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Fractional Chebyshev deep neural network (FCDNN) for solving differential models

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  • Hajimohammadi, Zeinab
  • Baharifard, Fatemeh
  • Ghodsi, Ali
  • Parand, Kourosh

Abstract

Differential and integral equations have been used vastly in modeling engineering and science problems. Solving these equations has been always an active and important area of research. In this paper, we propose the Fractional Chebyshev Deep Neural Network (FCDNN) for solving fractional differential models. Chebyshev orthogonal polynomials are basic functions in spectral methods. These functions are used as activation functions in FCDNN. The marching in time technique and the Gaussian method are applied in the fractional operations to simplify the calculations. We show how FCDNN can be used to solve fractional Fredholm integral equations (FFIEs). We also propose a solution to the extension of fractional time order partial differential equations (FPDEs). In this approach, fractional PDEs are first discretized by the finite difference and the marching in time methods and then are solved using FCDNN. Fractional Fredholm integral equations are also first approximated by the numerically Gaussian quadrature method and then are solved using FCDNN. A comparison between the results from FCDNN and some other methods is presented to validate the effectiveness and advance of the proposed method.

Suggested Citation

  • Hajimohammadi, Zeinab & Baharifard, Fatemeh & Ghodsi, Ali & Parand, Kourosh, 2021. "Fractional Chebyshev deep neural network (FCDNN) for solving differential models," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    • Handle: RePEc:eee:chsofr:v:153:y:2021:i:p2:s0960077921008845
    DOI: 10.1016/j.chaos.2021.111530
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    References listed on IDEAS

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    1. Hajimohammadi, Zeinab & Parand, Kourosh, 2021. "Numerical learning approximation of time-fractional sub diffusion model on a semi-infinite domain," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    2. Pei Chen & Rui Liu & Kazuyuki Aihara & Luonan Chen, 2020. "Autoreservoir computing for multistep ahead prediction based on the spatiotemporal information transformation," Nature Communications, Nature, vol. 11(1), pages 1-15, December.
    3. Kumar, Kamlesh & Pandey, Rajesh K. & Yadav, Swati, 2019. "Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    4. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    5. Sousa, J. Vanterler da C. & Oliveira, D.S. & Capelas de Oliveira, E., 2021. "A note on the mild solutions of Hilfer impulsive fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    6. Wubshet Ibrahim & Lelisa Kebena Bijiga, 2021. "Neural Network Method for Solving Time-Fractional Telegraph Equation," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-10, May.
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    Cited by:

    1. Hou, Jie & Ma, Zhiying & Ying, Shihui & Li, Ying, 2024. "HNS: An efficient hermite neural solver for solving time-fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    2. Wang, Chen & Zhang, Hai & Ye, Renyu & Zhang, Weiwei & Zhang, Hongmei, 2023. "Finite time passivity analysis for Caputo fractional BAM reaction–diffusion delayed neural networks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 424-443.

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