In this article, a new algorithm based on improved Legendre orthonormal basis for solving second-order BVPs is constructed. The method of this article is an improved Legendre orthogonal polynomial algorithm, and the $\epsilon$-best approximate solution is constructed skillfully in the reproducing kernel space. Compared with other methods, this method has the advantages of high approximation, $m$-order convergence and strong stability. Some numerical examples show the effectiveness and stability of the algorithm.

中文翻译：

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基于改进的Legendre正交基础的新算法求解二阶BVP

本文构建了一种基于改进的Legendre正交基础的求解二阶BVP的新算法。本文的方法是一种改进的Legendre正交多项式算法，$\epsilon$-在复制内核空间中熟练地构造最佳近似解决方案。与其他方法相比，该方法具有逼近度高的优点，$米$有序收敛性和较强的稳定性。数值例子表明了该算法的有效性和稳定性。