Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

TL;DR

This paper introduces a numerical test to determine the existence and uniqueness of solutions for boundary value problems on semi-infinite intervals, linking these properties to the zeros of an implicitly defined function.

Contribution

It defines a novel numerical test based on the iterative transformation method to assess well-posedness of semi-infinite boundary value problems.

Findings

Numerical test successfully applied to two examples.

Existence and uniqueness linked to zeros of a specific function.

Provides a practical approach for semi-infinite boundary problems.

Abstract

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context the numerical test is illustrated by two examples where we find meaningful numerical results.