Variable order R-L fractional calculus and its Applications
Keywords:
Fractional Calculus, Gamma Function, Mittag - Leffler FunctionAbstract
This paper presents a concise study of variable-order fractional calculus using the Riemann-Liouville approach. Specifically, we consider the Mittag-Leffler function with a single parameter as the order for both Riemann-Liouville fractional differentiation (FD) and fractional integration (FI). The study explores the impact of varying the parameter in the Mittag-Leffler (M-L) function and applies this variable-order fractional operator to polynomial functions of different degrees. For clarity and completeness, the behavior of the Mittag-Leffler-based Riemann-Liouville fractional calculus is examined both theoretically and graphically.
References
Francesco Mainardi.: Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific (2022)
Mainardi, F.: A tutorial on the basic special functions of Fractional Calculus. WSEAS Trans.Math., 19, 74–98 (2020)
Mainardi, F.; Consiglio, A.: The Wright functions of the second kind in Mathematical Physics. Mathematics, 8, 884 (2020)
AP Bhadane and KC Takale.: Basic developments of fractional calculus and its applications. Bulletin of Marathwada Mathematical Society, 12(2):1–17 (2011)
Coronel-Escamilla A, G´omez-Aguilar J, Torres L, Escobar-Jim´enez R, Valtierra-Rodr´ıguez M.: Synchronization of chaotic systems involving fractional operators of Liouville-Caputo type with variable-order. Physica A 487, 1–21 (2017). doi:10.1016/j.physa.2017.06.008
Sansit Patnaik, John P Hollkamp, and Fabio Semperlotti.: Applications of variable-order fractional operators: a review. Proceedings of the Royal Society A, 476(2234):20190498 (2020)
Ortigueira MD, Val´erio D, Machado JT.: Variable order fractional systems. Commun.Nonlinear Sci. Numer. Simul. 71, 231–243 (2019). doi:10.1016/j.cnsns.2018.12.003
Malesza W, Macias M, Sierociuk D.: Analytical solution of fractional variable order differential equations. J. Comput. Appl. Math. 348, 214–236 (2019). doi:10.1016/j.cam.2018.08.035
Yang J, Yao H, Wu B.: An efficient numerical method for variable order fractional functional differential equation. Appl. Math. Lett. 76, 221–226 (2018). doi:10.1016/j.aml.2017.08.020
Zahra W, Hikal M.: Non standard finite difference method for solving variable order fractional optimal control problems. J. Vib. Control 23, 948–958 (2017). doi:10.1177/1077546315586646
Moghaddam BP, Machado, J.A.T.: Extended algorithms for approximating variable order fractional derivatives with applications. J. Sci. Comput. 71, 1351–1374 (2017). doi:10.1007/s10915-016-0343-1
Sakrajda P, Sierociuk D.: Modeling heat transfer process in grid-holes structure changed in time using fractional variable order calculus. In Theory and applications of non-integer order systems (eds A Babiarz, A Czornik, J Klamka, M Niezabitowski), pp. 297–306, Cham, Switzerland: Springer International Publishing (2017)
Sahoo S, Saha Ray S, Das S, Bera RK.: The formation of dynamic variable order fractional differential equation. Int. J. Mod. Phys. C 27, 1650074 (2016). doi:10.1142/S0129183116500741
Sierociuk D, Malesza W, Macias M.: Derivation, interpretation, and analog modellingof fractional variable order derivative definition. Appl. Math. Model. 39, 3876–3888 (2015). doi:10.1016/j.apm.2014.12.009
Chen Y, Liu L, Li B, Sun Y.: Numerical solution for the variable order linear cable equation with Bernstein polynomials. Appl. Math. Comput. 238, 329–341 (2014). doi:10.1016/j.amc.2014.03.066
Rapai´c MR, Pisano A.: Variable-order fractional operators for adaptive order and parameter estimation. IEEE Trans. Autom. Control 59, 798–803 (2013). doi:10.1109/TAC.2013.2278136
Gorenflo, R.; Kilbas, A.A.; Mainardi, F.; Rogosin, S.: Mittag-Leffler Functions. Related Topics and Applications, 2nd ed.; Springer: Berlin, Germany (2020)
Mainardi, F.; Consiglio, A.: The pioneers of the Mittag-Leffler functions in dielectrical and mechanical relaxation processes. WSEAS Trans. Math. 19, 289–300 (2020)
Fernandez, A.; Baleanu, D.; Srivastava, H.M.: Series representations for models of fractional calculus involving generalised Mittag-Leffler functions. Commun. Nonlinear Sci. Numer. Simul. 67, 517–527 (2019)
Baleanu, D.; Fernandez, A.: On some new properties of fractional derivatives with Mittag-Leffler kernel. Commun. Nonlinear Sci. Numer. Simul. 59, 444–462 (2019)
Garra, R.; Garrappa, R.: The Prabhakar or three parameter Mittag-Leffler function: Theory and application. Commun. Nonlinear Sci. Numer. Simul. 56, 314–329 (2018)
KC Takale, VR Nikam, and AS Shinde.: Mittag leffler functions, its computations and application to differential equation of fractional order. In International Conference on Mathematical Modelling and Applied Soft Computing,(1), pages 561–575 (2012)
Hans J Haubold, Arak M Mathai, Ram K Saxena, et al.: Mittag-leffler functions and their applications. Journal of applied mathematics (2011)
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Copyright (c) 1970 Sayali Nikam, Dr. S. D. Manjarekar
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