Scaling Invariance and the Iterative Transformation Method for a Class of Parabolic Moving Boundary Problems

TL;DR

This paper uses scaling invariance to transform certain parabolic moving boundary problems into simpler ODE-based free boundary problems and applies an iterative transformation method for their numerical solution, demonstrating accuracy through examples.

Contribution

It introduces a novel application of scaling invariance analysis combined with the iterative transformation method for solving parabolic moving boundary problems.

Findings

Numerical solutions agree well with known exact or approximate solutions.

The method effectively reduces complex problems to simpler ODEs.

Demonstrated applicability on two illustrative problems.

Abstract

In this paper we apply a scaling invariance analysis to reduce a class of parabolic moving boundary problems to free boundary problems governed by ordinary differential equations. As well known free boundary problems are always non-linear and, consequently, their numerical solution is often obtained iteratively. Among the numerical methods, developed for the numerical solution of this kind of problems, we focus on the iterative transformation method that has been defined within scaling invariance theory. Then, as illustrative examples, we solve two problems of interest in the applications. The obtained numerical results are found in good agreement with exact or approximate ones.