Numerical solution of delay differential equation using two-derivative Runge-Kutta type method with Newton interpolation

Numerical approach of two-derivative Runge-Kutta type method with three-stage fifth-order (TDRKT3(5)) is developed and proposed for solving a special type of third-order delay dif-ferential equations (DDEs) with constant delay. An algorithm based on Newton interpolation and hybrid with the TDRKT method is built to approximate the solution of third-order DDEs. In this paper, three-stage fifth-order called TDRKT3(5) method with single third derivative and multiple evaluations of the fourth derivative is highlighted to solve third-order pantograph type delay differ-ential equations directly with the aid of the Newton interpolation method. Stability analysis of TDRKT3(5) method is investigated. The numerical experiments illustrate high efficiency and valid-ity of the new method for solving a special class of third-order DDEs and some future works are recommended by extending proposed method to solve fractional and singularly perturbed delay dif-ferential equations. (c) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

Automated summary

This paper develops a numerical approach based on the two-derivative Runge-Kutta type method for solving a special type of third-order delay differential equations with constant delay. The proposed method, named TDRKT3(5), utilizes Newton interpolation and demonstrates high efficiency and validity in solving third-order pantograph type delay differential equations. The stability analysis of the TDRKT3(5) method is also investigated. Numerical experiments confirm the effectiveness of the new method and suggest the possibility of extending it to solve fractional and singularly perturbed delay differential equations.