated integral transforms . Turkish J. Anal. Number Theory.,3 (5)(2015), 116-119. - HM. Srivastava, and PW. Karlsson. Multiple Gaussian hypergeometric series,
, Halsted Press (Ellis Horwood Limited), John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1985. - HM. Srivastava, D. Kumar, and J. Singh.
An efficient analytical technique for fractional model of vibration equation . Appl. Math. Model..,45 (2017), 192-204. - HM. Srivastava, and M. Saigo.
Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation . J. Math. Anal. Appl..,121 (2)(1987), 325-369. - HM. Srivastava, and RK. Saxena.
Operators of fractional integration and their applications . Appl. Math Comput.,118 (2001), l-52. - HM. Srivastava, RK. Saxena, and RK. Parmar.
Some families of the incomplete H-functions and the incomplete H?-functions and associated integral transforms and operators of fractional calculus with applications . Russ. J. Math. Phys..,25 (1)(2018), 116-138. - X.-J. Yang, D. Baleanu, and HM. Srivastava. Local fractional integral transforms and their applications,
, Elsevier/Academic Press, Amsterdam, 2016.