Some delta q−fractional linear dynamic equations and a generalized delta q−Mittag-Leffler function

Document Type : Research Paper


Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq.


In this paper, we introduce a generalized delta q−Mittag-Leffler function. Also, we solve some Caputo delta q−fractional dynamic equations and these solutions are expressed by means of the newly introduced delta q−Mittag-Leffler function. 


Main Subjects

  • [1] R. P. Agarwal, M. Chand, and S. Jain, Certain integrals involving generalized mittag-leffler functions, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 85(3) (2015), 359–371.
  • [2] R. P. Agarwal, Certain fractional q-integrals and q-derivatives, Mathematical Proceedings of the Cambridge Philosophical Society, 66(2) (1969), 365–370.
  • [3] W. A. Al-Salam and A. Verma, A Fractional Leibniz q-Formula, Pacific Journal of Mathematics, 60(2) (1975), 1–9.
  • [4] W. A. Al-Salam, Some Fractional q-Integrals and q-Derivatives. Proceedings of the Edinburgh Mathematical Society, 15(2) (1966), 135–140.
  • [5] W. A. Al-Salam, q-Analogues of Cauchy’s Formulas, Proceedings of the American Mathematical Society, 17(3) (1966), 616–621.
  • [6] R. Alchikh and S.A. Khuri, Numerical solution of a fractional differential equation arising in optics, Optik, 208 (2020), 163911.
  • [7] R. Almeida and N. Martins, Existence results for fractional q-difference equations of order with three-point boundary conditions, Communications in Nonlinear Science and Numerical Simulation, 19(6) (2014), 1675–1685.
  • [8] S. Arshad, D. Baleanu, and Y. Tang, Fractional differential equations with bio-medical applications, In Dumitru Bˇaleanu and Ant´onio Mendes Lopes, editors, Applications in Engineering, Life and Social Sciences, Part A, (2019), 1–20.
  • [9] M. Bohner and A. Peterson, Dynamic equations on time scales: An introduction with applications, Springer Science & Business Media, 2001.
  • [10] M. Bohner and A. C Peterson, Advances in dynamic equations on time scales, Springer Science & Business Media, 2002.
  • [11] M. H. Derakhshan, A numerical technique for solving variable order time fractional differential-integro equations, Communications in Mathematics, 32(1) (2023), 128–147.
  • [12] A. Din, F, M. Khan, Z, U. Khan, A, Yusuf, and T. Munir, The mathematical study of climate change model under nonlocal fractional derivative, Partial Differential Equations in Applied Mathematics, 5 (2022), 100204.
  • [13] T. Ernst, The history of q-calculus and a new method, Citeseer, 2000.
  • [14] V. S. Erturk, A. Ahmadkhanlu, P. Kumar, and V. Govindaraj, Some novel mathematical analysis on a corneal shape model by using caputo fractional derivative, Optik, 261 (2022), 169086.
  • [15] V. S. Erturk, A.K. Alomari, P. Kumar, and M. M. Arcila, Analytic solution for the strongly nonlinear multi-order fractional version of a BVP occurring in chemical reactor theory, Discrete Dynamics in Nature and Society, 2022 (2022), 1–9.
  • [16] V.S. Erturk, E. Godwe, D. Baleanu, P. Kumar, J. Asad, and A. Jajarmi, Novel fractional-order lagrangian to describe motion of beam on nanowire, Acta Physica Polonica A,140(3) (2021), 265–272.
  • [17] S. Hilger, Analysis on Measure Chains — A Unified Approach to Continuous and Discrete Calculus, Results in Mathematics, 18(1-2) (1990), 18–56. aug 1990.
  • [18] F. H. Jackson, XI.—On q-Functions and a certain Difference Operator, Transactions of the Royal Society of Edinburgh, 46(2) (1909), 253–281.
  • [19] A. F. A. Jalil and A. R. Khudair, Toward solving fractional differential equations via solving ordinary differential equations Computational and Applied Mathematics, 41(1) (2022), 37.
  • [20] A. S. Joujehi, M.H. Derakhshan, and H.R. Marasi, An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis Communications in Nonlinear Science and Numerical Simulation, 114 (2022), 106620.
  • [21] S. L. Khalaf and H. S. Flayyih. Analysis, predicting, and controlling the COVID-19 pandemic in iraq through SIR model, Results in Control and Optimization, 10 (2023), 100214.
  • [22] S. L. Khalaf and A. R. Khudair. Particular solution of linear sequential fractional differential equation with constant coefficients by inverse fractional differential operators Differential Equations and Dynamical Systems, 25(3) (2017), 373–383.
  • [23] S. L. Khalaf, M. S. Kadhim, and A. R. Khudair. Studying of COVID-19 fractional model: Stability analysis Partial Differential Equations in Applied Mathematics, 7 (2023), 100470.
  • [24] A. R. Khudair. Reliability of adomian decomposition method for high order nonlinear differential equations Applied Mathematical Sciences, 7 (2013), 2735–2743.
  • [25] A. R. Khudair. On solving non-homogeneous fractional differential equations of euler type, Computational and Applied Mathematics, 32(3) (2013), 577–584.
  • [26] A. A. Kilbas, Theory and applications of fractional differential equations, Elsevier, Amsterdam Boston, 2006.
  • [27] A. A Kilbas and M. Saigo, On solution of nonlinear abel–volterra integral equation, Journal of mathematical analysis and applications, 229(1) (1999), 41–60.
  • [28] A. A. Kilbasi and M. Saigo. On mittag-leffler type function, fractional calculas operators and solutions of integral equations, Integral Transforms and Special Functions, 4(4) (1996), 355–370.
  • [29] P. Kumar, V. Govindaraj, V. S. Erturk, and M. H. Abdellattif. A study on the dynamics of alkali–silica chemical reaction by using caputo fractional derivative, Pramana, 96(3) (2022), 128.
  • [30] Z. A. Lazima and S. L. Khalaf. Optimal control design of the in-vivo HIV fractional model, Iraqi Journal of Science, 63(9) (2022), 3877–3888.
  • [31] N. K. Mahdi and A. R. Khudair. Stability of nonlinear q−fractional dynamical systems on time scale, Partial Differential Equations in Applied Mathematics, 7 (2023), 100496.
  • [32] N. K. Mahdi and A. R. Khudair, The delta q-fractional gronwall inequality on time scale, Results in Control and Optimization, 12 (2023), 100247.
  • [33] Z. S. Mansour M. H. Annaby, q-Fractional Calculus and Equations, Springer-Verlag GmbH, 2012.
  • [34] H. R. Marasi and M. H. Derakhshan. Haar wavelet collocation method for variable order fractional integrodifferential equations with stability analysis, Computational and Applied Mathematics, 41(3) (2022), 106.
  • [35] H. Marasi and M. H. Derakhshan. A composite collocation method based on the fractional chelyshkov wavelets for distributed - order fractional mobile immobile advection-dispersion equation, Mathematical Modelling and Analysis, 27(4) (2022), 590–609.
  • [36] J. K. Mohammed and A. R. Khudair. Numerical solution of fractional integro-differential equations via fourthdegree hat functions, Iraqi Journal for Computer Science and Mathematics, 4(2) (2023), 10–30.
  • [37] J. K. Mohammed and A, R. Khudair. Solving volterra integral equations via fourth-degree hat functions, Partial Differential Equations in Applied Mathematics, 7 (2023), 100494.
  • [38] J. K. Mohammed and A. R. Khudair. A novel numerical method for solving optimal control problems using fourth-degree hat functions, Partial Differential Equations in Applied Mathematics, 7 (2023), 100507.
  • [39] J. K. Mohammed and A. R. Khudair. Integro-differential equations: Numerical solution by a new operational matrix based on fourth-order hat functions, Partial Differential Equations in Applied Mathematics, 8 (2023), 100529.
  • [40] J. K. Mohammed and A. R. Khudair. Solving nonlinear stochastic differential equations via fourth-degree hat functions, Results in Control and Optimization, 12 (2023), 100291.
  • [41] Z. M. Odibat. Analytic study on linear systems of fractional differential equations, Computers and Mathematics with Applications, 59(3) (2010), 1171–1183.
  • [42] J. Peng and K. Li. A note on property of the mittag-leffler function Journal of Mathematical Analysis and Applications, 370(2) (2010), 635–638.
  • [43] I. Podlubny. Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, 1999.
  • [44] R. Predrag, M. Sladjana, and S. Miomir. Fractional integrals and derivatives in q-calculus Applicable Analysis and Discrete Mathematics, 1(1) (2007), 311–323.
  • [45] P. Rajkovic, S. Marinkovic, and M. Stankovic. On q–Analogues of Caputo Derivative and Mittag–Leffler function, Fractional calculus and applied analysis, 10(4) (2007), 359–373.
  • [46] N. A. Rangaig and V. C. Convicto C. T. Pada. On the Existence of the Solution for q-Caputo Fractional Boundary Value Problem, Applied Mathematics and Physics, 5(3) (2017), 99–102.
  • [47] S.K Sharma and R Jain. On some properties of generalized q-mittag leffler function Mathematica Aeterna, 4(6) (2014), 613–619.
  • [48] A.K. Shukla and J.C. Prajapati. On a generalization of mittag-leffler function and its properties Journal of Mathematical Analysis and Applications, 336(2) (2007), 797–811.
  • [49] E Viera-Martin, JF G´omez-Aguilar, JE Sol´ıs-P´erez, JA Hern´andez-P´erez, and RF Escobar-Jim´enez. Artificial neural networks: a practical review of applications involving fractional calculus, The European Physical Journal Special Topics, 231(10) (2022), 2059–2095.
  • [50] Q. Yang, D. Chen, T. Zhao, and Y. Q. Chen, Fractional calculus in image processing: A review, Fractional Calculus and Applied Analysis, 19 (2016), 1222–1249.