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New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks

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Abstract.

In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.

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Authors and Affiliations

  1. Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico

    C. J. Zúñiga-Aguilar, A. Coronel-Escamilla & V. M. Alvarado-Martínez

  2. CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico

    J. F. Gómez-Aguilar

  3. Univ. Grenoble Alpes, F-38000, Grenoble, France

    H. M. Romero-Ugalde

  4. CEA LETI MINATEC Campus, F-38054, Grenoble, France

    H. M. Romero-Ugalde

Authors
  1. C. J. Zúñiga-Aguilar
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  2. A. Coronel-Escamilla
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  3. J. F. Gómez-Aguilar
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  4. V. M. Alvarado-Martínez
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Correspondence to J. F. Gómez-Aguilar.

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Zúñiga-Aguilar, C.J., Coronel-Escamilla, A., Gómez-Aguilar, J.F. et al. New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks. Eur. Phys. J. Plus 133, 75 (2018). https://doi.org/10.1140/epjp/i2018-11917-0

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