A feedforward neural network based on Legendre polynomial for solving linear Fredholm integro-differential equations

Abstract

This paper presents a method for solving linear Fredholm integro-differential equations using a feedforward neural network based on Legendre polynomials. Firstly, the Legendre polynomials are used to approximate the unknown function and the kernel function in the equations. Secondly, The roots of Legendre polynomials are obtained by using Newton iteration method to obtain Gaussian integration points, and the obtained Gaussian integration points are used as the input nodes of the neural network, and the corresponding weight is learned by the gradient descent method to obtain an approximate solution. Finally, through the numerical value Case analysis to verify the effectiveness of the method.