Search Advanced

AIMS Mathematics

2021, Volume 6, Issue 9: 10130-10163. doi: 10.3934/math.2021588
Previous Article Next Article
Research article Special Issues

Further study on the conformable fractional Gauss hypergeometric function

  • 1.
    Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
  • 2.
    Academy of Scientific Research and Technology (ASRT), Cairo, Egypt
  • 3.
    Mathematics Department, College of Science, King Khalid University, Abha, Saudi Arabia
  • Received: 21 May 2021 Accepted: 09 July 2021 Published: 09 July 2021

  • MSC : 26A33, 33C05, 33C90, 34K37

  • This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points $ x = 1 $ and $ x = \infty $. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF.

    Citation: Mahmoud Abul-Ez, Mohra Zayed, Ali Youssef. Further study on the conformable fractional Gauss hypergeometric function[J]. AIMS Mathematics, 2021, 6(9): 10130-10163. doi: 10.3934/math.2021588

    Related Papers:

  • This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points $ x = 1 $ and $ x = \infty $. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF.


    加载中


    [1] M. Abul-Ez, Bessel polynomial expansions in spaces of holomorphic functions, J. Math. Anal. Appl., 221 (1998), 177-190. doi: 10.1006/jmaa.1997.5840
    [2] M. Abdalla, M. Abul‐Ez, J. Morais, On the construction of generalized monogenic Bessel polynomials, Math. Method. Appl. Sci., 41 (2018), 9335-9348. doi: 10.1002/mma.5274
    [3] L. Aloui, M. Abul-Ez, G. Hassan, Bernoulli special monogenic polynomials with the difference and sum polynomial bases, Complex Var. Elliptic, 59 (2014), 631-650. doi: 10.1080/17476933.2012.750450
    [4] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
    [5] I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
    [6] R. Hilfer, Applications of fractional calculus in physics, Singapore: World scientific, 2000.
    [7] R. L. Magin, Fractional calculus in bioengineering, Crit Rev Biomed Eng., 32 (2004), 1-104. doi: 10.1615/CritRevBiomedEng.v32.10
    [8] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337. doi: 10.1007/s11071-004-3764-6
    [9] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
    [10] K. S. Nisar, O. A. Ilhan, J. Manafian, M. Shahriari, D. Soybaş, Analytical behavior of the fractional Bogoyavlenskii equations with conformable derivative using two distinct reliable methods, Results Phys., 22 (2021), 103975. doi: 10.1016/j.rinp.2021.103975
    [11] H. Tajadodi, Z. A. Khan, J. F. Gómez-Aguilar, A. Khan, H. Khan, Exact solutions of conformable fractional differential equations, Results Phys., 22 (2021), 103916. doi: 10.1016/j.rinp.2021.103916
    [12] R. W. Ibrahim, D. Altulea, R. M. Elobaid. Dynamical system of the growth of COVID-19 with controller, Adv. Differ. Equ., 1 (2021), 1-12.
    [13] J. Yang, M. Fečkan, J. Wang, Consensus problems of linear multi-agent systems involving conformable derivative, Appl. Math. Comput., 394 (2020), 125809.
    [14] M. J. Lazo, D. F. Torres, Variational calculus with conformable fractional derivatives, IEEE/CAA J. Automatic Sinica, 4 (2016), 340-352.
    [15] T. Chiranjeevi, R. K. Biswas, Closed-form solution of optimal control problem of a fractional order system, J. King Saud Univ. Sci., 31 (2019), 1042-1047. doi: 10.1016/j.jksus.2019.02.010
    [16] A. Atangana, M. A. Khan, Validity of fractal derivative to capturing chaotic attractors, Chaos Soliton. Fract., 126 (2019), 50-59. doi: 10.1016/j.chaos.2019.06.002
    [17] G. Consolini, M. Materassi, A stretched logistic equation for pandemic spreading, Chaos Soliton. Fract., 140 (2020), 110-113.
    [18] R. Agarwal, S. D. Purohit, Mathematical model for anomalous subdiffusion using conformable operator, Chaos Soliton. Fract., 140 (2020), 110-199.
    [19] M. Zayed, M. Abul-Ez, M. Abdalla, N. Saad, On the fractional order Rodrigues formula for the shifted Legendre-type matrix polynomials, Mathematics, 8 (2020), 136. doi: 10.3390/math8010136
    [20] E. Ünal, A. Gökdoğan, Uyumlu kesir mertebeden chebyshev diferensiyel denklemleri ve kesirsel chebyshev polinomları, Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 16 (2016), 576-584.
    [21] H. Ç. Yaslan, Numerical solution of the conformable space-time fractional wave equation, Chinese J. Phys., 56 (2018), 2916-2925. doi: 10.1016/j.cjph.2018.09.026
    [22] E. F. D. Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation, Math. Model. Anal., 21 (2016), 188-198. doi: 10.3846/13926292.2016.1145607
    [23] E. F. D. Goufo, Strange attractor existence for non-local operators applied to four-dimensional chaotic systems with two equilibrium points, Chaos, 29 (2019), 023117. doi: 10.1063/1.5085440
    [24] E. F. D. Goufo, M. Mbehou, M. M. K. Pene, A peculiar application of Atangana-Baleanu fractional derivative in neuroscience: Chaotic burst dynamics, Chaos Soliton. Fract., 115 (2018), 170-176. doi: 10.1016/j.chaos.2018.08.003
    [25] P. Agarwal, M. Chand, S. D. Purohit, A note on generating functions involving the generalized Gauss hypergeometric functions, Natl. Acad. Sci. Lett., 37 (2014), 457-459. doi: 10.1007/s40009-014-0250-7
    [26] M. A. Chaudhry, A. Qadir, M. Rafique, 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
    [27] M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589-602.
    [28] E. Özergin, Some properties of hypergeometric functions, Doctoral dissertation, Eastern Mediterranean University (EMU), 2011.
    [29] E. Özergin, M. A. Özarslan, A. Altın, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601-4610. doi: 10.1016/j.cam.2010.04.019
    [30] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. doi: 10.1016/j.cam.2014.01.002
    [31] M. A. Hammad, H. Alzaareer, H. Al-Zoubi, H. Dutta, Fractional Gauss hypergeometric differential equation, J. Interdiscip. Math., 22 (2019), 1113-1121. doi: 10.1080/09720502.2019.1706838
    [32] A. Ali, M. Islam, A. Noreen, Solution of fractional k-Hypergeometric differential equation, J. Math. Anal., 14 (2020), 125-132.
    [33] M. Abul-Ez, M. Zayed, A. Youssef, M. De la Sen, On conformable fractional Legendre polynomials and their convergence properties with applications, Alex. Eng. J., 59 (2020), 5231-5245. doi: 10.1016/j.aej.2020.09.052
    [34] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66. doi: 10.1016/j.cam.2014.10.016
    [35] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017), 903-917. doi: 10.1007/s10092-017-0213-8
    [36] H. Kiskinov, M. Petkova, A. Zahariev, About the Cauchy problem for nonlinear system with conformable derivatives and variable delays, AIP Conference Proceedings, 2172 (2019), 050006. doi: 10.1063/1.5133525
    [37] D. R. Anderson, D. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137.
    [38] A. A. Martynyuk, I. M. Stamova, Fractional-like derivative of Lyapunov-type functions and applications to the stability analysis of motion, Electron. J. Differ. Eq., 2018 (2018), 1-12. doi: 10.1186/s13662-017-1452-3
    [39] A. Martynyuk, G. Stamov, I. Stamova, Integral estimates of the solutions of fractional-like equations of perturbed motion, Nonlinear Anal-Model, 24 (2019), 138-149.
    [40] A. Martynyuk, G. Stamov, I. Stamova, Practical stability analysis with respect to manifolds and boundedness of differential equations with fractional-like derivatives, Rocky Mt. J. Math., 49 (2019), 211-233.
    [41] M. A. Hammad, R. Khalil, Systems of linear fractional differential equations, Asian Journal of Mathematics and Computer Research, 12 (2016), 120-126.
    [42] A. El-Ajou, A modification to the conformable fractional calculus with some applications, Alex. Eng. J., 59 (2020), 2239-2249. doi: 10.1016/j.aej.2020.02.003
    [43] S. Mubeen, G. Rahman, A. Rehman, M. Naz, Contiguous function relations for-hypergeometric functions, International Scholarly Research Notices, 2014 (2014), 410801.
    [44] E. D. Rainville, Special functions, Chelsea: The Macmillan Co., 1960.
    [45] K. S. Rao, V. Lakshminarayanan, Generalized hypergeometric functions, IOP Publishing, 2018.
    [46] S. B. Opps, N. Saad, H. M. Srivastava, Recursion formulas for Appell's hypergeometric function $F_{2}$ with some applications to radiation field problems, Appl. Math. Comput., 207 (2009), 545-558.
    [47] T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons, 2019.
    [48] W. Wang, H. Wang, Some results on convolved $(p, q)$-Fibonacci polynomials, Integr. Transf. Spec. F., 26 (2015), 340-356. doi: 10.1080/10652469.2015.1007502
    [49] N. Taskara, K. Uslu, H. H. Gulec, On the properties of Lucas numbers with binomial coefficients, Appl. Math. Lett., 23 (2010), 68-72. doi: 10.1016/j.aml.2009.08.007
    [50] H. H. Gulec, N. Taskara, K. Uslu, A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients, Appl. Math. Comput., 220 (2013), 482-486.
    [51] A. Erdlyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, Vols. I-III, New York: McGraw-Hill, 1953.
    [52] S. Flügge, Practical quantum mechanics I, Berlin: Springer-Verlag, 1971.
    [53] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, New York: Dover publications inc., 1972.
    [54] H. M. Srivastava, P. Agarwal, S. Jain, Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput., 247 (2014), 348-352.
    [55] L. U. Ancarani, G. Gasaneo, Derivatives of any order of the Gaussian hypergeometric function $_2F_1 (a, b, c; z)$ with respect to the parameters a, b and c, J. Phys. A Math. Theor., 42 (2009), 395208. doi: 10.1088/1751-8113/42/39/395208
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
2.2 3

Metrics

Article views(2535) PDF downloads(150) Cited by(2)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog