Abstract In this study, a new multiscale algorithm was proposed to solve the boundary value problems of second order differential equations. A multiscale basis consisting of two sets of multiscale functions was constructed in the reproducing kernel space, and the proposed multiscale basis was proved to be orthonormal. The e− approximate solution was defined, and then it was proved to be the optimal solution. In addition, the stability, convergence and complexity of this algorithm were discussed and illustrated in this study. Numerical examples verify the effectiveness and feasibility of the algorithm, and the results show that the proposed intelligent multiscale algorithm has advantages in accuracy and stability compared with other methods.

中文翻译：

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一种求解二阶边值问题的多尺度新算法

摘要 本研究提出一种新的多尺度算法来求解二阶微分方程的边值问题。在再生核空间中构造了由两组多尺度函数组成的多尺度基，并证明了所提出的多尺度基是正交的。定义了e-近似解，然后证明它是最优解。此外，本研究还讨论和说明了该算法的稳定性、收敛性和复杂性。数值算例验证了算法的有效性和可行性，结果表明所提出的智能多尺度算法与其他方法相比在准确性和稳定性上具有优势。