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

  • 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.

    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:

  • 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.



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    [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.
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