International journal of

ADVANCED AND APPLIED SCIENCES

EISSN: 2313-3724, Print ISSN:2313-626X

Frequency: 12
line decor
  
line decor

 Volume 6, Issue 2 (February 2019), Pages: 1-5

----------------------------------------------

 Original Research Paper

 Title: Fractional integral operator associated with extended Mittag-Leffler function

 Author(s): Aneela Nadir *, Adnan Khan

 Affiliation(s):

 Department of Mathematics, National College of Business Administration and Economics, 40-E\1, Gullberg III, Lahore, Pakistan

  Full Text - PDF          XML

 * Corresponding Author. 

  Corresponding author's ORCID profile: https://orcid.org/0000-0003-2475-1661

 Digital Object Identifier: 

 https://doi.org/10.21833/ijaas.2019.02.001

 Abstract:

Popular literature of Special Functions includes the generalization and extensions of functions like gamma function, beta function, Mittag-Leffler function, hypergeometric function and confluent hypergeometric function etc. This sequel deals with the extension of Mittag-Leffler functions and its properties. We aim to find the composition of fractional integration formula known as Ҏ𝛿 − transform with the extended Mittag-Leffler function. Some special cases and corollaries are pointed out which follow from our main results. 

 © 2019 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

 Keywords: Laplace transform, Elzaki transform, Hadamard product

 Article History: Received 23 July 2018, Received in revised form 8 December 2018, Accepted 12 December 2018

 Acknowledgement:

No Acknowledgement

 Compliance with ethical standards

 Conflict of interest:  The authors declare that they have no conflict of interest.

 Citation:

 Nadir A and Khan A (2019). Fractional integral operator associated with extended Mittag-Leffler function. International Journal of Advanced and Applied Sciences, 6(2): 1-5

 Permanent Link to this page

 Figures

 No Figure

 Tables

 No Table

----------------------------------------------

 References (26) 

  1. Agarwal P and Choi J (2016). Fractional calculus operators and their image formulas. Journal of the Korean Mathematical Society, 53(5): 1183-1210. https://doi.org/10.4134/JKMS.j150458   [Google Scholar]
  2. Chaudhry MA, Qadir A, Srivastava HM, and Paris RB (2004). Extended hypergeometric and confluent hypergeometric functions. Applied Mathematics and Computation, 159(2): 589-602. https://doi.org/10.1016/j.amc.2003.09.017   [Google Scholar]
  3. Choi J and Agarwal P (2014). Certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions. Abstract and Applied Analysis, 2014: Article ID 735946. https://doi.org/10.1155/2014/735946   [Google Scholar]
  4. Choi J and Agarwal P (2015). Certain integral transforms for the incomplete functions. Applied Mathematics and Information Sciences, 9(4): 2161-2167.   [Google Scholar]
  5. Desai R, Salehbhai IA, and Shukla AK (2016). Note on generalized Mittag-Leffler function. SpringerPlus, 5(1): 683-691. https://doi.org/10.1186/s40064-016-2299-x   [Google Scholar]
  6. Gehlot KS (2013). Integral representation and certain properties of m–series associated with fractional calculus. In International Mathematical Forum, 8(9): 415-426. https://doi.org/10.12988/imf.2013.13041   [Google Scholar]
  7. Gupta A and Parihar CL (2017). Saigo’s  K-fractional calculus operators. Malaya Journal of Matematik, 5(3): 494-504.   [Google Scholar]
  8. Kataria KK and Vellaisamy P (2015). The generalized K-Wright function and marichev-saigo-maeda fractional operators. Journal of Analysis, 23: 75–87.   [Google Scholar]
  9. Kilbas AA, Saigo M, and Saxena RK (2004). Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms and Special Functions, 15(1): 31-49. https://doi.org/10.1080/10652460310001600717   [Google Scholar]
  10. Kumar D (2013). Solution of fractional kinetic equation by a class of integral transform of pathway type. Journal of Mathematical Physics, 54(4): 043509. https://doi.org/10.1063/1.4800768   [Google Scholar]
  11. Mittal E, Pandey RM, and Joshi S (2016). On extension of Mittag-Leffler function. Applications and Applied Mathematics, 11(1): 307-316.   [Google Scholar]
  12. Nadir A and Khan A (2018a). Marichev-Saigo-Maeda differential operator and generalized incomplete hypergeometric functions. Journal of Mathematics, 50(3): 123-129.   [Google Scholar]
  13. Nadir A and Khan A (2018b). Caputo MSM fractional differentiation of extended Mittag-Leffler function. International Journal of Advanced and Applied Sciences. 5(10): 28-34. https://doi.org/10.21833/ijaas.2018.10.005   [Google Scholar]
  14. Nadir A, Khan A, and Kalim M (2014). Integral transforms of the generalized mittag-leffler function. Applied Mathematical Sciences, 8(103): 5145-5154. https://doi.org/10.12988/ams.2014.43218   [Google Scholar]
  15. Özarslan MA and Yılmaz B (2014). The extended Mittag-Leffler function and its properties. Journal of Inequalities and Applications, 2014(1): 85-95. https://doi.org/10.1186/1029-242X-2014-85   [Google Scholar]
  16. Parmar RK (2015). A Class of extended mittag–leffler functions and their properties related to integral transforms and fractional calculus. Mathematics, 3(4): 1069-1082. https://doi.org/10.3390/math3041069   [Google Scholar]
  17. Pohlen T (2009). The Hadamard product and universal power series. Ph.D. Dissertation, University in Trier, Trier, Germany.   [Google Scholar]
  18. Prabhakar TR (1971). A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Mathematical Journal, 19: 7–15.   [Google Scholar]
  19. Rahman G, Baleanu D, Qurashi MA, Purohit SD, Mubeen S and Arshad M (2017a). The extended Mittag-Leffler function via fractional calculus. Journal of Nonlinear Sciences and Applications. 10: 4244-4253. https://doi.org/10.22436/jnsa.010.08.19   [Google Scholar]
  20. Rahman G, Ghaffar A, Mubeen S, Arshad M, and Khan SU (2017b). The composition of extended Mittag-Leffler functions with pathway integral operator. Advances in Difference Equations, 2017(176): 1-10. https://doi.org/10.1186/s13662-017-1237-8   [Google Scholar]
  21. Rainville ED (1971). Special functions. 2nd Edition, Chelsea Publishing Company, Bronx, New York, USA.   [Google Scholar]
  22. Rao A, Garg M, and Kalla SL (2010). Caputo-type fractional derivative of a hypergeometric integral operator. Kuwait Journal of Science and Engineering, 37(1A): 15-29.   [Google Scholar]
  23. Saigo M (1978). A remark on integral operators involving the Gauss hypergeometric functions. Kyushu University, 11: 135-143.   [Google Scholar]
  24. Saigo M and Maeda N (1998). More generalization of fractional calculus. In: Rusev P, Dimovski I, and Kiryakova V (Eds.), Transform methods and special functions: 386-400. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Varna, Bulgaria.   [Google Scholar]  PMCid:PMC106901
  25. Samko SG, Kilbas AA, and Marichev OI (1993). Fractional integrals and derivatives: Theory and applications. Gordon and Breach, Yverdon, Switzerland.   [Google Scholar]
  26. Saxena RK and Parmar RK (2017). Fractional integration and differentiation of the generalized mathieu series. Axioms, 6(3): 18-29. https://doi.org/10.3390/axioms6030018   [Google Scholar]