Legendre Deep Neural Network (LDNN) and its application for approximation of nonlinear Volterra–Fredholm–Hammerstein integral equations
Authors: Kourosh Parand, Zeinab Hajimohammadi, Ali Ghodsi
Keywords: Deep neural network, Volterra–Fredholm–Hammerstein integral equations, Legendre orthogonal polynomials, Gaussian quadrature method, Collocation method
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 (V-F-H-IEs). LDNN utilizes Legendre orthogonal polynomials as activation functions of the Deep structure. We present how LDNN can be used to solve nonlinear V-F-H-IEs. We show using the Gaussian quadrature collocation method in combination with LDNN results in a novel numerical solution for nonlinear V-F-H-IEs. Several examples are given to verify the performance and accuracy of LDNN.