A Neural Computational Method for Solving Renewal Delay Integro-Differential Equations Constrained by the Half-Line

Abstract

This study aims to solve the renewal delay integro-differential equations constrained by the half-line, introducing a computational method composed of the matrix relations of the Stieltjes-Wigert polynomials at the collocation points. In order to mathematically interpret their robust integral part, the method is also fed neurally by the Stieltjes-Wigert polynomials and a hybrid polynomial dependent upon the alteration of the Taylor and exponential polynomial bases. Thus, the method easily gathers the matrix relations into a matrix equation and immediately produces a desired solution. An error bound analysis is established to discuss the accuracy of the method by employing the collaboration of the mentioned polynomials. A population model with time-lags (delays), the detection of the displaced atoms versus kinetic energy, an integral delay equation, and a delayed problem with functional kernel are firstly treated by the method. Consequently, it is evident that the method presents a novel consistent approach and is directly programmable on a mathematical software thanks to its neural structure.