Abstarct: ‎ Modeling electrical, power, chemical and mechanical, is described by a specific set of differential and algebraic equations called the differential-algebraic equations when these systems are exposed to delay. For example in the electrical and power systems this delay appears in the result of the internal interconnections of the circuits and transmission lines or when simulating the chemical processes while modeling the pipe flow. ‎ With regard to the numerous applications of the delayed differential-algebraic equations, analyzing and presenting appropriate solutions to these equations is important, but so far few studies have been conducted on the structure of these equations and their solutions. This paper first introduces the differential-algebraic equations and then the asymptotic stability of differential-algebraic equations and linear and nonlinear algebraic delay and considers asymptotic stability for numerical methods such as multi-step, Runge-Kutta, θmethod and… . Finally the semi-analytical variational iteration and Adomian decomposition methods are used to solve these equations.