The generalized p-k-Mittag-Leffler function and solution of fractional kinetic equations

Abstract

In this paper, we define the generalized p-k-Mittag-Leffler function and investigate its various important properties such as: Mellin-Barnes integral formula, integral transforms, and fractional calculus. The generalized p-k-Mittag-Leffler function is also discussed in terms of the solution of fractional kinetic equations. Certain interesting and useful examples are considered as special cases of generalized p-k-MLF to give the applications of our main results. We also point out their relevance with known results.

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References

  1. 1.

    Agarwal, P., M. Chand, D. Baleanu, D. O’Regan, and S. Jain. 2018. On the solution of certain fractional kinetic equations involving k-Mittag-Leffler function. Advances in Differential Equations 1–13. https://doi.org/10.1186/s13662-018-1694-8.

  2. 2.

    Agarwal, P. and J. Nieto. 2017. Some fractional integral formulas for the Mittag-Leffler type function with four parameters. Open Mathematics. https://doi.org/10.1515/math-2015.

  3. 3.

    Agarwal, P., S.K. Ntouyas, S. Jain, M. Chand, and G. Singh. 2017. Fractional kinetic equations involving genralized \(k\)-Bessel function via Sumudu transform. Alexndria Engineering Journal 1-6. https://doi.org/10.1016/j.aej.2017.03.046.

  4. 4.

    Ahmed, S. 2014. On the generalized fractional integrals of the generalized Mittag-Leffler function. Springer Plus 3: 1–5.

    Article  Google Scholar 

  5. 5.

    Chand, M., J.C. Prajaptai, and E. Bonyah. 2017. Fractional integral and solution of fractional kinetic equations involving k-Mittag-Leffler function. Transaction of A Razmadze Mathematical Institute 171: 144–166.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Choi, J., and D. Kumar. 2015. Solution of generalized fractional kinetic equations involving Alpha function. Mathematical Communication 20: 113–123.

    MATH  Google Scholar 

  7. 7.

    Fox, C. 1961. The G and H functions as symmetrical Fourier kernels. Transactions of the American Mathematical Society 98: 395–429.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Gehlot, K.S. 2017. Two parameter gamma function and it’s properties. 1-7. article ID: arXiv:1701.01052 [math.CA].

  9. 9.

    Haubold, H.J., and A.M. Mathai. 2000. The fractional kinetic equation and thermonuclear functions. Astrophysics Space Science 327: 53–63.

    Article  Google Scholar 

  10. 10.

    Khan, N.U., T. Usman, and M. Ghasuddin. 2016. Some integral associated with multiindex Mittag-Leffler functions. Journal of Applied Mathematics and Informatics 34 (3): 249–255.

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kilbas, A.A., and N. Sebstian. 2008. Generalized fractional integration of Bessel function of the first kind. Integral Transforms and Special Functions 19 (12): 869–883.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Luchko, Y., H. Martinez, and J. Trujillo. 2008. Fractional Fourier transform and some of its applications. Fractional Calculus and Applied Analysis 11 (4): 457–470.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Mathai, A.M., and R.K. Saxana. 2010. Generalized hypergeometric functions with applications. New York: Springer.

    Google Scholar 

  14. 14.

    Nisar, K.S., S.D. Purohit, and S.R. Mondal. 2016. Generalized fractional kinetic equations involving generalized Struve function of the first kind. Journal of King Saud University-Sciences 28: 167–171.

    Article  Google Scholar 

  15. 15.

    Pathak, R.S. 1966. Certain convergence theorems and asymptotic properties of a generalized of Lomel and maitland transformations. Proceedings of the National Academy of Sciences India A–36 (1): 81–86.

    Google Scholar 

  16. 16.

    Romero, L., R. Cerutti, and L. Luque. 2011. A new fractional Fourier transform and convolutions products. International Journal of Pure and Applied Mathematics 66: 397–408.

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Saichev, A.I., and G. Zaslavsky. 1997. Fractional kinetic equations; solutions and applications. Chaos 7 (4): 753–764.

    MathSciNet  Article  Google Scholar 

  18. 18.

    Saigo, M. 1978. A remark on integral operators involving the gauss hypergeometric functions. Mathematical Reports 11: 135–143.

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Saxana, R.K., J. Daiya, and A. Singh. 2014. Integral transform of the k-generalized Mittag-Leffler function \(E_{k,\alpha ,\beta }^{\gamma ,\tau }(z)\). LE MATHEMATIC HE LXIX: 7-16.

  20. 20.

    Saxena, R.K., and S.L. Kalla. 2008. On the solutions of certain fractional kinetic equations. Applied Mathematics and Computation 199: 504–511.

    MathSciNet  Article  Google Scholar 

  21. 21.

    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 (2): 797–811.

    MathSciNet  Article  Google Scholar 

  22. 22.

    Sneddon, I.N. 1979. The use of integral transforms. New York: Tata McGraw-Hill.

    Google Scholar 

  23. 23.

    Spiegel, M.R. 1965. Theory and problem of Laplace transforms, Schums Outline Series. New york: McGraw-Hill.

    Google Scholar 

  24. 24.

    Srivastava, H.M., and P.W. Karlsson. 1985. Multiple Gaussian Hypergeometric Series, Halsted Press(Ellis Horwood Limited, Chichester). New York: Wiley.

    Google Scholar 

  25. 25.

    Wiman, A. 1905. Uber den fundamental satz in der theorie der funktionen \(E_{\alpha }(x)\). Acta Mathematica 29 (1): 191–201.

    Google Scholar 

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Correspondence to Owais Khan.

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Kamarujjama, M., Khan, N.U. & Khan, O. The generalized p-k-Mittag-Leffler function and solution of fractional kinetic equations. J Anal 27, 1029–1046 (2019). https://doi.org/10.1007/s41478-018-0160-z

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Keywords

  • Integral transforms
  • Fractional calculus
  • Fox-Wright function
  • Generalized Mittag-Leffler function
  • Fractional kinetic equations

Mathematics Subject Classification

  • 42A38
  • 26A33
  • 33E12
  • 33C20