O‘zbekiston matematika jurnali Volume 70 Issue 2 (2026) · pp. 159-166
Mixed piecewise constant argument and the Runge-Kutta method for numerically solving first-order differential equations
Muminov, M., Jumaev, Z.
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.
Initial value problemDifferential equation with piecewise constant argumentApproximated solutionRunge-Kutta methodAbsolute error
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