In this paper, we consider a fully third-order nonlinear boundary value problem that is of great interest of many researchers. First, we establish the existence and uniqueness of solution. Next, we propose simple iterative methods on both continuous and discrete levels. We prove that the discrete methods are of second-order and third-order of accuracy due to the use of appropriate formulas for numerical integration and obtain estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative methods.

中文翻译：

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求解完全三阶非线性边值问题的二阶和三阶收敛的简单数值方法

在本文中，我们考虑了一个完全三阶非线性边值问题，这是许多研究人员都非常感兴趣的问题。首先，我们确定解决方案的存在性和唯一性。接下来，我们提出连续和离散级别的简单迭代方法。我们证明了由于使用了适当的公式进行了数值积分，因此离散方法具有二阶和三阶精度，并且可以获得总误差的估计值。一些实例证明了所获得的理论结果的有效性以及迭代方法的有效性。