Differential equation with piecewise constant arguments and applications for solving initial value problems

In this talk, Ordinary Differential Equations (ODE) with piecewise constant arguments will be considered. The existence conditions for a certain type of ODE will be demonstrated. Further, an initial value problem for ODE will be examined. New methods for approximately solving the considered initial value problems will be presented. First, a differential equation with piecewise constant arguments, corresponding to the considered initial value problem and depending on a positive integer $N$, is introduced. It is proven that this equation has a unique piecewise-smooth solution, which serves as an approximate solution to the initial value problem for large $N$. Numerical results are provided through examples, demonstrating the efficiency and high accuracy of the proposed method.