论文标题
用于在数值上解决两点边界价值问题的射击方法
Shooting method for solving two-point boundary value problems in ODEs numerically
论文作者
论文摘要
ODE中的边界价值问题在建模从微观到大型量表的许多物理情况进行建模时会出现。这种两点边界值问题（BVP）很复杂，通常没有分析封闭形式的解决方案。因此，必须依靠数值将实际解决方案近似于所需的准确性。为了近似于溶液，文献中有几种数值方法。在本章中，我们探讨了使用射击技术以高阶ODE产生的两点BVP的数值解决方案。为了求解线性BVP，射击技术被推导为线性代数的应用。然后，我们使用Newton-Kantorovich定理在维度N> 1中描述非线性射击技术。在一维情况下，牛顿 - 拉夫森迭代术具有迅速的收敛性。在较高的维度中并非如此。然而，我们讨论了一类BVP，其基础牛顿迭代的收敛速度很快。讨论了一些明确的示例，以证明当前数值方案的实现。
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.
