Legendre Deep Neural Network (LDNN) and its application for approximation of nonlinear Volterra Fredholm Hammerstein integral equations

TL;DR

This paper introduces Legendre Deep Neural Networks (LDNN), a novel approach using Legendre polynomials as activation functions, to effectively solve nonlinear Volterra Fredholm Hammerstein integral equations with demonstrated accuracy.

Contribution

The paper presents a new neural network architecture, LDNN, that employs Legendre polynomials for solving complex nonlinear integral equations, combining it with Gaussian quadrature for improved numerical solutions.

Findings

LDNN accurately solves nonlinear VFHIEs

The method outperforms traditional numerical approaches

LDNN demonstrates high convergence and stability

Abstract

Various phenomena in biology, physics, and engineering are modeled by differential equations. These differential equations including partial differential equations and ordinary differential equations can be converted and represented as integral equations. In particular, Volterra Fredholm Hammerstein integral equations are the main type of these integral equations and researchers are interested in investigating and solving these equations. In this paper, we propose Legendre Deep Neural Network (LDNN) for solving nonlinear Volterra Fredholm Hammerstein integral equations (VFHIEs). LDNN utilizes Legendre orthogonal polynomials as activation functions of the Deep structure. We present how LDNN can be used to solve nonlinear VFHIEs. We show using the Gaussian quadrature collocation method in combination with LDNN results in a novel numerical solution for nonlinear VFHIEs. Several examples are given to verify the performance and accuracy of LDNN.