A one-step hybrid block method which is applied for solving Lane-Emden second-order singular differential equations arising in astrophysics is developed. The present technique is obtained by considering three intra-step nodal points, which are chosen appropriately in order to get optimized errors of the main formulas approximating the solution and the first derivative at the final position of the block, and another approximation of the solution at an intermediate point. The stability properties and characteristics of the proposed strategy are analyzed. The new approach is combined with an appropriate algorithm which is applied to the first subinterval to circumvent the singular behaviour at the left endpoint of the integration interval. Some second-order singular differential problems are solved numerically to show the effectiveness and validity of the proposed technique, which has been compared with other strategies available in the recent literature. The presented results support the better performance of the developed scheme.

中文翻译：

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结合一对一步混合块 Nyström 方法解决天体物理学中出现的二阶奇异问题的数值解

开发了一种用于求解天体物理学中出现的Lane-Emden二阶奇异微分方程的一步混合块法。通过考虑三个步骤内节点点来获得本技术，以适当地选择，以便在梯形块的最终位置处获得近似溶液的主要公式的优化误差，以及解决方案的另一个近似值一个中间点。分析了所提出策略的稳定性和特性。新方法与适用于第一个子区间的适当算法相结合，以规避积分区间左端点处的奇异行为。一些二阶奇异微分问题被数值求解，以显示所提出技术的有效性和有效性，已将其与最近文献中可用的其他策略进行了比较。所呈现的结果支持所开发方案的更好性能。