Approximate Solution of Initial Value Problems for First-Order Differential Equations using a Combined Runge-Kutta And Piecewise Constant Argument Method
DOI:
https://doi.org/10.71310/pcam.3_73.2026.11Keywords:
initial value problem, piecewise constant argument, approximated solution, Runge-Kutta method, absolute errorAbstract
This study presents an efficient method for approximating the solutions of a certain class of first-order differential equations with variable coefficients. The approach constructs an auxiliary differential equation that combines the Runge-Kutta method with a piecewise constant argument, derived from the original initial value problem and parameterized by a positive integer ????. It is shown that, for sufficiently large ????, this auxiliary equation has a unique piecewise-smooth solution that approximates the considered initial value problem. Error estimates for the residual are derived to quantify the accuracy and to illustrate the influence of ????. The numerical results show that the method achieves higher accuracy with fewer computational steps than the classical Runge-Kutta and related schemes, and the framework extends to a broader class of nonlinear equations.
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