Operational matrix and their applications for solving time-varying delay systems

Document Type : Research Paper


1 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

2 Department of Mathematics, University of Texas at Dallas, Richardson, USA

3 Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran

4 Department of Mathematics, Khalkhal Branch, Islamic Azad University, Khalkhal, Iran


The purpose of this paper is to provide a generalized formulation for Hat basis functions and to present the operational matrices for finding the approximate solution for time-invariant delay systems‎. ‎From this prospect‎, ‎the hat basic functions of integration‎, ‎delay‎, ‎product‎, ‎and dual are derived‎, ‎which are utilized to practically reduce the time-varying delay systems solution to the simplest system of algebraic equations‎. ‎The numerical results compared and tabled with previous works showcase the method's simplicity‎, ‎clarity‎, ‎and effectiveness through the three examples.


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Volume 14, Issue 9
September 2023
Pages 79-88
  • Receive Date: 24 August 2022
  • Revise Date: 14 November 2022
  • Accept Date: 26 November 2022