Nonlinear Functional Analysis and Applications

The Basic Sciences Research Institute, Kyungnam University (경남대학교 기초과학연구소)
권호 표지
DOI QR코드

DOI QR Code

A TYPE OF FRACTIONAL KINETIC EQUATIONS ASSOCIATED WITH THE (p, q)-EXTENDED 𝜏-HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Khan, Owais (Department of Mathematics and Statistics Integral University) ;
  • Khan, Nabiullah (Department of Applied Mathematics, Faculty of Engineering and Technology Aligarh Muslim University) ;
  • Choi, Junesang (Department of Mathematics, Dongguk University) ;
  • Nisar, Kottakkaran Sooppy (Department of Mathematics, College of Arts and Sciences Prince Sattam bin Abdulaziz University)
  • Received : 2020.10.02
  • Accepted : 2020.11.01
  • Published : 2021.06.15
Download PDF
⟨ Previous Next ⟩

Abstract

During the last several decades, a great variety of fractional kinetic equations involving diverse special functions have been broadly and usefully employed in describing and solving several important problems of physics and astrophysics. In this paper, we aim to find solutions of a type of fractional kinetic equations associated with the (p, q)-extended 𝜏 -hypergeometric function and the (p, q)-extended 𝜏 -confluent hypergeometric function, by mainly using the Laplace transform. It is noted that the main employed techniques for this chosen type of fractional kinetic equations are Laplace transform, Sumudu transform, Laplace and Sumudu transforms, Laplace and Fourier transforms, P𝛘-transform, and an alternative method.

Keywords

Acknowledgement

The authors would like express their deep-felt thanks for the reviewers' favorable and constructive comments. The third-named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R111A1A01052440).

References

  1. G. Agarwal and R. Mathur, Solution of fractional kinetic equations by using integral transform, AIP Conference Proceedings, 2253, 020004 (2020). https://doi.org/10.1063/5.0019256
  2. P. Agarwal, M. Chand, D. Baleanu, D. O'Regan and S. Jain, On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function, Adv. Difference Equ., 2018 (2018), Article ID 249. https://doi.org/10.1186/s13662-018-1694-8
  3. G. Agarwal and K.S. Nisar, Certain fractional kinetic equations involving generalized K-functions, Analysis, 39(2) (2019), 65-70. https://doi.org/10.1515/anly-2019-0013
  4. J. Agnihotri and G. Agarwal, Solution of fractional kinetic equations by using generalized extended Mittag-Leffler functions, Int. J. Adv. Sci. Tech., 29(3s) (2020), 1475-1480. http://sersc.org/journals/index.php/IJAST/article/view/6147
  5. B.N. Al-Saqabi and V.K. Tuan, Solution of a fractional differintegral equation, Integral Transforms Spec. Funct., 4(4) (1996), 321-326. https://doi.org/10.1080/10652469608819118
  6. M.A. Asiru, Sumudu transform and the solution of integral equations of convolution type, Intern. J. Math. Edu. Sci. Tech., 32(6) (2001), 906-910. https://doi.org/10.1080/002073901317147870
  7. M.K. Bansal, D. Kumar, P. Harjule and J. Singh, Fractional kinetic equations associated with incomplete I-functions, Fractal Fract., 4(2) (2020), ID 19. https://doi.org/10.3390/fractalfract4020019
  8. M.K. Bansal, D. Kumar and R. Jain, A study of Marichev-Saigo-Maeda fractional integral operators associated with the S-generalized Gauss hypergeometric function, Kyungpook Math. J., 59(3) (2019), 433-443. https://doi.org/10.5666/kmj.2019.59.3.433
  9. L. Batalov and A. Batalova, Critical dynamics in systems controlled by fractional kinetic equations, Physica A, 392 (2013), 602-611. https://doi.org/10.1016/j.physa.2012.10.017
  10. F.B.M. Belgacem and A.A. Karaballi, Sumudu transform fundamental properties investigations and applications, J. Appl. Math. Stoch. Anal., Article ID 091083 (2006), 1-23. https://doi.org/10.1155/JAMSA/2006/91083
  11. F.B.M. Belgacem, A.A. Karaballi and S.L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Math. Prob. Eng., 3 (2003), 103-118.
  12. A.A. Bhat and R. Chauhan, Fractional kinetic equation involving integral transform, Proceedings of 10th International Conference on Digital Strategies for Organizational Success, (2019). https://ssrn.com/abstract=3328161 or http://dx.doi.org/10.2139/ssrn.3328161
  13. M. Chand, J.C. Prajapati and E. Bonyah, Fractional integrals and solution of fractional kinetic equations involving k-Mittag-Leffler function, Trans. A. Razmadze Math. Inst., 171 (2017), 144-166. https://doi.org/10.1016/j.trmi.2017.03.003
  14. M.A. Chaudhry, A. Qadir, H.M. Srivastava and R.B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159(2) (2004), 589-602. https://doi.org/10.1016/j.amc.2003.09.017
  15. M.A. Chaudhry and S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994), 99-124. https://doi.org/10.1016/0377-0427(94)90187-2
  16. M.A. Chaudhry and S.M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman and Hall), Boca Raton, FL, 2002.
  17. J. Choi and D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun., 20 (2015), 113-123.
  18. J. Choi, R.K. Parmar and T.K. Pogany, Mathieu-type series built by (p, q)-extended Gaussian hypergeometric function, Bull. Korean Math. Soc., 54(3) (2017), 789-797. https://doi.org/10.4134/BKMS.b160313
  19. J. Choi, A.K. Rathie and R.K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J. 36(2) (2014), 339-367. https://doi.org/10.5831/HMJ.2014.36.2.339
  20. A. Gaur and G. Agarwal, On β-Laplace integral transform and its properties, Int. J. Adv. Sci. Tech., 29(3s) (2020), 1481-1491. http://sersc.org/journals/index.php/IJAST/article/view/6148
  21. V.G. Gupta, B. Sharma and F.B.M. Belgacem, On the solutions of generalized fractional kinetic equations, Appl. Math. Sci., (17-20) 5 (2011), 899-910.
  22. H.J. Haubold and A.M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci., 273 (2000), 53-63. https://doi.org/10.1023/A:1002695807970
  23. A.A. Kilbas and M. Saigo, H-Transforms: Theory and Application, Chapman & Hall/CRC Press, Boca Raton, London, New York, 2004.
  24. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (NorthHolland) Science Publishers, Amsterdam, London and New York, 2006.
  25. D. Kumar, J. Choi and H.M. Srivastava, Solution of a general family of fractional kinetic equations associated with the generalized Mittag-Leffler function, Nonlinear Funct. Anal. Appl., 23(3) (2018), 455-471. https://doi.org/10.22771/NFAA.2018.23.03.04
  26. D. Marquez-Carreras, Generalized fractional kinetic equations: another point of view, Adv. Appl. Prob., 41 (2009), 893-910. https://doi.org/10.1239/aap/1253281068
  27. A.M. Mathai, R.K. Saxena and H.J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010.
  28. K.S. Nisar, Fractional integrations of a generalized Mittag-Leffler type function and its application, Mathematics, 7(12) (2019), ID 1230. https://doi.org/10.3390/math7121230
  29. R.K. Parmar, Extended τ-hypergeometric functions and associated properties, C. R. Acad. Sci. Paris, Ser.I, 353 (2015), 421-426. https://doi.org/10.1016/j.crma.2015.01.016
  30. R.K. Parmar, T.K. Pog'any and R.K. Saxena, On properties and applications of (p, q)-extended τ-hypergeometric functions, C. R. Acad. Sci. Paris, Ser.I, 356(3) (2018), 278-282. https://doi.org/10.1016/j.crma.2017.12.014
  31. T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
  32. R.K. Saxena and S.L. Kalla, On the solutions of certain fractional kinetic equations, Appl. Math. Comput., 199 (2008), 504-511. https://doi.org/10.1016/j.amc.2007.10.005
  33. R.K. Saxena, A.M. Mathai and H.J. Haubold, On fractional kinetic equations, Astrophys. Space Sci., 282 (2002), 281-287. https://doi.org/10.1023/A:1021175108964
  34. R.K. Saxena, A.M. Mathai and H.J. Haubold, Unified fractional kinetic equations and a fractional diffusion equation, Astrophys. Space Sci., 290 (2004), 299-310. https://doi.org/10.1023/B:ASTR.0000032531.46639.a7
  35. R.K. Saxena, A.M. Mathai and H.J. Haubold, On generalized fractional kinetic equations, Physica A, 344 (2004), 657-664. https://doi.org/10.1016/j.physa.2004.06.048
  36. R.K. Saxena, A.M. Mathai and H.J. Haubold, Solutions of certain fractional kinetic equations and a fractional diffusion equation, J. Math. Phys., 51 (2010), Article ID 103506. https://doi.org/10.1063/1.3496829
  37. R.K. Saxena, J. Ram and D. Kumar, Alternative derivation of generalized fractional kinetic equations, J. Fract. Calc. Appl., 4(2) (2013), 322-334.
  38. J.L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag, Berlin, Heidelberg and New York, 1999.
  39. H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  40. H.M. Srivastava, R. Jain and M.K. Bansal, A Study of the S-generalized Gauss hypergeometric function and its associated integral transforms, Turkish J. Anal. Number Theo., 3(5) (2015), 116-119. DOI: 10.12691/tjant-3-5-1
  41. H.M. Srivastava and P.W. Karlsson, Multiple Gaussian hypergeometric Series, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985.
  42. N. Sudland, B. Baumann and T. F. Nannenmacher, Open problem: Who knows about the N-function?, Appl. Anal., 1(4) (1998), 401-402.
  43. N. Virchenko, On some generalizations of the functions of hypergeometric type, Fract. Calc. Appl. Anal. 2 (1999), 233-244.
  44. N. Virchenko, S.L. Kalla and A. Al-Zamel, Some results on a generalized hypergeometric function, Integral Trans. Spec. Funct., 12(1) (2001), 89-100. https://doi.org/10.1080/10652460108819336
  45. G.K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Edu. Sci. Tech., 24 (1993), 35-43. https://doi.org/10.1080/0020739930240105
  46. G.K. Watugala, The Sumudu transform for function of two variables, Math. Eng. Ind., 8 (2002), 293-302.
  47. A. Wiman, Uber den fundamentalsatz in der theorie der funktionen Eα(x), Acta Math., 29 (1905), 191-201. https://doi.org/10.1007/BF02403202