Analysis of Linear Piecewise Constant Delay Systems Using a Hybrid Numerical Scheme

H R Marzban,S M Hoseini

doi:10.1155/2016/8903184

H R Marzban, S M Hoseini

Open Access

https://doi.org/10.1155/2016/8903184

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Abstract

An efficient computational technique for solving linear delay differential equations with a piecewise constant delay function is presented. The new approach is based on a hybrid of block-pulse functions and Legendre polynomials. A key feature of the proposed framework is the excellent representation of smooth and especially piecewise smooth functions. The operational matrices of delay, derivative, and product corresponding to the mentioned hybrid functions are implemented to transform the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the proposed numerical scheme.

Highlights

Delay differential equations (DDEs) naturally arise in diverse areas of science and engineering such as transmission lines, communication networks, biological models, population dynamics, and transportation systems [1, 2]
Owing to the nature of DDEs, none of the smooth basis functions is able to properly model the inherent behavior of this class of systems. This is due to the lack of smoothness of analytical solution associated with DDEs
An efficient procedure has been successfully developed for solving delay differential equations with a piecewise constant delay function

Summary

Introduction

Delay differential equations (DDEs) naturally arise in diverse areas of science and engineering such as transmission lines, communication networks, biological models, population dynamics, and transportation systems [1, 2]. Marzban and Shahsiah proposed an efficient numerical scheme for solving DDEs containing piecewise constant delay. Their method is based on a hybrid of block-pulse functions and Chebyshev polynomials. The time-delay systems considered in [15, 16] involve constant delay, while here we investigate linear piecewise constant delay systems The latter systems are a general class of constant DDEs. Third, the approach employed here is based on the derivative matrix corresponding to the mentioned hybrid functions while the method implemented in our earlier works is based on the operational matrix of integration.

Hybrid Functions

Operational Matrices

Problem Statement and Its Approximation

Illustrative Examples

Conclusion