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A neural networks-based numerical method for the generalized Caputo-type fractional differential equations

Author

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  • S M, Sivalingam
  • Kumar, Pushpendra
  • Govindaraj, Venkatesan

Abstract

The paper presents a numerical technique based on neural networks for generalized Caputo-type fractional differential equations with and without delay. We employ the theory of functional connection-based approximation and the physics-informed neural network with extreme learning machine-based learning to solve the differential equation. The proposed method uses the L1 finite difference scheme and the Volterra integral equation scheme to create the loss function. The novelty of this work is the proposal of the neural network-based scheme coupling the idea of the theory of functional connections and a new loss function for the solution of generalized Caputo-type differential equations. The proposed approach is applied to single differential equations and the system of differential equations with single and multiple delays.

Suggested Citation

  • S M, Sivalingam & Kumar, Pushpendra & Govindaraj, Venkatesan, 2023. "A neural networks-based numerical method for the generalized Caputo-type fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 213(C), pages 302-323.
    • Handle: RePEc:eee:matcom:v:213:y:2023:i:c:p:302-323
    DOI: 10.1016/j.matcom.2023.06.012
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    Citations

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    Cited by:

    1. Liu, Shuang & Xu, Tianwei & Wang, Qingyun & Yang, Yan, 2024. "The impulsive synchronization of multiplex networks with mixed delays and dual uncertainties," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 141-163.
    2. Admon, Mohd Rashid & Senu, Norazak & Ahmadian, Ali & Majid, Zanariah Abdul & Salahshour, Soheil, 2024. "A new modern scheme for solving fractal–fractional differential equations based on deep feedforward neural network with multiple hidden layer," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 311-333.
    3. Sivalingam, S M & Kumar, Pushpendra & Trinh, Hieu & Govindaraj, V., 2024. "A novel L1-Predictor-Corrector method for the numerical solution of the generalized-Caputo type fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 462-480.

    More about this item

    Keywords

    Generalized Caputo derivative; Neural network; L1 scheme; Nonlinear least squares;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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