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March  2020, 13(3): 539-560. doi: 10.3934/dcdss.2020030

## Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions

 1 Department of Mathematics, Baba Farid College, Bathinda-151001, India 2 Post Graduate Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388120, India 3 Department of Mathematics Education, University of Education, Winneba(Kumasi campus), Ghana 4 Department of Applied Sciences, Guru Kashi University, Bathinda-1513002, India

* Corresponding author: Jyotindra C. Prajapati

Received  July 2018 Revised  September 2018 Published  March 2019

The aim of the present paper is to establish certain new image formulae of family of some extended generalized Gauss hypergeometric functions by applying the operators of fractional derivative involving ${}_2F_1(.)$ due to Saigo. Furthermore, by employing some integral transforms on the resulting formulas, we obtained some more image formulas and also develop a new and further generalized form of the fractional kinetic equation involving the family of some extended generalized Gauss hypergeometric functions and the manifold generality of the family of functions is discussed in terms of the solution of the fractional kinetic equation. The results obtained here are quite general in nature.

Citation: Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 539-560. doi: 10.3934/dcdss.2020030
##### References:
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Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056. [6] A. Atangana, Blind in a commutative world: Simple illustrations with functions and chaotic attractors, Chaos, Solitons and Fractals, 114 (2018), 347-363.  doi: 10.1016/j.chaos.2018.07.022. [7] M. Axtell and M. E. Bise, Fractional Calculus Applications in Control Systems, In: Proceedings of the 1990 National Aerospace and Electronics Conference, Dayton, OH, USA, 1990. doi: 10.1109/NAECON.1990.112826. [8] M. Caputo, Linear models of dissipation whose $q$ is almost frequency independent Ⅱ, Geophys. J. Royal Astr. Soc., 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [9] M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math., 78 (1997), 19-32.  doi: 10.1016/S0377-0427(96)00102-1. [10] M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589-602.  doi: 10.1016/j.amc.2003.09.017. [11] V. B. L. Chaurasia and S. C. Pandey, On the new computable solution of the generalized fractional kinetic equations involving the generalized function for the fractional calculus and related functions, Astrophys. Space Sci., 317 (2008), 213-219. [12] Y. Chen, I. Petráš and D. Xue, Fractional Order Control, A Tutorial Proceedings of 2009 American Control Conference, St. Louis, MO, USA, 2009. [13] J. Choi, P. Agarwal, S. Mathur and S. D. Purohit, Certain new integral formulas involving the generalized Bessel function, Bull.Korean Math. Soc., 51 (2014), 995–1003. Available from: http://dx.doi.org/10.5831/HMJ.2013.35.4.667 doi: 10.4134/BKMS.2014.51.4.995. [14] J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, Filomat, 30 (2016), 1931–1939. Available from: https://www.jstor.org/stable/24898765 doi: 10.2298/FIL1607931C. [15] A. Chouhan and S. Sarswat, On solution of generalized kinetic equation of fractional order, Int. J. Math. Sci. Appl., 2 (2012), 813-818. [16] A. Chouhan, S. D. Purohit and S. Saraswat, An alternative method for solving generalized differential equations of fractional order, Kragujevac J. Math., 37 (2013), 299-306. [17] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, In: Tables of integral transforms, McGraw-Hill, New York-Toronto-London, 1 (1954). [18] A. C. Escamilla, J. F. Gómez-Aguilar, D. Baleanu, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, M. M. A. Qurashi, Bateman-feshbach tikochinsky and caldirolakanai oscillators with new fractional differentiation, Entropy, 19 (2017), 1–13. Available from: https://doi.org/10.3390/e19020055 [19] A. C. Escamilla, F. Torres, J. F. Gómez-Aguilar, R. F. Escobar-Jiménez and G. V. Guerrero-Ramírez, On the trajectory tracking control for an scara robot manipulator in a fractional model driven by induction motors with pso tuning, Multibody Syst Dyn., 43 (2018), 257-277.  doi: 10.1007/s11044-017-9586-3. [20] A. C. Escamilla, J. F. Gómez-Aguilar, L. Torres and R. F. Escobar-Jiménez, A numerical solution for a variable-order reaction diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A, 491 (2018), 406-424.  doi: 10.1016/j.physa.2017.09.014. [21] J. F. Gómez-Aguilar, Novel analytical solutions of the fractional Drude model, Optik, 168 (2018), 728–740. Available from: https://doi.org/10.1016/j.ijleo.2018.04.107 [22] J. F. Gómez-Aguilar, H. Yépez-Martnez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragozaand and J. Reyes-Reyes, Analytical and numerical solutions of electrical circuits described by fractional derivatives. Applied Mathematical Modelling, Applied Mathematical Modelling, 40 (2016), 9079-9094.  doi: 10.1016/j.apm.2016.05.041. [23] J. F. Gómez-Aguilar, Chaos in a nonlinear bloch system with atangana abaleanu fractional derivatives, Numer. Methods Partial Differential Eq., 34 (2018), 1716-1738.  doi: 10.1002/num.22219. [24] J. F. Gómez-Aguilar, H. Yépez-Martínez, J. Torres-Jiménez, T. Córdova-Fraga, R. F. Escobar-Jiménez and V. H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Advances in Difference Equations, 68 (2017), 1-18.  doi: 10.1186/s13662-017-1120-7. [25] V. G. Gupta, B. Sharma and F. B. M. Belgacem, On the solutions of generalized fractional kinetic equations, Appl. Math. Sci., 5 (2011), 899-910. [26] A. Gupta and C. L. Parihar, On solutions of generalized kinetic equations of fractional order, Bol. Soc. Paran. Mat., 32 (2014), 183-191.  doi: 10.5269/bspm.v32i1.18146. [27] R. E. Gutiérrez, J. M. Rosário and J. T. Machado, Fractional order calculus: Basic concepts and engineering applications, Mathematical Problems in Engineering, 2010 (2010), Article ID 375858, 19 pages. doi: 10.1155/2010/375858. [28] S. E. Hamamci, Stabilization using fractional order pi and pid controllers, Nonlinear Dynamics, 51 (2008), 329-343. [29] S. E. Hamamci and M. Koksal, Calculation of all stabilizing fractional-order pd controllers for integrating time delay systems, Computers and Mathematics with Applications, 59 (2010), 1621-1629.  doi: 10.1016/j.camwa.2009.08.049. [30] H. J. Haubold and A. M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci., 273 (2000), 53-63. [31] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 204 (2006), 7-10. [32] A. A. Kilbas and N. Sebastian, Generalized fractional integration of Bessel function of the first kind, Int. Transf. Spec. Funct., 19 (2008), 869-883.  doi: 10.1080/10652460802295978. [33] H. Kober, On fractional integrals and derivatives, Quart. J. Math. Oxford Ser., 11 (1940), 193-212.  doi: 10.1093/qmath/os-11.1.193. [34] D. Kumar, S. D. Purohit, A. Secer and A. Atangana, On generalized fractional kinetic equations involving generalized Bessel, Mathematical Problems in Engineering, 2015 (2015), Article ID 289387, 7 pages. doi: 10.1155/2015/289387. [35] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Functions: Theory and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-0916-9. [36] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, USA, 1993. [37] G. M. Mittag-Leffler, Sur la representation analytique d'une branche uniforme d'une fonction monogene, Acta. Math., 29 (1905), 101-181.  doi: 10.1007/BF02403200. [38] K. B. Oldham and J. Spanier, The fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, New York, 1974. [39] E. $\ddot{O}$zergin, M. A. $\ddot{O}$zarslan and A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601-4610.  doi: 10.1016/j.cam.2010.04.019. [40] E.$\ddot{O}$zergin, Some Properties of Hypergeometric Functions, Ph.D. Thesis, Eastern Mediterranean University, North Cyprus, 2011. Available from: http://hdl.handle.net/11129/217 [41] R. K. Parmar, A new generalization of Gamma, Beta, hypergeometric and confluent hypergeometric functions, Matematiche (Catania), 68 (2013), 33-52. [42] I. Petráš, Stability of fractional-order systems with rational orders, A survey, Fractional Calculus & Applied Analysis, 12 (2009), 269–298. Available from: https://arXiv.org/pdf/0811.4102 [43] I. Podlubny, Fractional Differential Equations, New York: Academic Press. San Diego, CA, USA, 1999. [44] T. Pohlen, The Hadamard product and universal power series (Dissertation), Universit$\ddot {\rm {a}}$t Trier, (2009). [45] Y. N. Rabotnov, Creep Problems in Structural Members, North-Holland Series in Applied Mathematics and Mechanics, 1969. [46] E. D. Rainville, Special Functions, Macmillan Company, New York, Reprinted by Chelsea Publishing Company, Bronx, New York, 1971. [47] J. J. Rosales, M. Guía, F. Gmez, F. Aguilar and J. Martínez, Two dimensional fractional projectile motion in a resisting medium, Cent. Eur. J. 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show all references

##### References:
 [1] P. Agarwal, M. Chand and G. Singh, Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Engineering Journal, 55 (2016), 3053-3059.  doi: 10.1016/j.aej.2016.07.025. [2] P. Agarwal, S. K. Ntouyas, S. Jain, M. Chand and G. Singh, Fractional kinetic equations involving generalized k-Bessel function via Sumudu transform, Alexandria Engineering Journal, 57 (2018), 1937-1942.  doi: 10.1016/j.aej.2017.03.046. [3] M. A. Al-Bassam and Y. K. Luchko, On generalized fractional calculus and its application to the solution of integro-differential equations, J. Fract. Calc., 7 (1995), 69-88. [4] A. Atangana and J. F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166.  doi: 10.1140/epjp/i2018-12021-3. [5] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056. [6] A. Atangana, Blind in a commutative world: Simple illustrations with functions and chaotic attractors, Chaos, Solitons and Fractals, 114 (2018), 347-363.  doi: 10.1016/j.chaos.2018.07.022. [7] M. Axtell and M. E. Bise, Fractional Calculus Applications in Control Systems, In: Proceedings of the 1990 National Aerospace and Electronics Conference, Dayton, OH, USA, 1990. doi: 10.1109/NAECON.1990.112826. [8] M. Caputo, Linear models of dissipation whose $q$ is almost frequency independent Ⅱ, Geophys. J. Royal Astr. Soc., 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [9] M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math., 78 (1997), 19-32.  doi: 10.1016/S0377-0427(96)00102-1. [10] M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589-602.  doi: 10.1016/j.amc.2003.09.017. [11] V. B. L. Chaurasia and S. C. Pandey, On the new computable solution of the generalized fractional kinetic equations involving the generalized function for the fractional calculus and related functions, Astrophys. Space Sci., 317 (2008), 213-219. [12] Y. Chen, I. Petráš and D. Xue, Fractional Order Control, A Tutorial Proceedings of 2009 American Control Conference, St. Louis, MO, USA, 2009. [13] J. Choi, P. Agarwal, S. Mathur and S. D. Purohit, Certain new integral formulas involving the generalized Bessel function, Bull.Korean Math. Soc., 51 (2014), 995–1003. Available from: http://dx.doi.org/10.5831/HMJ.2013.35.4.667 doi: 10.4134/BKMS.2014.51.4.995. [14] J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, Filomat, 30 (2016), 1931–1939. Available from: https://www.jstor.org/stable/24898765 doi: 10.2298/FIL1607931C. [15] A. Chouhan and S. Sarswat, On solution of generalized kinetic equation of fractional order, Int. J. Math. Sci. Appl., 2 (2012), 813-818. [16] A. Chouhan, S. D. Purohit and S. Saraswat, An alternative method for solving generalized differential equations of fractional order, Kragujevac J. Math., 37 (2013), 299-306. [17] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, In: Tables of integral transforms, McGraw-Hill, New York-Toronto-London, 1 (1954). [18] A. C. Escamilla, J. F. Gómez-Aguilar, D. Baleanu, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, M. M. A. Qurashi, Bateman-feshbach tikochinsky and caldirolakanai oscillators with new fractional differentiation, Entropy, 19 (2017), 1–13. Available from: https://doi.org/10.3390/e19020055 [19] A. C. Escamilla, F. Torres, J. F. Gómez-Aguilar, R. F. Escobar-Jiménez and G. V. Guerrero-Ramírez, On the trajectory tracking control for an scara robot manipulator in a fractional model driven by induction motors with pso tuning, Multibody Syst Dyn., 43 (2018), 257-277.  doi: 10.1007/s11044-017-9586-3. [20] A. C. Escamilla, J. F. Gómez-Aguilar, L. Torres and R. F. Escobar-Jiménez, A numerical solution for a variable-order reaction diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A, 491 (2018), 406-424.  doi: 10.1016/j.physa.2017.09.014. [21] J. F. Gómez-Aguilar, Novel analytical solutions of the fractional Drude model, Optik, 168 (2018), 728–740. Available from: https://doi.org/10.1016/j.ijleo.2018.04.107 [22] J. F. Gómez-Aguilar, H. Yépez-Martnez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragozaand and J. Reyes-Reyes, Analytical and numerical solutions of electrical circuits described by fractional derivatives. Applied Mathematical Modelling, Applied Mathematical Modelling, 40 (2016), 9079-9094.  doi: 10.1016/j.apm.2016.05.041. [23] J. F. Gómez-Aguilar, Chaos in a nonlinear bloch system with atangana abaleanu fractional derivatives, Numer. Methods Partial Differential Eq., 34 (2018), 1716-1738.  doi: 10.1002/num.22219. [24] J. F. Gómez-Aguilar, H. Yépez-Martínez, J. Torres-Jiménez, T. Córdova-Fraga, R. F. Escobar-Jiménez and V. H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Advances in Difference Equations, 68 (2017), 1-18.  doi: 10.1186/s13662-017-1120-7. [25] V. G. Gupta, B. Sharma and F. B. M. Belgacem, On the solutions of generalized fractional kinetic equations, Appl. Math. Sci., 5 (2011), 899-910. [26] A. Gupta and C. L. Parihar, On solutions of generalized kinetic equations of fractional order, Bol. Soc. Paran. Mat., 32 (2014), 183-191.  doi: 10.5269/bspm.v32i1.18146. [27] R. E. Gutiérrez, J. M. Rosário and J. T. Machado, Fractional order calculus: Basic concepts and engineering applications, Mathematical Problems in Engineering, 2010 (2010), Article ID 375858, 19 pages. doi: 10.1155/2010/375858. [28] S. E. Hamamci, Stabilization using fractional order pi and pid controllers, Nonlinear Dynamics, 51 (2008), 329-343. [29] S. E. Hamamci and M. Koksal, Calculation of all stabilizing fractional-order pd controllers for integrating time delay systems, Computers and Mathematics with Applications, 59 (2010), 1621-1629.  doi: 10.1016/j.camwa.2009.08.049. [30] H. J. Haubold and A. M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci., 273 (2000), 53-63. [31] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 204 (2006), 7-10. [32] A. A. Kilbas and N. Sebastian, Generalized fractional integration of Bessel function of the first kind, Int. Transf. Spec. Funct., 19 (2008), 869-883.  doi: 10.1080/10652460802295978. [33] H. Kober, On fractional integrals and derivatives, Quart. J. Math. Oxford Ser., 11 (1940), 193-212.  doi: 10.1093/qmath/os-11.1.193. [34] D. Kumar, S. D. Purohit, A. Secer and A. Atangana, On generalized fractional kinetic equations involving generalized Bessel, Mathematical Problems in Engineering, 2015 (2015), Article ID 289387, 7 pages. doi: 10.1155/2015/289387. [35] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Functions: Theory and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-0916-9. [36] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, USA, 1993. [37] G. M. Mittag-Leffler, Sur la representation analytique d'une branche uniforme d'une fonction monogene, Acta. Math., 29 (1905), 101-181.  doi: 10.1007/BF02403200. [38] K. B. Oldham and J. Spanier, The fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, New York, 1974. [39] E. $\ddot{O}$zergin, M. A. $\ddot{O}$zarslan and A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601-4610.  doi: 10.1016/j.cam.2010.04.019. [40] E.$\ddot{O}$zergin, Some Properties of Hypergeometric Functions, Ph.D. Thesis, Eastern Mediterranean University, North Cyprus, 2011. Available from: http://hdl.handle.net/11129/217 [41] R. K. 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