Authors:
Ahmed. O. M. Abubaker,DOI NO:
https://doi.org/10.26782/jmcms.2025.12.00008Keywords:
Runge-Kutta method,fourth-order,systems of first-order differential equations,positive solutions,Abstract
The Runge-Kutta method, and especially its fourth-order variant (RK4), is perhaps the most widely adopted method for solving ordinary differential equations (ODEs) and their systems. This paper deals specifically with the RK4 method to explain a system of first-order differential equations, and the ability of the method to converge and stabilize positive solutions. It is well known that standard RK4 is both accurate and stable, but to particularly maintain positivity of solutions, where the model represents physical quantities that must be non-negative, such as populations or concentrations, often requires extra techniques. This paper discusses theoretically the RK4 method and systems, their execution, the need for retention of positivity, and methodologies for retention of positivity. Several illustrative examples are included to demonstrate the application of the method and the difficulty of maintaining positivity as well.Refference:
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