Legendre Neural Network Method for Several Classes of Singularly Perturbed Differential Equations Based on Mapping and Piecewise Optimization Technology

Abstract

In this paper, we develop a novel neural network model with mapping and piecewise optimization technology for several classes of the linear singularly perturbed initial value and boundary value differential equations with variable coefficients. First, the Legendre polynomials are selected as the activation function of the artificial neural network, the mapping technology is employed to transform the original uniform partition points and the piecewise optimization technology is used to improve the calculation accuracy. Then, the solution of the linear singularly perturbed differential equations is solved by using the extreme learning machine optimization algorithm. Finally, the numerical experiments show that the developed method can effectively improve the accuracy of the calculation.