Seventh‐Order, Variable Step Size, Two‐Step Runge–Kutta–Nyström Pairs for Linear Inhomogeneous Problems

ABSTRACT This study presents a seventh‐algebraic‐order explicit Numerov‐type numerical method, specifically tailored for the efficient solution of linear inhomogeneous differential equations. To optimize its computational efficiency, a cost‐effective step size adjustment algorithm is integrated. This adaptive strategy dynamically modifies the step length at each iteration, maintaining the current size, halving it, or doubling it, based on established criteria. Any necessary off‐step points for the method's implementation are accurately determined via a local interpolation. Comprehensive numerical experiments across diverse problem sets rigorously demonstrate the substantial efficiency enhancements achieved by this augmented approach. The method proves particularly efficacious for differential equations exhibiting oscillatory behavior, as it consistently maintains high accuracy while requiring a reduced number of function evaluations. This advancement holds significant implications for applications demanding precise numerical solutions over extended intervals, notably in fields such as physics and engineering. Furthermore, the paper provides a complete MATHEMATICA implementation, thereby promoting ease of use and facilitating further research within the numerical analysis community. By simultaneously addressing both computational efficacy and solution accuracy, this work furnishes a valuable tool for numerical analysts.