The application of numerical methods to solve linear systems with a time delay

This paper considers the application of modified numerical methods for solving differential equations with a delay which linearly depends on time. Since the delay increases indefinitely, it is also necessary to apply asymptotic methods to analyze the behavior of the solutions of such systems. The paper establishes the asymptotic properties of the systems under study, which significantly affect the accuracy of the numerical calculation. Given the unbounded delay and the instability of the solutions and to clarify the properties of the solution of such systems, it is useful to know the asymptotic properties of the derivatives having an order greater than one. Under the conditions formulated in the article, these derivatives tend to zero as $t\to \infty$. This property makes it possible to apply finite-order numerical methods (such as the Runge-Kutta method and the modified Euler method). As an illustration of the effectiveness of the methods developed, the article calculates the vertical oscillations of a locomotive pantograph moving at a constant speed when interacting with the contact wire. The numerical methods allow the study of the asymptotic behavior of more complex systems containing both constant and linear delay. Note that the use of numerical methods for calculating the solution reveal the instability of the solution of the systems under study and can be used to stabilize some systems containing an unlimited (not necessarily linear) delay.