The solution of singular two-point boundary value problem is usually not sufficiently smooth at one or two endpoints of the interval, which leads to a great difficulty when the problem is solved numerically. In this paper, an algorithm is designed to recognize the singular behavior of the solution and then solve the equation efficiently. First, the singular problem is transformed to a Fredholm integral equation of the second kind via Green’s function. Second, the truncated fractional series of the solution about the singularity is formulated by using Picard iteration and implementing series expansion for the nonlinear function. Third, a suitable variable transformation is performed by using the known singular information of the solution such that the solution of the transformed equation is sufficiently smooth. Fourth, the Chebyshev collocation method is used to solve the deduced equation to obtain approximate solution with high precision. Fifth, the convergence analysis of the collocation method is conducted in weighted Sobolev spaces for linear singular equations. Sixth, numerical examples confirm the effectiveness of the algorithm. Finally, the Thomas–Fermi equation and the Emden–Fowler equation as some applications are accurately solved by the method.

中文翻译：

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非线性奇异两点边值问题的级数展开和切比雪夫搭配方法

奇异两点边值问题的求解通常在区间的一个或两个端点处不够平滑，这给数值求解带来了很大的困难。在本文中，设计了一种算法来识别解的奇异行为，然后有效地求解方程。首先，奇异问题通过格林函数转化为第二类 Fredholm 积分方程。其次，通过使用皮卡德迭代和对非线性函数进行级数展开，得到关于奇点解的截断分数级数。第三，利用解的已知奇异信息进行适当的变量变换，使得变换方程的解足够平滑。第四，采用切比雪夫搭配法求解推导方程，得到高精度的近似解。第五，在线性奇异方程的加权Sobolev空间中进行了搭配方法的收敛性分析。第六，数值例子证实了算法的有效性。最后，Thomas-Fermi 方程和 Emden-Fowler 方程作为一些应用得到了精确求解。