Abstract
In the work variable-, fractional-order backward difference of the Grünwald-Letnikov type is presented. The backward difference is used to generate simulated experimental data to which additional noise signal is added. Using prepared data four different algorithms of finding the parameter of the order function (assuming that the general family of the function is known) and constant \(\lambda \) coefficient are compared. The algorithms are: trust region algorithm, particle swarm algorithm, simulated annealing algorithm and genetic algorithm.
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Almeida, R., Bastos, N.R.O., Monteiro, M.T.T.: A fractional Malthusian growth model with variable order using an optimization approach. Published online in International Academic Press (2018). https://doi.org/10.19139/soic.v6i1.465
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Company, Singapore (2000). https://doi.org/10.1142/3779
Kaczorek, T.: Fractional positive linear systems. Kybernetes 38(7/8), 1059–1078 (2009). https://doi.org/10.1108/03684920910976826
May, R.: Simple mathematical models with very complicated dynamics. Nature (1976). https://doi.org/10.1038/261459a0
Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1998)
Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of ICNN 1995 - International Conference on Neural Networks, Perth, Australia, vol. 4, pp. 1942–1948 (1995). https://doi.org/10.1109/ICNN.1995.488968
Mozyrska, D., Wyrwas, M.: The Z-transform method and delta type fractional difference operators. Discrete Dyn. Nat. Soc. 25 (2015). https://doi.org/10.1155/2015/852734
Mozyrska, D., Wyrwas, M.: Systems with fractional variable-order difference operator of convolution type and its stability. ELEKTRONIKA IR ELEKTROTECHNIKA (2018). https://doi.org/10.5755/j01.eie.24.5.21846
Mozyrska, D., Ostalczyk, P.: Generalized fractional-order discrete-time integrator. Complexity 2017, 1–11 (2017). Article ID 3452409. https://doi.org/10.1155/2017/3452409
Nikolaev, A.G., Jacobson, S.: Simulated annealing. In: Handbook of Metaheuristics, vol. 146, pp. 1–39 (2010). https://doi.org/10.1007/978-1-4419-1665-5_1
Podlubny, I.: Fractional Differential Equations. Mathematics in Sciences and Engineering, vol. 198. Academic Press, San Diego (1999)
Yuan, Y.: Nonlinear optimization: trust region algorithms. State Key Laboratory of Scientific and Engineering Computing, Academia Sinica, Beijing (1999)
Yuan, Y.: A review of trust region algorithms for optimization. State Key Laboratory of Scientific and Engineering Computing, Academia Sinica, Beijing (1999). 10.1.1.45.9964
Acknowledgment
The work was supported by Polish founds of National Science Center, granted on the basis of decision DEC-2016/23/B/ST7/03686.
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Oziablo, P. (2020). Numerical Simulations for Fitting Parameters of Linear and Logistic-Type Fractional-, Variable-Order Equations - Comparision of Methods. In: Malinowska, A., Mozyrska, D., Sajewski, Ł. (eds) Advances in Non-Integer Order Calculus and Its Applications. RRNR 2018. Lecture Notes in Electrical Engineering, vol 559. Springer, Cham. https://doi.org/10.1007/978-3-030-17344-9_6
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DOI: https://doi.org/10.1007/978-3-030-17344-9_6
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