New he-natural homotopy analysis transform method for nonlinear delay differential equations

Abstract

Delay differential equations (DDEs) are a type of functional differential equations that arise in numerous sciences, engineering, and many other fields of studies. These equations play a vital role in the mathematical modelling of real-life phenomena. Accordingly, systems of these equations provide a significant impact in different settings of applications. Many methods have been used to obtain a solution of various forms of DDEs. However, most of these methods that have been used give the difficulties in finding a convergent approximate analytical solution of nonlinear DDEs. These include divergence of the result as time increases, linearization, restrictive assumptions, and over-dependency of small or large parameters. Therefore, the analytical approximation of nonlinear DDEs has become a challenging task, especially the higher-order system of these equations. In this research, a new analytical method is introduced for solving different classes of nonlinear DDEs. The introduced method is based on the Homotopy analysis method and Natural transform, where the nonlinear terms are simply calculated as a series of He’s polynomial. Firstly, this approach is established based on retarded and neutral DDEs with constant and variable coefficients for both proportional and constant delays. The idea is extended to the systems of these types of DDEs, where the He’s polynomial is modified to suit the computation of the nonlinear terms of the systems. Secondly, the generalization of the method to the order nonlinear single and systems of these equations are provided. In addition, the convergence analysis of each developed algorithm is investigated to guarantee the convergence of the series solution produced by the approach. The established method gives the solution in the form of a rapidly convergent series, leading to the exact or approximate solution of at most three number of iterations of computational terms. Furthermore, unlike some existing methods, the developed method obtains solution without linearization, perturbation, or restrictive assumptions. Solutions to some problems of nonlinear DDEs from real-life applications are obtained to illustrate the effectiveness of the method. Finally, the obtained results are compared to the existing ones as well as the exact solutions that show the method adjusts the interval of convergence for the series solution to avoid round-off of errors and reduces the computational size compared to the reference methods. Hence, the approach is reliable and efficient in solving certain classes of nonlinear problems of DDEs.