Analysis of linear proportional delay systems via hybrid functions method

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.