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

Abstract

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.