REGULAR PAPER Analysis of linear proportional delay systems via hybrid functions method Sayyed Mohammad Hoseini Department of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran Correspondence Sayyed Hoseini, Department of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran. Email: smhoseiny@gmail.com Abstract In this paper, an approximate method for solving delayed differential equations involving pantograph-type delay is proposed. The method is based upon hybrid function approximations. The properties of hybrid functions of block-pulse functions and the well-known Legendre polynomials are provided. The operational matrix of pantograph-type delay corresponding to the method is presented. The associated operational matrices of delay, derivative, and product are utilized to reduce the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the applicability and efficiency of the proposed method. KEYWORDS approximate method, hybrid function, Legendre polynomial, pantograph, proportional delay 1 | INTRODUCTION Pantograph equation arises as a mathematical idealiza- tion and simplification of an industrial problem involving collection of current by the overhead of an electric locomotive [1,2]. The analysis gives rise to a system of differential equations in which the argument of some of the dependent variables is multiplied by some factors, say pantograph-type delays. The pantograph-type delay differential equations, which are also called proportional or scaled delay differential equations, are frequently encountered in various area of applications such as con- trol, electrodynamics, astrophysics, and economy [3,4]. In the recent years, there has been an increasing interest in the analysis and control of problems for the delay differ- ential equations. The attendance of proportional delay in differential equations make solving these equations much more difficult. The linear pantograph-type delay differential equations have been analyzed by many researchers [5-7]. We review a limited number of research works briefly here. Block-pulse functions were used in Chen [5] to analyze the linear pantograph-type delay systems. The Adomian decomposition method was applied on the pantograph equation in Dehghan and Shakeri [8] and the convergence of the method was established. The pantograph-type delay differential equations were investi- gated using the perturbation-iteration algorithms by Bahs ¸i and Çevik [9]. The existence and uniqueness theorems for multipantograph equations have been given by Liu and Li [6]. They also proved that the θ-methods for 1 2 < θ ≤ 1 are asymptotically stable. In Zhao [10], the Runge–Kutta method was used for multipantograph equations of neutral type and the stability of the method was investigated. In Bellen [11], a method based on a particular choice of the mesh was proposed for solving the pantograph differential equation. Their suggested scheme is analogue of the h-method that was introduced by Bellen et al. [12]. In Ishiwata [13] and Zhao et al. [14], the neutral functional differential equation involving pro- portional delay was considered. They analyzed the attain- able order of m-stage implicit Runge–Kutta methods. The sufficient condition of the asymptotic stability for analytic and approximate solutions was investigated in other Received: 30 June 2019 Revised: 6 July 2020 Accepted: 30 July 2020 DOI: 10.1002/asjc.2425 © 2020 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd Asian J Control. 2020;1–11. wileyonlinelibrary.com/journal/asjc 1