In this paper, a new algorithm is presented to solve the nonlinear second-order differential equations. The approach combines the Quasi-Newton’s method and the multiscale orthogonal bases. It is worth mentioning that the convergence of Newton’s method is verified for solving the nonlinear differential equations. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the efficiency and feasibility of the proposed method.

中文翻译：

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基于多尺度正交基的非线性边值问题的新方法

本文提出了一种求解非线性二阶微分方程的新算法。该方法结合了拟牛顿法和多尺度正交基。值得一提的是，通过牛顿法的收敛性可以解决非线性微分方程的问题。还证明了数值解及其导数的一致收敛性。数值算例表明了该方法的有效性和可行性。