In this study, a general framework of the optimized decomposition method, which suggests an optimal construction of the series solutions, for analytic treatment of IVPs for second order differential equations is proposed. The presented framework expands the application of the method to several classes of IVPs for nonlinear differential equations. Illustrative comparisons between the optimized decomposition method and the Adomian decomposition method for solving some nonlinear IVPs is then presented. The results confirm that the optimized decomposition method produces better performance and gives better accuracy compared to Adomian decomposition method. The study highlights the strength and advantages of the method in dealing with nonlinear differential equations.

在这项研究中，提出了优化分解方法的一般框架，该框架提出了级数解的最优构造，用于分析处理二阶微分方程的 IVP。所提出的框架将该方法的应用扩展到非线性微分方程的几类 IVP。然后介绍了优化分解方法和 Adomian 分解方法之间用于解决一些非线性 IVP 的说明性比较。结果证实，与 Adomian 分解方法相比，优化的分解方法产生更好的性能并提供更好的准确性。该研究突出了该方法在处理非线性微分方程方面的优势和优势。