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Approximate solutions of fractional Riccati equations using the Adomian decomposition method. (English) Zbl 1474.65264

Summary: The fractional derivative equation has extensively appeared in various applied nonlinear problems and methods for finding the model become a popular topic. Very recently, a novel way was proposed by J.-S. Duan [Appl. Math. Comput. 216, No. 4, 1235–1241 (2010; Zbl 1190.65031)] to calculate the Adomian series which is a crucial step of the Adomian decomposition method. In this paper, it was used to solve some fractional nonlinear differential equations.

MSC:

65L99 Numerical methods for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations

Citations:

Zbl 1190.65031
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References:

[1] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports A: Review Section of Physics Letters, 339, 1, 1-77 (2000) · Zbl 0984.82032
[2] Sun, H.; Chen, W.; Chen, Y., Variable-order fractional differential operators in anomalous diffusion modeling, Physica A: Statistical Mechanics and its Applications, 388, 21, 4586-4592 (2009)
[3] Bagley, R. L.; Torvik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27, 3, 201-210 (1983) · Zbl 0515.76012
[4] Bagley, R. L.; Torvik, P. J., Fractional calculus-a different approach to the analysis of viscoelastically damped structures, AIAA Journal, 21, 5, 741-748 (1983) · Zbl 0514.73048
[5] Wu, G.-C.; Baleanu, D., Discrete fractional logistic map and its chaos, Nonlinear Dynamics, 75, 1-2, 283-287 (2014) · Zbl 1281.34121
[6] Wu, G.-C.; Baleanu, D., Chaos synchronization of the discrete fractional logistic map, Signal Processing, 102, 96-99 (2014)
[7] Baleanu, D.; Wu, G.-C.; Duan, J.-S.; Tenreiro Machado, D. B. J. A.; Luo, A. C. J., Some analytical techniques in fractional calculus: realities and challenges, Discontinuity and Complexity in Nonlinear Physical Systems, 35-62 (2014), New York, NY, USA: Springer, New York, NY, USA · Zbl 1315.26004
[8] Duan, J.-S., Recurrence triangle for Adomian polynomials, Applied Mathematics and Computation, 216, 4, 1235-1241 (2010) · Zbl 1190.65031
[9] Duan, J.-S., Convenient analytic recurrence algorithms for the Adomian polynomials, Applied Mathematics and Computation, 217, 13, 6337-6348 (2011) · Zbl 1214.65064
[10] Duan, J.-S.; Rach, R., New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods, Applied Mathematics and Computation, 218, 6, 2810-2828 (2011) · Zbl 1478.65071
[11] Duan, J.-S.; Chaolu, T.; Rach, R., Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method, Applied Mathematics and Computation, 218, 17, 8370-8392 (2012) · Zbl 1245.65087
[12] Duan, J. S.; Rach, R.; Baleanu, D.; Wazwaz, A.-M., A review of the Adomian decomposition method and its applications to fractional differential equations, Communications in Fractional Calculus, 3, 2, 73-99 (2012)
[13] Caputo, M., Linear models of dissipation whose Q is almost frequency independent—II, Geophysical Journal International, 13, 5, 529-539 (1967)
[14] Podlubny, I., Fractional Differential Equations (1999), Academic Press · Zbl 0918.34010
[15] Daftardar-Gejji, V.; Jafari, H., Adomian decomposition: a tool for solving a system of fractional differential equations, Journal of Mathematical Analysis and Applications, 301, 2, 508-518 (2005) · Zbl 1061.34003
[16] Ray, S. S.; Bera, R. K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Applied Mathematics and Computation, 167, 1, 561-571 (2005) · Zbl 1082.65562
[17] Sakar, M. G.; Erdogan, F., The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian’s decomposition method, Applied Mathematical Modelling, 37, 20-21, 8876-8885 (2013) · Zbl 1427.65323
[18] Daftardar-Gejji, V.; Jafari, H., Solving a multi-order fractional differential equation using Adomian decomposition, Applied Mathematics and Computation, 189, 1, 541-548 (2007) · Zbl 1122.65411
[19] Li, C.; Wang, Y., Numerical algorithm based on Adomian decomposition for fractional differential equations, Computers and Mathematics with Applications, 57, 10, 1672-1681 (2009) · Zbl 1186.65110
[20] El-Tawil, M. A.; Bahnasawi, A. A.; Abdel-Naby, A., Solving Riccati differential equation using Adomian’s decomposition method, Applied Mathematics and Computation, 157, 2, 503-514 (2004) · Zbl 1054.65071
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