Free Boundary Formulation for Boundary Value Problems on Semi-Infinite Intervals: An up to Date Review

TL;DR

This review paper discusses the free boundary formulation for boundary value problems on semi-infinite intervals, illustrating its effectiveness through examples and extending it to general higher-order problems with promising numerical results.

Contribution

It provides a comprehensive review of the free boundary formulation, including new applications to higher-order BVPs and validation through numerical experiments.

Findings

Numerical results agree well with existing literature.

Effective application of iterative transformation and Keller's methods.

Extension of free boundary formulation to n-order differential equations.

Abstract

In this paper, we propose a review of the free boundary formulation for BVPs defined on semi-infinite intervals. The main idea and theorem are illustrated, for the reader convenience, by using a class of second-order BVPs. Moreover, we are able to show the effectiveness of the proposed approach using two examples where the exact solution both for the BVPs and their \FBF \ are available. Then, we describe the free boundary formulation for a general class of BVPs governed by an $n$-order differential equation. In this context, we report three problems solved using the free boundary formulation. The reported numerical results, obtained by the iterative transformation method or Keller's second-order finite difference method, are found to be in very good agreement with those available in the literature. The last result of this research is that, in order to orient the interested reader, we provide an extensive bibliography. Of course, we may aspect further and more interesting applications of the free boundary formulation in the future.