Analytical solutions of $ q $-fractional differential equations with proportional derivative

Aisha Abdullah Alderremy; Mahmoud Jafari Shah Belaghi; Khaled Mohammed Saad; Tofigh Allahviranloo; Ali Ahmadian; Shaban Aly; Soheil Salahshour; Aisha Abdullah Alderremy; Mahmoud Jafari Shah Belaghi; Khaled Mohammed Saad; Tofigh Allahviranloo; Ali Ahmadian; Shaban Aly; Soheil Salahshour

doi:10.3934/math.2021338

AIMS Mathematics

2021, Volume 6, Issue 6: 5737-5749. doi: 10.3934/math.2021338

Previous Article Next Article

Research article Special Issues

Analytical solutions of $ q $-fractional differential equations with proportional derivative

1.
Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Kingdom of Saudi Arabia
2.
Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey
3.
Department of Mathematics, College of Arts and Sciences, Najran University, Najran, Kingdom of Saudi Arabia
4.
Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen
5.
Institute of IR 4.0, The National University of Malaysia, 43600 UKM, Bangi, Malaysia
6.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71511, Egypt

Received: 06 November 2020 Accepted: 18 January 2021 Published: 26 March 2021
MSC : 26A33, 39A13

In this paper, we aim to propose a novel $ q $-fractional derivative in the Caputo sense included proportional derivative. To this end, we firstly introduced a new concept of proportional $ q $-derivative and discussed its properties in detail. Then, we add this definition in the concept of Caputo derivative to state a new type of dynamical system with $ q $-calculus. For analytically solving this system, $ q $-Laplace transform has been successfully applied to obtain the solutions. Indeed, the bivariate Mittag-Leffler function has an essential role in this regard. Two illustrative examples are also given in detail.
- $ q $-calculus,
- $ q $-proportional,
- proportional Caputo $ q $-derivative,
- $ q $-Laplace transform
Citation: Aisha Abdullah Alderremy, Mahmoud Jafari Shah Belaghi, Khaled Mohammed Saad, Tofigh Allahviranloo, Ali Ahmadian, Shaban Aly, Soheil Salahshour. Analytical solutions of $ q $-fractional differential equations with proportional derivative[J]. AIMS Mathematics, 2021, 6(6): 5737-5749. doi: 10.3934/math.2021338

Related Papers:

Abstract

In this paper, we aim to propose a novel $ q $-fractional derivative in the Caputo sense included proportional derivative. To this end, we firstly introduced a new concept of proportional $ q $-derivative and discussed its properties in detail. Then, we add this definition in the concept of Caputo derivative to state a new type of dynamical system with $ q $-calculus. For analytically solving this system, $ q $-Laplace transform has been successfully applied to obtain the solutions. Indeed, the bivariate Mittag-Leffler function has an essential role in this regard. Two illustrative examples are also given in detail.

References

[1]	F. H. Jackson, q-Difference equations, Am. J. Math., 32 (1910), 305–314. doi: 10.2307/2370183
[2]	R. D. Carmichael, The general theory of linear q-difference equations, Am. J. Math., 34 (1912), 147–168. doi: 10.2307/2369887
[3]	T. E. Mason, On properties of the solutions of linear q-difference equations with entire function coefficients, Am. J. Math., 37 (1915), 439–444. doi: 10.2307/2370216
[4]	C. R. Adams, On the linear ordinary q-difference equation, Ann. Math., 30 (1928), 195–205. doi: 10.2307/1968274
[5]	W. J. Trjitzinsky, Analytic theory of linear q-difference equations, Acta Math., 61 (1933), 1–38. doi: 10.1007/BF02547785
[6]	T. Ernst, A new notation for q-calculus and a new q-Taylor formula, Uppsala University, Department of Mathematics, 1999.
[7]	R. Floreanini, L. Vinet, q-Gamma and q-beta functions in quantum algebra representation theory, J. Comput. Appl. Math., 68 (1996), 57–68. doi: 10.1016/0377-0427(95)00253-7
[8]	V. Kac, P. Cheung, Quantum calculus, Springer Science & Business Media, 2001.
[9]	R. Finkelstein, E. Marcus, Transformation theory of the q-oscillator, J. Math. Phys., 36 (1995), 2652–2672. doi: 10.1063/1.531057
[10]	R. Finkelstein, The q-Coulomb problem, J. Math. Phys., 37 (1996), 2628–2636. doi: 10.1063/1.531532
[11]	R. Floreanini, L. Vinet, Automorphisms of the q-oscillator algebra and basic orthogonal polynomials, Phys. Lett. A, 180 (1993), 393–401. doi: 10.1016/0375-9601(93)90289-C
[12]	R. Floreanini, L. Vinet, Symmetries of the q-difference heat equation, Lett. Math. Phys., 32 (1994), 37–44. doi: 10.1007/BF00761122
[13]	R. Floreanini, L. Vinet, Quantum symmetries of q-difference equations, J. Math. Phys., 36 (1995), 3134–3156. doi: 10.1063/1.531017
[14]	P. G. Freund, A. V. Zabrodin, The spectral problem for theq-Knizhnik-Zamolodchikov equation and continuousq-Jacobi polynomials, Commun. Math. Phys., 173 (1995), 17–42. doi: 10.1007/BF02100180
[15]	M. Marin, On existence and uniqueness in thermoelasticity of micropolar bodies, Comptes Rendus de L Academie des Sciences Serie Ii Fascicule B-mecanique Physique Astronomie, 321 (1995), 475–480.
[16]	M. Marin, C. Marinescu, Thermoelasticity of initially stressed bodies, asymptotic equipartition of energies, Int. J. Eng. Sci., 36 (1998), 73–86. doi: 10.1016/S0020-7225(97)00019-0
[17]	G. Han, J. Zeng, On a q-sequence that generalizes the median Genocchi numbers, Ann. Sci. Math. Québec, 23 (1999), 63–72.
[18]	M. Marin, Lagrange identity method for microstretch thermoelastic materials, J. Math. Anal. Appl., 363 (2010), 275–286. doi: 10.1016/j.jmaa.2009.08.045
[19]	P. M. Rajković, S. D. Marinković, M. S. Stanković, Fractional integrals and derivatives in q-calculus, Appl. Anal. Discr. Math., 1 (2007), 311–323. doi: 10.2298/AADM0701072C
[20]	Z. S. Mansour, Linear sequential q-difference equations of fractional order, Fract. Calc. Appl. Anal., 12 (2009), 159–178.
[21]	Y. Zhao, H. Chen, Q. Zhang, Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions, Adv. Differ. Equ., 2013 (2013), 48–63. doi: 10.1186/1687-1847-2013-48
[22]	T. Abdeljawad, B. Benli, D. Baleanu, A generalized q-Mittag-Leffler function by q-Captuo fractional linear equations, Abstr. Appl. Anal., 2012 (2012), 1–11.
[23]	I. Koca, A method for solving differential equations of q-fractional order, Appl. Math. Comput., 266 (2015), 1–5. doi: 10.1016/j.amc.2015.05.049
[24]	T. Zhang, Y. Tang, A difference method for solving the q-fractional differential equations, Appl. Math. Lett., 98 (2019), 292–299. doi: 10.1016/j.aml.2019.06.020
[25]	Y. Tang, T. Zhang, A remark on the q-fractional order differential equations, Appl. Math. Comput., 350 (2019), 198–208. doi: 10.1016/j.amc.2019.01.008
[26]	Z. Noeiaghdam, T. Allahviranloo, J. J. Nieto, q-Fractional differential equations with uncertainty, Soft Comput., 23 (2019), 9507–9524. doi: 10.1007/s00500-019-03830-w
[27]	T. Zhang, Q. Guo, The solution theory of the nonlinear q-fractional differential equations, Appl. Math. Lett., 104 (2020), 106282. doi: 10.1016/j.aml.2020.106282
[28]	P. Lyu, S. Vong, An efficient numerical method for q-fractional differential equations, Appl. Math. Lett., 103 (2020), 106156. doi: 10.1016/j.aml.2019.106156
[29]	H. Vu, N. Van Hoa, Uncertain fractional differential equations on a time scale under Granular differentiability concept, Comput. Appl. Math., 38 (2019), 110. doi: 10.1007/s40314-019-0873-x
[30]	M. H. Annaby, Z. S. Mansour, q-Fractional calculus and equations, Springer-Verlag Berlin Heidelberg, 2012.
[31]	F. M. Atici, P. W. Eloe, Fractional q-calculus on a time scale, J. Nonlinear Math. Phy., 14 (2007), 341–352. doi: 10.2991/jnmp.2007.14.3.4
[32]	T. Abdeljawad, D. Baleanu, Caputo q-fractional initial value problems and a q-analogue Mittag–Leffler function, Commun. Nonlinear Sci., 16 (2011), 4682–4688. doi: 10.1016/j.cnsns.2011.01.026
[33]	G. Gasper, M. Rahman, Basic hypergeometric series, Cambridge University Press, 2004.
[34]	H. Exton, q-Hypergeometric functions and applications, Halsted Press, 1983.
[35]	T. Ernst, A comprehensive treatment of q-calculus, Birkhäuser Basel, 2012.
[36]	B. Abdalla, T. Abdeljawad, J. J. Nieto, A monotonicity result for the q-fractional operator, 2016, arXiv: 1602.07713.
[37]	H. W. Gould, The q-series generalization of a formula of Sparre Andersen, Math. Scand., 9 (1961), 90–94. doi: 10.7146/math.scand.a-10626
[38]	F. H. Jackson, A basic-sine and cosine with symbolical solutions of certain differential equations, P. Edinburgh Math. Soc., 22 (1903), 28–39. doi: 10.1017/S0013091500001930
[39]	W. Hahn, Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr., 2 (1949), 340–379. doi: 10.1002/mana.19490020604
[40]	F. H. Jackson, On basic double hypergeometric functions, Quart. J. Math., 1 (1942), 69–82.
[41]	M. J. S. Belaghi, Some properties of the q-exponential functions, J. Comput. Anal. Appl., 29 (2021), 737–741.
[42]	W. A. Al-Salam, q-Analogues of Cauchy's Formulas, P. Am. Math. Soc., 17 (1966), 616–621.
[43]	W. A. Al-Salam, Some fractional q-integrals and q-derivatives, P. Am. Math. Soc., 15 (1966), 135–140.
[44]	R. P. Agarwal, Certain fractional q-integrals and q-derivatives, Math. Proc. Cambridge, 66 (1969), 365–370. doi: 10.1017/S0305004100045060
[45]	D. R. Anderson, D. J. Ulness, Newly defined conformable derivatives, Advances in Dynamical Systems and Applications, 10 (2015), 109–137.

Reader Comments

Your name:*

Email:*
© 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)