Artificial Neural Network Method for Solving a Fractional Order Differential Equation with the Cauchy-Type Problem

Abstract

Differential equations with Cauchy type initial conditions and boundary conditions play a very important role in mathematics, physics and other sciences. In this article, we have developed an artificial neural network (ANN) method for finding solutions to the Cauchy problem for fractional order differential equations (FODEs). The fractional derivative is considered to be of the Riemann-Liouville type. Here, we used a three-layer feedforward neural architecture (one input layer, one hidden layer and one output layer), L-BFGS (Broyden-Fletcher-Goldfarb-Shanno) optimization method to minimize the error function and change the parameters (weights and biases). Illustrative examples demonstrating the accuracy and efficiency of this method are given. Compare the results of the current method with the mathematical results.