An efficient numerical method to solve ordinary differential equations using Fibonacci neural networks

Abstract

The authors present a method to solve differential equations with any kind of initial and boundary conditions using the Fibonacci neural network (FNN). Fibonacci polynomial has been used as an activation function in the middle layer to construct the FNN. The trial solution of the differential equation is considered as the output of the feed-forward neural network, which consists of adjustable parameters (weights). The weights are adjusted with Newtons’ like method for equality constraints. The authors have also shown the stability and convergence of the weights with iteration through the graphs. The application of the current method is range from single ordinary differential equations (ODEs) to system of ODE’s. The authors have implemented the method to solve a variety of differential equations and established the exactness of the current method by comparison of the solution obtained by previously solved methods. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.