Modeling of dynamic delayed systems by using general runge-kutta methods

A system of differential equations with several variable delays is considered, which is a mathematical model of many technical processes with time delay. Most often, numerical Runge-Kutta methods and the Taylor series expansion by delay method are used to model such systems. Previously, Runge-Kutta methods were used for systems of differential equations with one variable delay. A generalization of the Runge-Kutta methods for systems of differential equations with a finite number of variable delays is obtained, which significantly expands the class of problems for which this method is applicable. Also, this method has advantages over the Taylor expansion method in terms of delay since it is applicable to systems with several variable delays, is convenient for programming, and has no restrictions on the value of delay. To generalize the Runge-Kutta methods for systems with delay, we used Runge-Kutta methods for systems of differential equations without delay in time and interpolation and extrapolation of the redistribution. The approximation properties of the generalization of Runge-Kutta methods for systems with delay are established. Namely, the connection between the approximation order of the Runge-Kutta methods for systems with delay and the approximation order of the Runge-Kutta methods for systems without delay and the approximation order of interpolation and extrapolation of the model history is shown. Thus, it was established that the approximation order of the Runge-Kutta method for a system with delay will be the minimum of the approximation orders of the Runge-Kutta method for a system without delay and the approximation order of interpolation and extrapolation of the model history will be increased by one. Based on this statement, it is concluded that if we use the Lagrange polynomial in the number of nodes that coincides with the approximation order of the Runge-Kutta method for a system without delay, then, with a sufficiently small step, the Runge-Kutta method for a system with delay retains the approximation order of the method Runge-Kutta without delay. Without limiting the step length when using the Euler method for extrapolating a solution, it was found that in this case the approximation order of the Runge-Kutta methods for a system with a delay of more than two.