Eur. Phys. J. Plus (2018) 133: 75
https://doi.org/10.1140/epjp/i2018-11917-0

Regular Article

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

¹ Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico
² CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico
³ Univ. Grenoble Alpes, F-38000, Grenoble, France
⁴ CEA LETI MINATEC Campus, F-38054, Grenoble, France

^* e-mail: jgomez@cenidet.edu.mx

Received: 4 January 2018
Accepted: 29 January 2018
Published online: 26 February 2018

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.