Second-Order Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach

Yongtao Xuan,Zohaib Khan,Imad Ullah,Shah Nazir,Rohul Amin,Fakhar Zaman,Rahim Khan

doi:10.1155/2021/5551497

Yongtao Xuan, Zohaib Khan + Show 5 more

Open Access

https://doi.org/10.1155/2021/5551497

Copy DOI

Abstract

In this article, an efficient numerical approach for the solution of second-order delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.

Highlights

Delay differential equations (DDEs) are type of DEs in which the solution of the unknown function is given in the previous time interval
Haar Collocation Technique (HCT) scheme is devoted for the solution of second-order DDEs to deal with the experimentation of IIoT
The Haar technique is applied to linear DDEs for dealing with the experimentation of the Internet of Industrial Things

Summary

Introduction

Delay differential equations (DDEs) are type of DEs in which the solution of the unknown function is given in the previous time interval. This article studies the solutions of second-order DDEs, that is, we develop numerical technique using Haar wavelet with constant delay. We discuss the solution of the second-order DDEs using a HCT to deal with the experimentation of IIoT, consider linear DDEs as w′′ðtÞ = aðtÞw′ðtÞ + bðtÞwðtÞ + cðtÞuðtÞ + dðtÞwðt − τÞ + eðtÞuðt − τÞ, BB@ w′ð0Þ = α1, wð0Þ = α2, ð1Þ wðtÞ = φðtÞ,−τ ≤ t < 0, where uðtÞ is a control function, φðtÞ is the delay condition, and wðtÞ is a state function.

Haar Wavelet

Numerical Method

Numerical Examples

Discussion

Conclusion