2021/06/27 by Zeinab Hajimohammadi, Hajimohammadi, Zeinab, Kourosh Parand +3 · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Control Systems Design #FOS: Computer and information sciences #FOS: Mathematics #Fractional Differential Equations Solutions #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2106.14320
openalex publication_date 2021/06/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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.