An iterative algorithm for nonlinear fractional-order oscillators with modified Riemann-Liouville derivative

[1] A. Atangana and J.F. Gomez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu, Numer. Methods Partial Differential Eq. 36 (2017), no. 4, 1502–1523.

[2] A. Atangana and J.F. Gomez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus 133 (2018), 166–187.

[3] A. Big-Alabo, Periodic solutions of Duffing-type oscillators using continuous piecewise linearization method, Mech. Eng. Res. 8 (2018), no. 1, 41–52.

[4] A. Big-Alabo, Approximate periodic solution and qualitative analysis of non-natural oscillators based on the restoring force, Eng. Res. Exp. 2 (2020), no. 1, 015029. 14 pp.

[5] A. Big-Alabo, A simple cubication method for approximate solution nonlinear Hamiltonian oscillators, Int’l J. Mech. Eng. Educ. 48 (2020), no. 3, 241–254.

[6] A. Big-Alabo, Continuous piecewise linearization method for approximate periodic solution of the relativistic oscillator, Int’l J. Mech. Eng. Educ. 48 (2020), no. 2, 178–194.

[7] A. Big-Alabo, P. Harrison and M.P. Cartmell, Algorithm for the solution of elastoplastic half-space impact: force indentation linearisation method, Proc. IMechE, C: J. Mech. Eng. Sci. 229 (2015), no. 5, 850–858.

[8] A. Big-Alabo, C.O. Ogbodo and C.V. Ossia, Semi-analytical treatment of complex nonlinear oscillations arising in the slider-crank mechanism, World Sci. News 142 (2020), 1–24.

[9] A. Big-Alabo and C.V. Ossia, Periodic Solution of Nonlinear Conservative Systems, Progress in Relativity, 2020.

[10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), 73-–85.

[11] M. Di Paola and M. Zingales, Exact mechanical models of fractional hereditary materials, J. Rheol. 56 (2012), no. 5, 983–1004.

[12] R. Garrappa, Gr¨unwald–Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun. Nonlinear Sci. Numer. Simulat. 38 (2016), 178-–191.

[13] J.F. Gomez-Aguilar, H. Y´epez-Mart´ınez, C. Calder´on-Ram´on, I. Cruz-Ordu˜na, R.F. Escobar-Jim´enez and V.H. Olivares-Peregrino, modelling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel, Entropy 17 (2015), 6289–6303.

[14] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Meth. Appl. Mech. Eng. 167 (1998), 57—68.

[15] J.H. He, A new fractal derivation, Therm. Sci. 15 (2011), 145—147.

[16] J.H. He, Asymptotic methods for Solitary Solutions and Compactons, Abstr. Appl. Anal. 2012 (2012), Article ID: 916793, 1–130.

[17] J.H. He, A tutorial review on fractal spacetime and fractional calculus, Int. J. Theor. Phys. 53 (2014), 3698—3718.

[18] J.H. He, Fractal calculus and its geometrical explanation, Results Phys. 10 (2018), 272–276.

[19] G. Jumarie, Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-differentiable Functions Further Results, Comp. Math. Appl. 51 (2006), 1137—1376.

[20] U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comp. 218 (2011), no. 3, 860–865.

[21] U.N. Katugampola, A new approach to generalized fractional derivatives, Bul. Math. Anal. Appl. 6 (2014), no. 4, 1–15.

22] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.

[23] F.M.S. Lima, Simple but accurate periodic solutions for the nonlinear pendulum equation, Rev. Bras. Ens. F´ıs. 41 (2019), no. 1, e20180202-1-6.

[24] Y. Liu, Y.W. Du, H. Li and J.F. Wang, A two-grid finite element approximation for a nonlinear time-fractional Cable equation, Phys. Lett. A 85 (2016), 2535-–2548.

[25] Q.X. Liu, J.K. Liu and Y.M. Chen, An analytical criterion for jump phenomena in fractional Duffing oscillators, Chaos, Solits. Fractals 98 (2017), 216—219.

[26] Y. Luchko, A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis, Math. Model. Nat. Phenom. 11 (2017), no. 3, 1—17.

[27] P.M. Mathews and M. Lakshmanan, On a unique nonlinear oscillator, Q. Appl. Math. 32 (1974), 215—218.

[28] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Eur. Phys. J. Plus 365 (2007), 345-–350.

[29] V.F. Morales-Delgado, J.F. Gomez-Aguilar, K.M. Saad, M.A. Khan and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Phys. A 523 (2019), 48–65.

[30] Y. Wang and J.Y. An, Amplitude–frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion, J. Low Freq. Noise Vib. Active Control 38 (2019), no. 3-4, 1008–1012.

[31] Y. Wang, Y.F. Zhang and W.J. Rui, Shallow water waves in porous medium for coast protection, Therm. Sci. 21 (2017), 145-–151.

[32] X.J. Yang, General Fractional Derivatives - Theory, Methods and Applications, CRC Press, Taylor and Francis Group, Boca Raton, 2019.

[33] X.J. Yang and D. Baleanu, Fractal heat conduction problem solved by local fractional variational iteration method, Therm. Sci. 17 (2013), no. 2, 625–628.