Purpose

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

Design/methodology/approach

The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.

Findings

When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.

Research limitations/implications

Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.

Practical implications

Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.

Originality/value

Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

中文翻译：

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用迭代方法求解带非线性边界条件的二阶非线性边界值问题

目的

本文的目的是通过迭代数值方法解决具有非线性边界条件的二阶非线性边界值问题。

设计/方法/方法

作者介绍了特征函数作为检验函数，从而推导了弱形式的积分方程。通过扩展加权本征函数的数值解，并使用本征函数相对于权函数的正交性，并结合非分离/混合边界条件，可以借助Drazin获得闭合形式的膨胀系数反演公式。

发现

当作者开发迭代算法时，去掉时变项和右侧的非线性项，以解决非线性边值问题，它收敛速度非常快，并且提供了非常精确的数值解。

研究局限/意义

基本上，作者对迭代数值算法的策略是将时变项和非线性项放在右侧。

实际影响

从具有零值的初始猜测开始，作者使用封闭形式的公式快速生成新的解决方案，直到满足收敛要求。

创意/价值

通过六个数值实验的测试，作者证明了该迭代算法适用于具有非线性边界条件的高度复杂的非线性边界值问题。由于系数矩阵是在迭代循环之外建立的，并且由于闭式展开系数的性质，所提出的迭代算法非常节省时间并且收敛非常快。