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Solving differential equations of fractional order using an optimization technique based on training artificial neural network

Author

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  • Pakdaman, M.
  • Ahmadian, A.
  • Effati, S.
  • Salahshour, S.
  • Baleanu, D.

Abstract

The current study aims to approximate the solution of fractional differential equations (FDEs) by using the fundamental properties of artificial neural networks (ANNs) for function approximation. In the first step, we derive an approximate solution of fractional differential equation (FDE) by using ANNs. In the second step, an optimization approach is exploited to adjust the weights of ANNs such that the approximated solution satisfies the FDE. Different types of FDEs including linear and nonlinear terms are solved to illustrate the ability of the method. In addition, the present scheme is compared with the analytical solution and a number of existing numerical techniques to show the efficiency of ANNs with high accuracy, fast convergence and low use of memory for solving the FDEs.

Suggested Citation

  • Pakdaman, M. & Ahmadian, A. & Effati, S. & Salahshour, S. & Baleanu, D., 2017. "Solving differential equations of fractional order using an optimization technique based on training artificial neural network," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 81-95.
    • Handle: RePEc:eee:apmaco:v:293:y:2017:i:c:p:81-95
    DOI: 10.1016/j.amc.2016.07.021
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    Citations

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    Cited by:

    1. 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.
    2. Biswas, Chetna & Singh, Anup & Chopra, Manish & Das, Subir, 2023. "Study of fractional-order reaction-advection-diffusion equation using neural network method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 15-27.
    3. Navickas, Z. & Telksnys, T. & Marcinkevicius, R. & Ragulskis, M., 2017. "Operator-based approach for the construction of analytical soliton solutions to nonlinear fractional-order differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 625-634.
    4. Sabir, Zulqurnain & Raja, Muhammad Asif Zahoor & Khalique, Chaudry Masood & Unlu, Canan, 2021. "Neuro-evolution computing for nonlinear multi-singular system of third order Emden–Fowler equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 799-812.
    5. Jafarian, Ahmad & Measoomy Nia, Safa & Khalili Golmankhaneh, Alireza & Baleanu, Dumitru, 2018. "On artificial neural networks approach with new cost functions," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 546-555.
    6. Dimitrios Kartsonakis Mademlis & Nikolaos Dritsakis, 2021. "Volatility Forecasting using Hybrid GARCH Neural Network Models: The Case of the Italian Stock Market," International Journal of Economics and Financial Issues, Econjournals, vol. 11(1), pages 49-60.
    7. Muhammad Imran Asjad & Saif Ur Rehman & Ali Ahmadian & Soheil Salahshour & Mehdi Salimi, 2021. "First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface," Mathematics, MDPI, vol. 9(12), pages 1-18, June.
    8. Thanon Korkiatsakul & Sanoe Koonprasert & Khomsan Neamprem, 2019. "New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions," Mathematics, MDPI, vol. 7(9), pages 1-20, September.
    9. Qu, Haidong & She, Zihang & Liu, Xuan, 2021. "Neural network method for solving fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    10. Li, Qiaoping & Liu, Sanyang & Chen, Yonggang, 2018. "Combination event-triggered adaptive networked synchronization communication for nonlinear uncertain fractional-order chaotic systems," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 521-535.
    11. 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).
    12. Zarepour, Mazyar & Loghmani, Ghasem Barid, 2021. "Numerical solution of arbitrary-order linear partial differential equations using an optimal control technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 77-96.

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