Abstract For a second-order nonlinear singularly perturbed boundary value problem (SPBVP), we develop two novel algorithms to find the solution, which automatically satisfies the Robin boundary conditions. For the highly singular nonlinear SPBVP the Robin boundary functions are hard to be fulfilled exactly. In the paper we first introduce the new idea of boundary shape function (BSF), whose existence is proven and it can automatically satisfy the Robin boundary conditions. In the BSF, there exists a free function, which leaves us a chance to develop new algorithms by adopting two different roles of the free function. In the first type algorithm we let the free functions be the exponential type bases endowed with fractional powers, which not only satisfy the Robin boundary conditions automatically, but also can capture the singular behavior to find accurate numerical solution by a simple collocation technique. In the second type algorithm we let the BSF be solution and the free function be another variable, such that we can transform the boundary value problem to an initial value problem (IVP) for the new variable, which can quickly find accurate solution for the nonlinear SPBVP through a few iterations. Research Highlights – Two novel algorithms are developed for a second-order nonlinear singularly perturbed boundary value problem which automatically satisfies the Robin boundary conditions. – The new idea of boundary shape function is first introduced, whose existence is proven and it can automatically satisfy the Robin boundary conditions. – In the first type algorithm the free functions are the exponential type bases endowed with fractional powers, which satisfy the Robin boundary conditions automatically, and can capture the singular behavior. – In the second one we let the free function be another variable, such that the boundary value problem can be transformed to an initial value problem for the new variable, which can quickly find accurate solution for the nonlinear problem through a few iterations.

中文翻译：

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求解具有Robin边界条件的非线性奇异摄动问题的边界形函数方法

摘要 对于二阶非线性奇异摄动边值问题 (SPBVP)，我们开发了两种新算法来寻找自动满足 Robin 边界条件的解。对于高度奇异的非线性 SPBVP，Robin 边界函数很难精确满足。在论文中我们首先介绍了边界形状函数（BSF）的新思想，它的存在性得到了证明并且可以自动满足Robin边界条件。在 BSF 中，存在一个自由函数，这让我们有机会通过采用自由函数的两个不同角色来开发新算法。在第一类算法中，我们让自由函数为具有分数幂的指数型基，不仅自动满足 Robin 边界条件，但也可以通过简单的搭配技术捕获奇异行为以找到准确的数值解。在第二种算法中，我们让 BSF 为解，自由函数为另一个变量，这样我们就可以将边值问题转化为新变量的初始值问题（IVP），这样可以快速找到非线性问题的准确解SPBVP 通过几次迭代。研究亮点 - 针对二阶非线性奇异摄动边界值问题开发了两种新算法，该问题自动满足 Robin 边界条件。– 首次引入边界形函数的新思想，证明其存在性，可自动满足Robin边界条件。– 在第一类算法中，自由函数是具有分数幂的指数型基，它自动满足罗宾边界条件，并且可以捕获奇异行为。– 在第二个中，我们让自由函数成为另一个变量，这样边值问题就可以转化为新变量的初值问题，这样可以通过几次迭代快速找到非线性问题的准确解。