Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks

In this paper, we approximate the solution of fractional differential equations using a new approach of artificial neural network. We consider fractional differential equations of variable-order with Mittag-Leffler kernel in Liouville-Caputo sense. With this new neural network approach, it is obtained an approximate solution of the fractional differential equation and this solution is optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional differential equations, the Willamowski-Rössler oscillator and a multi-scroll system. 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.