: In this thesis an efficient and effective numerical method is successfully developed for solving multi-delay systems with piecewise constant functions. Time delays arise in various fields of applications including population dynamics, chemical processes, power systems, biological models, physics, robotics, economics, communication networks and information technology. It is known that except for some special cases, it is difficult or impossible to obtain a closed-form solution for delay differential equations. The situation becomes more complicated when time-delay is a piecewise constant function. So far, many papers have been devoted to the numerical solution of time-delay systems with constant delays. To the best of our knowledge, a few research works have been dedicated to the numerical treatment of piecewise constant delay systems. The purpose of this thesis is to develop an efficient procedure to numerically solve piecewise constant multi-delay systems. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. Among the piecewise constant basis functions, block-pulse functions have a simple structure and are easy to use. Up to now, different types of hybrid functions have been developed in the literature. The hybrid of block-pulse functions and Taylor’s polynomials is much simpler than that of those available in the literature. The nice properties of the hybrid functions and the associated operational matrices of integration, product and delay are employed to transform the problem under consideration into a system of algebraic equations. The sparsity of the mentioned matrices makes the presented method computationally attractive. It should be mentioned that the exact solution corresponding to a piecewise constant delay system is a piecewise smooth function. As a result, continuous basis functions such as Legendre’s polynomials, Chebyshev’s polynomials or piecewise constant basis functions such as block-pulse functions are not able to provide a satisfactory approximation to this ltr"
