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Piecewise Picard iteration method for solving nonlinear fractional differential equation with proportional delays

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  • Chen, Zhong
  • Gou, QianQian

Abstract

In this paper, a numerical method for solving a class of nonlinear fractional differential equation with proportional delays is proposed. In order to overcome the strongly nonlinear case, we propose the piecewise Picard iteration method(PPIM). The convergence proof and error estimations of the Picard and the PPIM are obtained. Meanwhile, a sufficient condition for the stability of the PPIM is also given. Some numerical examples confirm the validity of the PPIM. It’s worth noting that the PPIM is quite effective for solving linear, weakly nonlinear and some strongly nonlinear fractional differential equations with proportional delays.

Suggested Citation

  • Chen, Zhong & Gou, QianQian, 2019. "Piecewise Picard iteration method for solving nonlinear fractional differential equation with proportional delays," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 465-478.
    • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:465-478
    DOI: 10.1016/j.amc.2018.10.058
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    References listed on IDEAS

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    1. Brajesh Kumar Singh & Pramod Kumar, 2017. "Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay," International Journal of Differential Equations, Hindawi, vol. 2017, pages 1-11, March.
    2. Zhen Wang, 2013. "A Numerical Method for Delayed Fractional-Order Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, May.
    3. Saeed, Umer & Rehman, Mujeeb ur & Iqbal, Muhammad Asad, 2015. "Modified Chebyshev wavelet methods for fractional delay-type equations," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 431-442.
    4. Ghasemi, M. & Fardi, M. & Khoshsiar Ghaziani, R., 2015. "Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 815-831.
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    Cited by:

    1. Peiyao Wang & Shangwen Peng & Yihao Cao & Rongpei Zhang, 2024. "The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform," Mathematics, MDPI, vol. 12(7), pages 1-14, April.

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