Orthogonal polynomial-based neural network solution for differential equations with implicit boundary conditions

Purpose This paper aims to provide the approximate solution to variety of differential equations with implicit boundary conditions. Specifically, the focus is on solving higher-order differential equations and system of differential equations, fluid mechanics problem using an orthogonal polynomial-based neural network with extreme learning machine (ELM) algorithm. Design/methodology/approach The authors use neural network constructed with four different types of orthogonal polynomials: Legendre polynomial, Laguerre polynomial, Chebyshev polynomial and Hermite polynomial and train it using ELM algorithm. The neural network consists of single hidden layer replaced by functional expansion block which uses orthogonal polynomial to extract its features. The result of neural network provides approximate solutions to the problems. To show the capability and efficacy of the approach, obtained solutions are compared with the exact solutions and those derived from traditional numerical technique. Findings Numerical and comparative studies show that the neural network solutions are obtained with better accuracy. Moreover, the used approach is simple to implement and offer robust framework for solving differential equations with complex boundary conditions. Originality/value Implicit boundary value problems are successfully addressed, yielding closed form solutions that are highly valuable for real-world applications. Further, the proposed approach for solving implicit problems enhances the applicability of orthogonal polynomial-based neural network with ELM.