Free Boundary Formulation for BVPs on a Semi-Infinite Interval and Non-Iterative Transformation Methods

TL;DR

This paper demonstrates exact solutions for free boundary formulations of BVPs on semi-infinite intervals, introduces non-iterative initial value methods based on Lie group invariance, and discusses their application as benchmarks and in various problems.

Contribution

It provides exact solutions for specific free boundary problems and develops non-iterative methods using Lie invariance, offering new tools for solving semi-infinite BVPs.

Findings

Exact solutions serve as benchmarks for numerical methods.

Non-iterative methods derived from Lie invariance are effective.

Application potential in various literature problems.

Abstract

This paper is concerned with two examples on the application of the free boundary formulation to BVPs on a semi-infinite interval. In both cases we are able to provide the exact solution of both the BVP and its free boundary formulation. Therefore, these problems can be used as benchmarks for the numerical methods applied to BVPs on a semi-infinite interval and to free BVPs. Moreover, we emphasize how for two classes of free BVPs, we can define non-iterative initial value methods, whereas BVPs are usually solved iteratively. These non-iterative methods can be deduced within Lie's group invariance theory. Then, we show how to apply the non-iterative methods to the two introduced free boundary formulations in order to obtain meaningful numerical results. Finally, we indicate several problems from the literature where our non-iterative transformation methods can be applied.