Neural Network Method for Solving Fractional Differential Equations α with the Dirichlet Problem

In this paper, we have developed an artificial neural network (ANN) method for finding solutions to the Dirichlet problem for fractional order differential equations (FODEs) 0 <α<1 using the definition of a conformable fractional derivative. Here, we used a feedforward neural architecture, L-BFGS (Broyden – Fletcher – Goldfarb - Shanno) optimization method to minimize the error function and change the parameters (weights and biases). The main idea is that if the sum of the norms of the residuals of the equation on the domain of definition and the boundary conditions tends to zero when the unknown function y(x) is replaced by its neural network approximation N(x), then N(x) is an approximate solution of the differential equation. Some illustrative examples are given demonstrating the accuracy and efficiency of this method and comparing the results of the current method with mathematical results.