Fractional shifted Morgan-Voyce neural networks for solving fractal-fractional pantograph differential equations

We provide an effective numerical strategy for fractal-fractional pantograph differential equations (FFPDEs). The fractal-fractional derivative is considered in the Atangana-Riemann-Liouville sense. The scheme is based on fractional shifted Morgan-Voyce neural network (FShM-VNN). We introduce a new class of functions called fractional-order shifted Morgan-Voyce and some useful properties of these functions for the first time. The FShM-VNN method is utilized the fractional-order shifted Morgan-Voyce functions (FShM-VFs) and Sinh function as activation functions of the hidden layer and output layer of the neural network (NN), respectively. The approximate function contains the FShM-VFs with unknown weights. Using the classical optimization method and Newton's iterative scheme, the weights are adjusted such that the approximate function satisfies the under study problem. Convergence analysis of the mentioned strategy is discussed. The scheme yields very accurate outcomes. The obtained numerical examples support this assertion.