A Semi-Analytical Solution of a Two-Dimensional Moving Boundary Problem Using the Homotopy Analysis Method

Abstract

This study presents a semi-analytical solution to a two-dimensional moving boundary problem governed by the classical heat equation. The physical model represents heat conduction within a rectangular domain, where the boundary evolves dynamically over time due to phase change phenomena. A Stefan-type condition is imposed at the moving interface to capture the effect of latent heat exchange. The Homotopy Analysis Method (HAM) is employed to construct convergent series solutions for both the temperature field and the moving boundary function. Unlike standard applications of HAM where convergence control is achieved via an auxiliary parameter, the convergence of the series solution in this work is inherently guided by the imposed boundary conditions. This boundary-driven convergence ensures consistency between the evolving interface and the thermal field without requiring external tuning of convergence control parameters. The analytical results are compared with numerical benchmarks to validate the accuracy and reliability of the proposed approach. The study demonstrates that HAM provides a robust framework for analyzing two-dimensional moving boundary problems with analytically tractable and physically consistent solutions.