颂歌
迭代函数
射击方法
牛顿法
边值问题
收敛速度
应用数学
数值分析
数学
非线性系统
趋同(经济学)
边界(拓扑)
点(几何)
常微分方程
数学分析
数学优化
计算机科学
微分方程
几何学
计算机网络
频道(广播)
物理
量子力学
经济
经济增长
出处
期刊:Cornell University - arXiv
日期:2022-08-28
标识
DOI:10.48550/arxiv.2208.13221
摘要
Boundary value problems in ODEs arise in modelling many physical situations from microscale to mega scale. Such two-point boundary value problems (BVPs) are complex and often possess no analytical closed form solutions. So, one has to rely on approximating the actual solution numerically to a desired accuracy. To approximate the solution numerically, several numerical methods are available in the literature. In this chapter, we explore on finding numerical solutions of two-point BVPs arising in higher order ODEs using the shooting technique. To solve linear BVPs, the shooting technique is derived as an application of linear algebra. We then describe the nonlinear shooting technique using Newton-Kantorovich theorem in dimension n>1. In the one-dimensional case, Newton-Raphson iterates have rapid convergence. This is not the case in higher dimensions. Nevertheless, we discuss a class of BVPs for which the rate of convergence of the underlying Newton iterates is rapid. Some explicit examples are discussed to demonstrate the implementation of the present numerical scheme.
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