<![CDATA[This research aims to develop a numerical method that can accurately estimate the relationship between cosmic energy (E) and the expansion rate of the universe (H), taking into account the complex interactions between ordinary matter, dark matter, and dark energy. Numerical approaches based on Euler, Runge-Kutta, and Adams-Bashforth integration methods will be refined to evaluate the correlation. The limitation of this study is to a flat universe (k = 0 geometry), but it has the potential to be extended to other geometries. This effective numerical method can revolutionize cosmology by allowing accurate testing of cosmological theories and improving predictive capabilities. This study not only deepens our understanding of the behavior of the universe, but also opens up opportunities for further exploration. While there has been research on the Friedmann equation and the evolution of the universe, this study fills the gap by comparing three numerical methods, promising a more comprehensive and accurate analysis. This research demonstrates significant advances in cosmological methodology, with the potential to change the cosmological paradigm through efficient numerical approaches. By improving the understanding of cosmic energy and the expansion rate of the universe, this research not only contributes to the current knowledge of cosmology, but also paves the way for impactful follow-up research in this field.]]>
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