Application of the Harmonic Runge–Kutta with Forward–Backward Technique by Parallelism

The studies that deal with various kinds of mean instead of the arithmetic standard mean in the Runge–Kutta technique have been researched considerably recently, and what has been studied here is a harmonic Runge–Kutta method of the second order that derives with forward and backward techniques to obtain a new parallel method in two stages that is presented for solving initial value problems. This proposed method was verified by numerical examples, and the results obtained are accurate enough compared with some exact solutions.