Numerical Algorithm for Nonlinear Delayed Differential Systems of $n$th Order

TL;DR

This paper introduces a semi-analytical numerical method combining the method of steps and differential transformation for solving complex delayed differential systems, demonstrating accuracy and efficiency through practical examples.

Contribution

A novel semi-analytical algorithm for efficiently approximating solutions of nonlinear delayed differential systems of arbitrary order.

Findings

The method achieves high accuracy compared to existing algorithms.

It demonstrates reliable convergence and computational efficiency.

Applicable to systems with multiple delay types, including neutral and pantograph equations.

Abstract

The purpose of this paper is to propose a semi-analytical technique convenient for numerical approximation of solutions of the initial value problem for $p$-dimensional delayed and neutral differential systems with constant, proportional and time varying delays. The algorithm is based on combination of the method of steps and the differential transformation. Convergence analysis of the presented method is given as well. Applicability of the presented approach is demonstrated in two examples: A system of pantograph type differential equations and a system of neutral functional differential equations with all three types of delays considered. Accuracy of the results is compared to results obtained by the Laplace decomposition algorithm, the residual power series method and Matlab package DDENSD. Comparison of computing time is done too, showing reliability and efficiency of the proposed technique.