Mixed piecewise constant argument and the Runge-Kutta method for numerically solving first-order differential equations

Abstract

This work introduces an efficient methodology for approximating solutions to first-order non-linear differential equations. The approach is based on formulating a differential equation with piecewise constant arguments associated with the parameters of the classical Runge-Kutta (RK) discrete equations depended on a positive integer parameter $n$. It is demonstrated that the constructed equation has a unique piecewise-smooth solution and for sufficiently large values of $n$, this solution approximating solution to the original problem. Numerical results are provided through examples, demonstrating the efficiency and high accuracy of the proposed method.