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An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations

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  • Postavaru, Octavian

Abstract

The Fibonacci sequence is significant because of the so-called golden ratio, which describes predictable patterns for everything. Fibonacci polynomials are related to Fibonacci numbers, and in this paper we extend their applicability by using them to solve fractional differential equations (FDEs) and systems of fractional differential equations (SFDEs). With the help of the Riemann–Liouville fractional integral operator for the fractional-order hybrid function of block-pulse functions and the Fibonacci polynomials defined in this paper, the solution of the considered FDE and SFDE is reduced to a system of algebraic equations, which can be solved by Newton’s iterative method. The fractional order is obtained by transforming x into xα, with α>0. Compared to other models, our method in some situations is better by twelve orders of magnitude. There are situations when we get the exact solution. The presented method proves to be simple and effective in solving nonlinear problems with given initial values.

Suggested Citation

  • Postavaru, Octavian, 2023. "An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 406-422.
    • Handle: RePEc:eee:matcom:v:212:y:2023:i:c:p:406-422
    DOI: 10.1016/j.matcom.2023.04.028
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    References listed on IDEAS

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    1. Dehestani, H. & Ordokhani, Y. & Razzaghi, M., 2018. "Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 433-453.
    2. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    3. Postavaru, Octavian & Toma, Antonela, 2022. "A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 269-284.
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