Solving Oscillatory Delay Differential Equations Using Block Hybrid Methods

Sufia Zulfa Ahmad,Norazak Senu,Fudziah Ismail

doi:10.1155/2018/2960237

Sufia Zulfa Ahmad, Norazak Senu + Show 1 more

Open Access

https://doi.org/10.1155/2018/2960237

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Journal: Journal of Mathematics	Publication Date: Oct 1, 2018
Citations: 4	License type: CC BY 4.0

Affiliation: Universiti Putra Malaysia

Abstract

A set of order condition for block explicit hybrid method up to order five is presented and, based on the order conditions, two-point block explicit hybrid method of order five for the approximation of special second order delay differential equations is derived. The method is then trigonometrically fitted and used to integrate second-order delay differential equations with oscillatory solutions. The efficiency curves based on the log of maximum errors versus the CPU time taken to do the integration are plotted, which clearly demonstrated the superiority of the trigonometrically fitted block hybrid method.

Highlights

Differential equations with a time delay are used to model the process which does depend on the current state of a system and the past states
Methods such as Runge-Kutta (RK), Runge-Kutta Nystrom (RKN), hybrid, and multistep are widely used for solving delay differential equations (DDEs)
In Problems 1, 3, and 4, TF-BEHM3(5) has better accuracy compared to all the methods in comparison

Summary

Introduction

Differential equations with a time delay are used to model the process which does depend on the current state of a system and the past states. DDEs have become an important criteria to investigate the complexities of the real-world problems concerning infectious diseases, biotic population, neuronal networks, and population dynamics. Methods such as Runge-Kutta (RK), Runge-Kutta Nystrom (RKN), hybrid, and multistep are widely used for solving DDEs. Ismail et al [1] used RK method and Hermite interpolation to solve first-order DDEs. Taiwo and Odetunde [2] worked on decomposition method as an integrator for delay differential equations. From the order conditions we constructed a two-point threestage fifth-order block explicit hybrid method which is trigonometrically fitted so that it is suitable for solving.

Derivation of Order Condition for Block Explicit Hybrid Method

Construction of Trigonometrically Fitted Block Explicit Hybrid Method

Problems Tested and Numerical Results

Discussion and Conclusion