## 8 Discussion and Conclusions  
In this study, we introduced an innovative third-order method designed for solving initial value problems (IVPs). Our approach is rooted in a novel adaptation of the standard formulation employed in Runge-Kutta methods, incorporating Taylor series expansion. To validate the effectiveness of this new method, we employed two distinct numerical models, effectively showcasing its fundamental capabilities. It is important to underscore that all our numerical findings, including the accompanying tables and figures, were calculated using MATLAB (R2022a) software on a dedicated computer system operating with Windows 11 Pro. The system uses an 11 th Generation Intel(R) Core (TM) i7-11800H processor running at 2.30 GHz, backed by 16.0 GB of RAM (15.7 GB usable).  
A comprehensive numerical assessment was conducted using Tables 1-6, which present an intricate comparison of absolute errors across various step sizes, specifically \(h=10^{-1},\ h=5\times 10^{-2},\ \text{and}\ h=25\times 10^{-3}\). Through the graphical representations found in Figures 2-7, we were able to discern that our novel method, referred to as VRK3, consistently outperformed several benchmark techniques including [PERSON]'s scheme, RK3 scheme, ARK3 scheme, and [PERSON]'s scheme. This superiority primarily stems from the reduced local truncation error of VRK3. Additionally, our investigation revealed a significant insight regarding the impact of step size on accuracy. As we decreased the step size, the error progressively approached zero, strongly indicating that precision increased with smaller step sizes. This observation reinforces the importance of carefully selecting step sizes to achieve higher levels of accuracy in numerical solutions. Turning our attention to computational efficiency, Tables 7 and 8 provided valuable insights. The VRK3 scheme consistently demonstrated reduced CPU time compared to its counterparts, further validating its utility in practical applications. Furthermore, Figure 1 depicts the stability region of our third-order VRK3 scheme, establishing its equivalence to similar methodologies. Importantly, we substantiated the convergence of our VRK3 scheme, as it satisfies both the consistency and stability criteria.  
In conclusion, our newly proposed third-order method exhibits a commendable blend of efficiency and reliability. The method's stability and high accuracy render it particularly robust for a wide range of applications. This research contributes to the field of numerical methods for IVPs by presenting an innovative approach that holds promise for improving computational accuracy and efficiency. Future research directions might explore the extension of this method to more complex problems or its integration into broader computational frameworks.  
Conceptualization, N.Y.A.-H., Z.J.K. and A.H.A.; formal analysis, N.Y.A.-H., Z.J.K. and A.H.A.; investigation, N.Y.A.-H., Z.J.K. and A.H.A.; methodology, N.Y.A.-H., Z.J.K. and A.H.A.; software, A.H.A.; supervision, N.Y.A.-H.; writing--original draft, N.Y.A.-H. and Z.J.K.; writing--review and editing, A.H.A. All authors have read and agreed to the published version of the manuscript.  
The author received no direct funding for this work.  
The authors declare that they have no competing interests.