Three new and applicable approaches based on quasi-linearization technique, wavelet-homotopy analysis method, spectral methods, and converting two-point boundary value problem to Fredholm–Urysohn integral equation are proposed for solving a special case of strongly nonlinear two-point boundary value problems, namely Troesch problem. A quasi-linearization technique is utilized to reduce the nonlinear boundary value problem to a sequence of linear equations in the first method. Second method is devoted to applying generalized Coiflet scaling functions based on the homotopy analysis method for approximating the numerical solution of Troesch equation. In the third method we use an interesting technique to convert the boundary value problem to Urysohn–Fredholm integral equation of the second kind; afterwards generalized Coiflet scaling functions and Simpson quadrature are employed for solving the obtained integral equation. Introduced methods are new and computationally attractive, and applications are demonstrated through illustrative examples. Comparing the results of the presented methods with the results of some other existing methods for solving this kind of equations implies the high accuracy and efficiency of the suggested schemes.

中文翻译：

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求解一类强非线性两点边值问题的三种新方法

提出了基于拟线性化技术、小波同伦分析法、谱法以及将两点边值问题转化为 Fredholm-Urysohn 积分方程的三种新的适用方法来求解强非线性两点边值的特殊情况问题，即 Troesch 问题。在第一种方法中，使用拟线性化技术将非线性边值问题简化为线性方程序列。第二种方法致力于应用基于同伦分析方法的广义 Coiflet 标度函数来逼近 Troesch 方程的数值解。在第三种方法中，我们使用一种有趣的技术将边值问题转换为第二类 Ury​​sohn-Fredholm 积分方程；之后采用广义 Coiflet 标度函数和 Simpson 求积来求解所获得的积分方程。引入的方法是新的并且在计算上具有吸引力，并且通过说明性示例演示了应用程序。将所提出方法的结果与求解此类方程的其他一些现有方法的结果进行比较，表明所建议方案的准确性和效率很高。