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An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results

Original Research Paper

Abstract

In the present paper, we first introduce and investigate the generalized extended Mittag-Leffler (GEML) function which is represented in the following manner:
$$\begin{aligned}&E_{\delta , \kappa }^{\vartheta ; d}(z; q, \rho , \zeta )= \sum \limits _{n=0}^{\infty }\frac{B_{q}^{(\rho , \zeta )}(\vartheta +n, d-\vartheta )}{B(\vartheta , d-\vartheta )}\frac{(d)_{ n}}{\Gamma (\delta n+ \kappa )} \frac{z^{n}}{n!}\\ &\left( \begin{array}{cc} {\mathfrak{R}}(d)> {\mathfrak{R}}(\vartheta )> 0, {\mathfrak{R}}(\delta )> 0,{\mathfrak{R}}(\kappa )> 0, \\ {\mathfrak{R}}(q) \ge 0,\, \text{min}\left\{ {\mathfrak{R}}(\vartheta +n), {\mathfrak{R}}(d-\vartheta ), {\mathfrak{R}}(\rho ), {\mathfrak{R}}(\zeta )\right\} > 0 \end{array}\right) \end{aligned}$$
and propose some of it’s integral representations. Next, we present fractional calculus of function of our study. Further, we introduce and study an integral operator whose kernel is generalized extended Mittag-Leffler (GEML) function and point out it’s known special cases. Next, we derive some properties of aforementioned integral operator which includes it’s composition relationship with right-sided Riemann–Liouville fractional integral operator \(I^{\gamma }_{a+}\) and boundedness. Finally, we obtain image of \((\tau -a)^{\alpha -1}\Phi _{l_{j};\upsilon _{j}Q}^{k_{j};\varrho _{j}P}(\beta \tau ,s,a)\) under integral operator of our study. The results derived in this paper generalizes the results obtained by Özarslan and Yilmaz (J Inequal Appl 85:1–10, 2014) and Rahman et al. (Sociedad Matemática Mexican.  https://doi.org/10.1007/s40590-017-0167-5, 2017).

Keywords

Mittag-Leffler function Generalized Beta function Hilfer derivative Integral operator 

Mathematics Subject Classification

33E12 33B15 36A33 47G10 

Notes

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interest regarding the publication of this article.

Human/animals participation

The authors declare that there is no research involving human participants and/or animals in the content this paper.

References

  1. 1.
    Camargo, R.F., E. Capelas de Oliveira, and J. Vaz. 2012. On the generalized Mittag-Leffler function and its application in a fractional telegraph equation. Mathematical Physics Analysis and Geometry 15 (1): 1–16.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Džrbašjan, M.M. 1966. Integral Transforms and Representations of Functions in the Complex Domain. Moscow: Nauka (in Russian).Google Scholar
  3. 3.
    Gorenflo, R., and F. Mainardi. 1997. Fractional calculus: integral and differential equations of fractional order. In Fractals and Fractional Calculus in Continuum Mechanics. Springer Series on CSM Courses and Lectures, vol. 378, ed. A. Carpinteri, and F. Mainardi, 223–276. Vienna: Springer.CrossRefGoogle Scholar
  4. 4.
    Gorenflo, R., F. Mainardi, and H.M. Srivastava. 1998. Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena. In Proceedings of the Eighth International Colloquium on Differential Equations (Plovdiv, Bulgaria; August 18–23, 1997), ed. D. Bainov, 195–202. Utrecht: VSP Publishers.Google Scholar
  5. 5.
    Hilfer, R. (ed.). 2000. Applications of Fractional Calculus in Physics. Singapore: World Scientific Publishing Company.MATHGoogle Scholar
  6. 6.
    Hilfer, R. 2000. Fractional time evolution. In Applications of Fractional Calculus in Physics, ed. R. Hilfer. Singapore: World Scientific Publishing Company.CrossRefGoogle Scholar
  7. 7.
    Kilbas, A.A., and M. Saigo. 1996. On Mittag-Leffler type function, fractional calculus operators and solutions of integral equations. Integral Transforms and Special Functions 4: 355–370.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kilbas, A.A., H.M. Srivastava, and J.J. Trujillo. 2006. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, vol. 204. Amsterdam: Elsevier (North-Holland) Science Publishers.Google Scholar
  9. 9.
    Kilbas, A.A., M. Saigo, and R.K. Saxena. 2004. Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms and Special Functions 15: 31–49.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kumar, D., J. Singh, and D. Baleanu. 2018. A new numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses. Nonlinear Dynamics 91: 307–317.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kumar, D., J. Singh, and D. Baleanu. 2018. Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel. Physica A 492: 155–167.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mathai, A.M., and H.J. Haubold. 2010. Special Functions for Applied Sciences. Berlin: Springer.Google Scholar
  13. 13.
    Mittag-Leffler, G.M. 1903. Sur la nouvelle fonction \(E_{\alpha }(x)\). Comptes Rendus de l’Académie des sciences Paris 137: 554–558.MATHGoogle Scholar
  14. 14.
    Miller, K.S., and B. Ross. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: A Wiley-Interscience Publication, Wiley.MATHGoogle Scholar
  15. 15.
    Özarslan, M.A., and B. Yilmaz. 2014. The extended Mittag Leffler function and its properties. Journal of Inequalities and Applications 85: 1–10.MathSciNetMATHGoogle Scholar
  16. 16.
    Özergin, E., M.A. Özarslan, and A. Altin. 2011. Extension of gamma, beta and hypergeometric functions. Journal of Computational and Applied Mathematics 235: 4601–4610.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Prabhakar, T.R. 1971. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Mathematical Journal 19: 7–15.MathSciNetMATHGoogle Scholar
  18. 18.
    Rahman, G., P. Agarwal, S. Mubeen, and M. Arshad. 2017. Fractional integral operators involving extended Mittag-Leffler function as its kernel. Sociedad Matemática Mexican.  https://doi.org/10.1007/s40590-017-0167-5.Google Scholar
  19. 19.
    Rainville, E.D. 1960. Special Functions. New York: Macmillan.MATHGoogle Scholar
  20. 20.
    Singh, J., D. Kumar, and D. Baleanu. 2018. An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation. Applied Mathematics and Computation 335: 12–24.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Singh, J., D. Kumar, and D. Baleanu. 2017. On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag-Leffler type kernel. Chaos 27: 103113.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Samko, S.G., A.A. Kilbas, and O.I. Marichev. 1993. Fractional Integrals and Derivatives, Theory and Applications. Yverdon: Gordon and Breach Science Publishers.MATHGoogle Scholar
  23. 23.
    Shukla, A.K., and J.C. Prajapati. 2007. On a generalization of Mittag-Leffler function and its properties. Journal of Mathematical Analysis and Applications 336: 797–811.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Srivastava, H.M. 2011. Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Applied Mathematics and Information Sciences 5: 390–444.MathSciNetGoogle Scholar
  25. 25.
    Srivastava, H.M., P. Agarwal, and S. Jain. 2014. Generating functions for the generalized Gauss hypergeometric functions. Applied Mathematics and Computation 247: 348–352.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Srivastava, H.M., and J. Choi. 2001. Series Associated with the Zeta and Related Functions. Dordrecht: Kluwer Academic Publishers.CrossRefMATHGoogle Scholar
  27. 27.
    Srivastava, H.M., and J. Choi. 2012. Zeta and q-Zeta Functions and Associated Series and Integrals. Amsterdam: Elsevier Science Publishers.MATHGoogle Scholar
  28. 28.
    Srivastava, H.M., D. Jankov, T.K. Pogány, and R.K. Saxena. 2011. Two-sided inequalities for the extended Hurwitz–Lerch Zeta function. Computers & Mathematics with Applications 62: 516–522.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Srivastava, H.M., R.K. Saxena, T.K. Pogány, and R. Saxena. 2011. Integral and computational representations of the extended Hurwitz–Lerch Zeta function. Integral Transforms and Special Functions 22: 487–506.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Srivastava, H.M., and Ž. Tomovski. 2009. Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Applied Mathematics and Computation 211: 198–210.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wiman, A. 1905. Über den fundamental Satz in der Theorie der Funktionen \(E_{\alpha }(x)\). Acta Mathematica 29: 191–201.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2018

Authors and Affiliations

  1. 1.Department of Applied SciencesGovernment Engineering CollegeBanswaraIndia
  2. 2.Department of MathematicsMalaviya National Institute of TechnologyJaipurIndia
  3. 3.Department of MathematicsUniversity of RajasthanJaipurIndia

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