Moving boundary problems with Ermakov symmetry reduction: nonlinear superposition principle and reciprocal transformation applications

TL;DR

This paper introduces a method to solve complex moving boundary problems using Ermakov symmetry reduction, revealing a nonlinear superposition principle and reciprocal transformations for exact solutions.

Contribution

It presents a novel approach applying Ermakov symmetry reduction to Stefan-type problems, enabling exact solutions and transformations for nonlinear moving boundary equations.

Findings

Exact Airy-type solutions obtained for a third order nonlinear evolution equation.

Application of nonlinear superposition principle to moving boundary problems.

Derivation of exactly solvable reciprocal boundary problems.

Abstract

Moving boundary problems of Stefan-type for a novel third order nonlinear evolution equation with temporal modulation are here shown to be amenable to exact Airy-type solution via a classical Ermakov equation with its admitted nonlinear superposition principle. Application of the latter together with a class of involutory transformations sets the original moving boundary problem in a wide class with temporal modulation. As an appendix, reciprocally associated exactly solvable moving boundary problems are derived.