In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs.

中文翻译：

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非线性Painlevé-I超验进化三次样条智能求解器设计

在本研究论文中，基于进化三次样条法（CSM）的创新仿生算法已被用于估计非线性常微分方程 Painlevé-I 的数值结果。计算机制用于支持所提出的技术 CSM，并使用与顺序二次规划 (SQP) 混合的全局搜索技术遗传算法 (GA) 优化获得的结果，以实现快速细化。Painlevé-I 由提出的技术 CSM-GASQP 解决。在这个过程中，针对各种场景实现了样条的变化。CSM-GASQP 产生一个连续到其二阶导数的插值函数。此外，样条曲线被证明比拟合到所有点的单个多项式更稳定，并减少了列表点之间的摆动。该方法通过利用 GA-SQP 算法的性能进行全局搜索，为样条未知系数的自适应提供了一种可靠且出色的过程。通过对几次独立运行的统计分析，检验了所提出方案的收敛性、准确性和准确性。