An Extreme Learning Machine Technique for the Numerical Solution of Fractional Differential Equations

This research intends to create a novel approach for solving fractional differential equations (FDEs) of both linear and nonlinear types utilizing the fractional shifted Legendre neural network alongside the extreme machine learning (ML) algorithm. The neural network serves as a trial solution to formulate the loss function where it is used to train the neural network through an extreme learning machine (ELM) to obtain the solution. The new approach is applied to the linear differential equations and the unknown coefficient matrix is identified using the psuedoinverse of the activation function output matrix whereas the nonlinear equations are handled by employing nonlinear least square perturb algorithm. An algorithm is provided for solving the FDEs. Theoretical analysis such as convergence, stability, and error estimates are studied. The proposed method is tested on various problems and is compared with the other algorithms discussed in the literature.