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.
翻译：从微观规模到超大型规模,在模拟许多物理情况时都会出现ODE的边界值问题。这种两点边界值问题(BVPs)很复杂,往往没有分析封闭形式的解决方案。因此,人们必须依靠将实际解决办法的数值与期望的准确性相近。要从数字上比较,文献中可以提供几种数字方法。在本章中,我们探索如何寻找使用射击技术在更高层次的ODE中产生的两点BVP的数值解决办法。为了解决线性BVPs,射击技术是作为线性代数的一种应用。然后我们用牛顿-Kantorovich 等值来描述非线性射击技术。1 在一维情况中,Newton-Raphson Iterates迅速趋同。在更高层面中情况并非如此。然而,我们讨论的是一类BVPs的数值,其基础的趋同率是迅速的。我们讨论过一些明确的例子,以展示目前数字计划的执行情况。
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