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

The purpose of this paper is to provide a numerical method based on the hat basis functions as well as the operational matrices for finding the approximate solution of time-invariant delay systems. from this point of view, the operational matrices of integration, delay, product, and dual of hat basic functions are derived, which are utilized to practically reduce the time-varying delay systems solution to the simplest system of algebraic equations. the numerical results have been compared and tabulated with previous results. the simplicity, clarity and effectiveness of the proposed method are shown through three examples.