Numerical Solution of the Modified Radioactive Decay Rate Equation Using the Runge-Kutta Fourth Order Method

Abstract

The standard radioactive decay equation describes exponential decay but neglects possible production mechanisms that may arise in complex systems. This study investigates a modified radioactive decay rate equation that incorporates a quadratic production term, allowing for the modeling of coupled decay–production dynamics. The governing nonlinear ordinary differential equation was solved analytically for limiting cases and numerically using the fourth-order Runge–Kutta (RK4) method. Numerical simulations were performed for varying values of the production parameter β, while keeping the decay constant fixed, allowing for a direct comparison between standard decay and modified decay behaviors. The results show close agreement between analytical and numerical solutions at low β values, confirming the accuracy of the RK4 method. As β increases, deviations from simple exponential decay become significant, with the emergence of steady-state solutions where production balances decay. These steady states occur when the production term becomes comparable to the linear decay term. The study demonstrates that the modified model captures decay dynamics not represented by the standard equation and highlights the sensitivity of system behaviors to the quadratic production parameter. The findings confirm the suitability of the RK4 method for solving nonlinear radioactive decay models and provide a framework for extending the analysis to more complex decay systems involving spatial dependence or coupled nuclide chains.