## 1 Introduction  
Numerical analysis is the area of mathematics that deals with computational techniques for studying and finding solutions to math problems. It is an offshoot of computer science and mathematics that develops, analyzes, and generates algorithms for numerically solving mathematical problems. Numerical methods are typically centered on the implementation of numerical algorithms. The goal of these numerical methods is to provide systematic procedures for numerically solving mathematical problems. The ordinary differential equation (ODE) is a mathematical equation that is used in natural physical processes, chemistry, engineering, and other sciences. Ordinary differential equations are notoriously difficult to approximate analytically, so the numerical treatment is crucial because it offers a powerful alternate solution method for resolving the differential equation.  
We frequently use initial value problems (IVPs), such that  
\[y^{\prime}=f(x,y(x)),\;y(x_{0})=y_{0} \tag{1}\]  
where \(x\) is the independent variable, which might also indicate time in physical problems, and \(y(x)\) is the solution. Furthermore, because \(y(x)\) can be a vector-valued function with N-dimensions, the domain and range of \(f\) and the solution \(y\) are given by  
\[f\,:\mathbb{R}\times\mathbb{R}^{\;N}\rightarrow\mathbb{R}\]  
\[y:\mathbb{R}\rightarrow\mathbb{R}^{\;N}\]  
Furthermore, Equation (1), in which \(f\) is a function both of \(x\) and \(y\), is known as a \"non-autonomous\" equation. However, by simply adding an extra variable, which is always equal to \(x\), it can be easily rewritten in an equivalent \"autonomous\" form below, in which \(f\) is a function of \(y\) only:  
\[y^{\prime}=f(y(x)),\;y(x_{0})=y_{0} \tag{2}\]  
There are numerous well-known numerical methods that can be used to solve the IVP in Equation (1). One category of these is third-order methods, like [PERSON]'s third-order Runge-Kutta method (Ralston's scheme) [1], the third-order Runge-Kutta method (RK3 scheme) [2], the third-order Runge-Kutta method based on the arithmetic mean (ARK3 scheme) [3], the third-order Heun's method ([PERSON]'s scheme) [4], and several other methods. These numerical methods have been constructed to solve Equations (1) and (2) using different techniques such as the Taylor series expansion technique, homotopy analysis technique, quadrature formulas technique, and Adomian decomposition technique; for more details, see [5; 6; 7].  
In this study, we construct a new numerical method based on a variation of the standard formulation of the Runge-Kutta method using Taylor series expansion. The rest of this paper is divided into the following sections. In Section 2, we recall some definitions and auxiliary results that we will use in our work. The derivation of the new method is described in Section 3. Section 4 provides details on the local truncation error. Section 5 discusses the stability analysis of the suggested technique. The consistency of the new method and its convergence are detailed in Section 6. Several numerical models are shown in Section 7 to show the efficiency of this method and compare it with other relevant methods. Finally, Section 8 offers the discussion and the conclusions.