Shooting method for solving two-point boundary value problems in ODEs numerically

TL;DR

This paper discusses the shooting method for numerically solving two-point boundary value problems in ordinary differential equations, emphasizing linear and nonlinear cases, convergence properties, and practical implementation examples.

Contribution

It provides a detailed exploration of the shooting technique, including linear algebra application, Newton-Kantorovich approach for nonlinear problems, and convergence analysis in higher dimensions.

Findings

Shooting method effectively solves linear BVPs in ODEs.

Newton-Raphson exhibits rapid convergence in 1D cases.

Certain classes of BVPs have fast Newton iterates convergence.

Abstract

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.