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http://dx.doi.org/10.22771/nfaa.2021.26.02.10

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)
Publication Information
Nonlinear Functional Analysis and Applications / v.26, no.2, 2021 , pp. 381-392 More about this Journal
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
Fractional calculus; fractional kinetic equations; (generalized) Mittag-Leffler functions; H-function; ${\aleph}$-function; generalized M-series; (p, q)-extended ${\tau}$-hypergeometric function; (p, q)-extended ${\tau}$-confluent hypergeometric function; Laplace transform; Sumudu transform;
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Times Cited By KSCI : 1  (Citation Analysis)
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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   DOI
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   DOI
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 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   DOI
6 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.
7 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   DOI
8 N. Sudland, B. Baumann and T. F. Nannenmacher, Open problem: Who knows about the N-function?, Appl. Anal., 1(4) (1998), 401-402.
9 N. Virchenko, On some generalizations of the functions of hypergeometric type, Fract. Calc. Appl. Anal. 2 (1999), 233-244.
10 R.K. Saxena, J. Ram and D. Kumar, Alternative derivation of generalized fractional kinetic equations, J. Fract. Calc. Appl., 4(2) (2013), 322-334.
11 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   DOI
12 J.L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag, Berlin, Heidelberg and New York, 1999.
13 G.K. Watugala, The Sumudu transform for function of two variables, Math. Eng. Ind., 8 (2002), 293-302.
14 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.
15 B.N. Al-Saqabi and V.K. Tuan, Solution of a fractional differintegral equation, Integral Transforms Spec. Funct., 4(4) (1996), 321-326.   DOI
16 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.   DOI
17 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.   DOI
18 L. Batalov and A. Batalova, Critical dynamics in systems controlled by fractional kinetic equations, Physica A, 392 (2013), 602-611.   DOI
19 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.   DOI
20 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.   DOI
21 A. Wiman, Uber den fundamentalsatz in der theorie der funktionen Eα(x), Acta Math., 29 (1905), 191-201.   DOI
22 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.   DOI
23 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   DOI
24 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.
25 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.   DOI
26 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.
27 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
28 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
29 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.   DOI
30 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.   DOI
31 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.
32 H.J. Haubold and A.M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci., 273 (2000), 53-63.   DOI
33 A.A. Kilbas and M. Saigo, H-Transforms: Theory and Application, Chapman & Hall/CRC Press, Boca Raton, London, New York, 2004.
34 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.   DOI
35 D. Marquez-Carreras, Generalized fractional kinetic equations: another point of view, Adv. Appl. Prob., 41 (2009), 893-910.   DOI
36 A.M. Mathai, R.K. Saxena and H.J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010.
37 T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
38 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   DOI
39 R.K. Parmar, Extended τ-hypergeometric functions and associated properties, C. R. Acad. Sci. Paris, Ser.I, 353 (2015), 421-426.   DOI
40 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.   DOI
41 R.K. Saxena and S.L. Kalla, On the solutions of certain fractional kinetic equations, Appl. Math. Comput., 199 (2008), 504-511.   DOI
42 R.K. Saxena, A.M. Mathai and H.J. Haubold, On fractional kinetic equations, Astrophys. Space Sci., 282 (2002), 281-287.   DOI
43 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.   DOI
44 R.K. Saxena, A.M. Mathai and H.J. Haubold, On generalized fractional kinetic equations, Physica A, 344 (2004), 657-664.   DOI
45 M.A. Chaudhry and S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994), 99-124.   DOI
46 J. Choi and D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun., 20 (2015), 113-123.
47 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.