Asian Research Journal of Mathematics

This paper introduces a new two-step hybrid block method for the direct numerical solution of third-order initial value problems (IVPs) without reduction to systems of first-order equations. The method is derived using collocation and interpolation techniques with a power series basis function, incorporating optimized -step points within each interval. The scheme is proven to be zero-stable, consistent, and converges at order nine, with an error constant of C10 ≈ 2.14 × 10−8, which confirms that the method is A-stable. Numerical experiments conducted on linear and nonlinear third-order IVPs demonstrate the method’s superior accuracy and computational efficiency compared to existing standard methods such as the classical Runge–Kutta fourth-order method (RK4), Adams–Bashforth–Moulton (ABM), and other hybrid block methods in the literature. The proposed method is self-starting, generates solutions simultaneously at multiple points, and is recommended for solving challenging third-order IVPs in engineering and physics.