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Study of fractional-order reaction-advection-diffusion equation using neural network method

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  • Biswas, Chetna
  • Singh, Anup
  • Chopra, Manish
  • Das, Subir

Abstract

In the present article the spatio-temporal fractional-order nonlinear reaction-advection-diffusion equation is solved using the neural network method (NNM). Shifted Legendre orthogonal polynomials with variable coefficients are used in the network’s construction. The characteristics of a fractional-order derivative are used to determine the loss function of a neural network. The permissible learning rate range is discussed in detail, assuming that the Lipschitz hypothesis is accurate for the nonlinearity in reaction term. We have demonstrated the application of the NNM on two numerical examples by utilizing the neural networks which had been repeatedly trained on the training set. In other words, we have validated the effectiveness of the method for such problems. The effects of reaction term and also the degree of nonlinearity in reaction and advection terms on the solution profile are visualized through graphical presentations for specific test cases.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:208:y:2023:i:c:p:15-27
    DOI: 10.1016/j.matcom.2022.12.032
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    References listed on IDEAS

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    1. Qu, Hai-Dong & Liu, Xuan & Lu, Xin & ur Rahman, Mati & She, Zi-Hang, 2022. "Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    2. 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.
    3. Saffarian, Marziyeh & Mohebbi, Akbar, 2022. "Finite difference/spectral element method for one and two-dimensional Riesz space fractional advection–dispersion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 348-370.
    4. Singh, Anup & Das, Subir & Ong, S.H., 2022. "Study and analysis of nonlinear (2+1)-dimensional solute transport equation in porous media," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 491-500.
    5. Tinoco-Guerrero, G. & Domínguez-Mota, F.J. & Tinoco-Ruiz, J.G., 2020. "A study of the stability for a generalized finite-difference scheme applied to the advection–diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 176(C), pages 301-311.
    6. Das, S., 2009. "A note on fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2074-2079.
    7. 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.
    8. Raja, Muhammad Asif Zahoor & Samar, Raza & Manzar, Muhammad Anwar & Shah, Syed Muslim, 2017. "Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley–Torvik equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 139-158.
    9. Hafez, Ramy M. & Zaky, Mahmoud A. & Hendy, Ahmed S., 2021. "A novel spectral Galerkin/Petrov–Galerkin algorithm for the multi-dimensional space–time fractional advection–diffusion–reaction equations with nonsmooth solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 678-690.
    10. Tber, Moulay Hicham, 2023. "A semi-Lagrangian mixed finite element method for advection–diffusion variational inequalities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 202-215.
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