New Numerical Methods for Solving the Initial Value Problem Based on a Symmetrical Quadrature Integration Formula Using Hybrid Functions

In this study, we construct new numerical methods for solving the initial value problem (IVP) in ordinary differential equations based on a symmetrical quadrature integration formula using hybrid functions. The proposed are designed to provide an efficient and accurate solution IVP more suitable problems with non-smooth solutions. key idea behind is combine advantages of traditional methods, such as Runge–Kutta Taylor’s series strengths modern Furthermore, discuss accuracy stability analysis these methods. resulting can handle wide range problems, including those singularities, discontinuities, other features. Finally, demonstrate validity several examples illustrate efficiency