Solution of fractional kinetic equations involving extended $ (k, \tau) $-Gauss hypergeometric matrix functions

Muajebah Hidan; Mohamed Akel; Hala Abd-Elmageed; Mohamed Abdalla; Muajebah Hidan; Mohamed Akel; Hala Abd-Elmageed; Mohamed Abdalla

doi:10.3934/math.2022798

AIMS Mathematics

2022, Volume 7, Issue 8: 14474-14491. doi: 10.3934/math.2022798

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Solution of fractional kinetic equations involving extended $ (k, \tau) $-Gauss hypergeometric matrix functions

1.
Mathematics Department, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia
2.
Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt

Received: 10 December 2021 Revised: 16 May 2022 Accepted: 24 May 2022 Published: 06 June 2022
MSC : 33B15, 33C05, 33C20, 34A05

In this work, we define an extension of the k-Wright ($ (k, \tau) $-Gauss) hypergeometric matrix function and obtain certain properties of this function. Further, we present this function to achieve the solution of the fractional kinetic equations.
- $ (k, \tau) $-Gauss hypergeometric matrix function,
- Mellin transform,
- fractional kinetic equation
Citation: Muajebah Hidan, Mohamed Akel, Hala Abd-Elmageed, Mohamed Abdalla. Solution of fractional kinetic equations involving extended $ (k, \tau) $-Gauss hypergeometric matrix functions[J]. AIMS Mathematics, 2022, 7(8): 14474-14491. doi: 10.3934/math.2022798

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Abstract

In this work, we define an extension of the k-Wright ($ (k, \tau) $-Gauss) hypergeometric matrix function and obtain certain properties of this function. Further, we present this function to achieve the solution of the fractional kinetic equations.

References

[1]	P. Agarwal, R. P. Agarwal, M. Ruzhansky, Special functions and analysis of differential equations, New York: Chapman and Hall/CRC, 2020. https://doi.org/10.1201/9780429320026
[2]	N. Kaiblinger, Product of two hypergeometric functions with power arguments, J. Math. Anal. Appl., 479 (2019), 2236–2255. https://doi.org/10.1016/j.jmaa.2019.07.053 doi: 10.1016/j.jmaa.2019.07.053
[3]	M. Hidan, S. M. Boulaaras, B. Cherif, M. Abdalla, Further results on the $(p, k)$-analogue of hypergeometric functions associated with fractional calculus operators, Math. Probl. Eng., 2021 (2021), 1–10. https://doi.org/10.1155/2021/5535962 doi: 10.1155/2021/5535962
[4]	T. H. Zhao, M. K. Wang, G. J. Hai, Y. M. Chu, Landen inequalities for Gaussian hypergeometric function, RACSAM, 116 (2022), 1–23. https://doi.org/10.1007/s13398-021-01197-y doi: 10.1007/s13398-021-01197-y
[5]	T. H. Zhao, Z. Y. He, Y. M. Chu, On some refinements for inequalities involving zero-balanced hypergeometric function, AIMS Math., 5 (2020), 6479–6495. https://doi.org/10.3934/math.2020418 doi: 10.3934/math.2020418
[6]	T. H. Zhao, M. K. Wang, W. Zhang, Y. M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018 (2018), 1–15. https://doi.org/10.1186/s13660-018-1848-y doi: 10.1186/s13660-018-1848-y
[7]	N. Virchenko, S. L. Kalla, A. Al-Zamel, Some results on a generalized hypergeometric function, Integr. Trans. Spec. Funct., 12 (2001), 89–100. https://doi.org/10.1080/10652460108819336 doi: 10.1080/10652460108819336
[8]	N. Khan, T. Usman, M. Aman, S. Al-Omari, S. Araci, Computation of certain integral formulas involving generalized Wright function, Adv. Differ. Equ., 2020 (2020), 1–10. https://doi.org/10.1186/s13662-020-02948-8 doi: 10.1186/s13662-020-02948-8
[9]	A. Ghaffar, A. Saif, M. Iqbal, M. Rizwan, Two classes of integrals involving extended Wright type generalized hypergeometric function, Commun. Math. Appl., 10 (2019), 599–606. https://doi.org/10.26713/cma.v10i3.1190 doi: 10.26713/cma.v10i3.1190
[10]	S. B. Rao, J. C. Prajapati, A. D. Patel, A. K. Shukla, Some properties of Wright-type generalized hypergeometric function via fractional calculus, Adv. Differ. Equ., 2014 (2014), 1–11. https://doi.org/10.1186/1687-1847-2014-119 doi: 10.1186/1687-1847-2014-119
[11]	N. U. Khan, T. Usman, M. Aman, Some properties concerning the analysis of generalized Wright function, J. Comput. Appl. Math., 376 (2020), 112840. https://doi.org/10.1016/j.cam.2020.112840 doi: 10.1016/j.cam.2020.112840
[12]	R. K. Parmar, Extended $\tau$-hypergeomtric functions and associated properties, C. R. Math., 353 (2015), 421–426. https://doi.org/10.1016/j.crma.2015.01.016 doi: 10.1016/j.crma.2015.01.016
[13]	R. K. Gupta, B. S. Shaktawat, D. Kumar, Some results associted with extended $\tau$-Gauss hypergeomtric functions, Ganita Sandesh, 28 (2014), 55–60.
[14]	L. Jódar, J. C. Cortés, Some properties of Gamma and Beta matrix functions, Appl. Math. Lett., 11 (1998), 89–93. https://doi.org/10.1016/S0893-9659(97)00139-0 doi: 10.1016/S0893-9659(97)00139-0
[15]	L. Jódar, J. C. Cortés, On the hypergeometric matrix function, J. Comput. Appl. Math., 99 (1998), 205–217. https://doi.org/10.1016/S0377-0427(98)00158-7 doi: 10.1016/S0377-0427(98)00158-7
[16]	M. Abdalla, Special matrix functions: Characteristics, achievements and future directions, Linear Multilinear Algebra, 68 (2020), 1–28. https://doi.org/10.1080/03081087.2018.1497585 doi: 10.1080/03081087.2018.1497585
[17]	S. Mubeen, G. Rahman, M. Arshad, k-gamma, k-beta matrix functions and their properties, J. Math. Comput. Sci., 5 (2015), 647–657.
[18]	G. Rahman, A. Ghaffar, S. D. Purohit, S. Mubeen, M. Arshad, On the hypergeomtric matrix $k$-functions, Bull. Math. Anal. Appl., 8 (2016), 98–111.
[19]	G. S. Khammash, P. Agarwal, J. Choi, Extended $k$-gamma and $k$-beta functions of matrix arguments, Mathematics, 8 (2020), 1–13. https://doi.org/10.3390/math8101715 doi: 10.3390/math8101715
[20]	A. Bakhet, F. L. He, M. M. Yu, On the matrix version of extended Bessel functions and its application to matrix differential equations, Linear Multilinear Algebra, 2021. https://doi.org/10.1080/03081087.2021.1923629
[21]	M. Abdalla, S. A. Idris, I. Mekawy, Some results on the extended hypergeometric matrix functions and related functions, J. Math., 2021 (2021), 1–12. https://doi.org/10.1155/2021/2046726 doi: 10.1155/2021/2046726
[22]	M. Abdalla, A. Bakhet, Extended Gauss hypergeometric matrix functions, Iran. J. Sci. Technol. Trans. Sci., 42 (2018), 1465–1470. https://doi.org/10.1007/s40995-017-0183-3 doi: 10.1007/s40995-017-0183-3
[23]	F. L. He, A. Bakhet, M. Abdalla, M. Hidan, On the extended hypergeometric matrix functions and their applications for the derivatives of the extended Jacobi matrix polynomial, Math. Probl. Eng., 2020 (2020), 1–8. https://doi.org/10.1155/2020/4268361 doi: 10.1155/2020/4268361
[24]	M. Akel, A. Bakhet, M. Abdalla, F. L. He, On degenerate gamma matrix functions and related functions, Linear Multilinear Algebra, 2022. https://doi.org/10.1080/03081087.2022.2040942
[25]	M. Abdalla, H. Abd-Elmageed, M. Abul-Ez, M. Zayed, Further investigations on the two variables second Appell hypergeometric matrix function, Quaest. Math., 2022. https://doi.org/10.2989/16073606.2022.2034680
[26]	R. Dwivedi, V. Sahai, Lie algebras of matrix difference differential operators and special matrix functions, Adv. Appl. Math., 122 (2021), 102109. https://doi.org/10.1016/j.aam.2020.102109 doi: 10.1016/j.aam.2020.102109
[27]	M. Hidan, A. Bakhet, H. Abd-Elmageed, M. Abdalla, Matrix-valued hypergeometric Appell-type polynomials, Electron. Res. Arch., 30 (2022), 2964–2980. https://doi.org/10.3934/era.2022150 doi: 10.3934/era.2022150
[28]	A. Verma, J. Younis, H. Aydi, On the Kampé de Fériet hypergeometric matrix function, Math. Probl. Eng., 2021 (2021), 1–11. https://doi.org/10.1155/2021/9926176 doi: 10.1155/2021/9926176
[29]	A. Bakhet, Y. Jiao, F. L. He, On the Wright hypergeometric matrix functions and their fractional calculus, Integr. Trans. Spec. Funct., 30 (2019), 138–156. https://doi.org/10.1080/10652469.2018.1543669 doi: 10.1080/10652469.2018.1543669
[30]	M. Abdalla, Fractional operators for the Wright hypergeometric matrix functions, Adv. Differ. Equ., 2020 (2020), 1–14. https://doi.org/10.1186/s13662-020-02704-y doi: 10.1186/s13662-020-02704-y
[31]	R. Diaz, E. Pariguan, On hypergeometric functions and Pochhammer $k$-symbol, Divulg. Mat., 15 (2007), 179–192.
[32]	H. Abd-Elmageed, M. Hosny, S. Boulaaras, Results on the $(k, \tau)$-analogue of hypergeometric matrix functions and $k$-fractional calculus, Fractals, 2022, In press.
[33]	M. A. Chaudhry, S. M. Zubair, On a class of incomplete gamma functions with applications, New York: Chapman and Hall/CRC, 2002. https://doi.org/10.1201/9781420036046
[34]	S. Naz, M. N. Naeem, On the generalization of $k$-fractional Hilfer-Katugampola derivative with Cauchy problem, Turk. J. Math., 45 (2021), 110–124.
[35]	M. Abdalla, M. Hidan, S. M. Boulaaras, B. Cherif, Investigation of extended $k$-hypergeometric functions and associated fractional integrals, Math. Probl. Eng., 2021 (2021), 1–11. https://doi.org/10.1155/2021/9924265 doi: 10.1155/2021/9924265
[36]	D. L. Suthar, D. Baleanu, S. D. Purohit, F. Uçar, Certain $k$-fractional calculus operators and image formulas of $k$-Struve function, AIMS Math., 5 (2020), 1706–1719. https://doi.org/10.3934/math.2020115 doi: 10.3934/math.2020115
[37]	R. Yilmazer, K. Ali, Discrete fractional solutions to the $k$-hypergeometric differential equation, Math. Methods Appl. Sci., 44 (2020), 7614–7621. https://doi.org/10.1002/mma.6460 doi: 10.1002/mma.6460
[38]	R. K. Saxena, A. M. Mathai, H. J. Haubold, On fractional kinetic equations, Astrophys. Space Sci., 282 (2002), 281–287. https://doi.org/10.1023/A:1021175108964 doi: 10.1023/A:1021175108964
[39]	R. K. Saxena, A. M. Mathai, H. J. Haubold, On generalized fractional kinetic equations, Phys. A, 344 (2004), 657–664. https://doi.org/10.1016/j.physa.2004.06.048 doi: 10.1016/j.physa.2004.06.048
[40]	M. Samraiz, M. Umer, A. Kashuri, T. Abdeljawad, S. Iqbal, N. Mlaiki, On weighted $(k, s)$-Riemann-Liouville fractional operators and solution of fractional kinetic equation, Fractal Fract., 5 (2021), 1–18. https://doi.org/10.3390/fractalfract5030118 doi: 10.3390/fractalfract5030118
[41]	M. Abdalla, M. Akel, Contribution of using Hadamard fractional integral operator via Mellin integral transform for solving certain fractional kinetic matrix equations, Fractal Fract., 6 (2022), 1–14. https://doi.org/10.3390/fractalfract6060305 doi: 10.3390/fractalfract6060305
[42]	P. Agarwal, S. K. Ntouyas, S. Jain, M. Chand, G. Singh, Fractional kinetic equations involving generalized $k$-Bessel function via Sumudu transform, Alex. Eng. J., 57 (2018), 1937–1942. https://doi.org/10.1016/j.aej.2017.03.046 doi: 10.1016/j.aej.2017.03.046
[43]	K. S. Nisar, A. Shaikh, G. Rahman, D. Kumar, Solution of fractional kinetic equations involving class of functions and Sumudu transform, Adv. Differ. Equ., 2020 (2020), 1–11. https://doi.org/10.1186/s13662-020-2513-6 doi: 10.1186/s13662-020-2513-6
[44]	R. Garrappa, M. Popolizio, Computing the matrix Mittag-Leffler function with applications to fractional calculus, J. Sci. Comput., 77 (2018), 129–153. https://doi.org/10.1007/s10915-018-0699-5 doi: 10.1007/s10915-018-0699-5
[45]	A. Sadeghi, J. R. Cardoso, Some notes on properties of the matrix Mittag-Leffler function, Appl. Math. Comput., 338 (2018), 733–738. https://doi.org/10.1016/j.amc.2018.06.037 doi: 10.1016/j.amc.2018.06.037
[46]	M. Hidan, M. Akel, S. M. Boulaaras, M. Abdalla, On behavior Laplace integral operators with generalized Bessel matrix polynomials and related functions, J. Funct. Spaces, 2021 (2021), 1–10. https://doi.org/10.1155/2021/9967855 doi: 10.1155/2021/9967855

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