Lie Symmetry Analysis and Similarity Solutions for Two-Dimensional Heat and Wave Equations

Abstract

This paper employs Lie symmetry theory to derive similarity solutions for the two-dimensional heat equation and wave equation. By identifying the Lie point symmetriesof these partial differential equations (PDEs), we perform symmetry reductions to transform the PDEs into ordinary differential equations (ODEs). The resulting ODEs are solved to obtain similarity solutions, which are invariant under specific symmetry transformations. We present explicit solutions for both equations, highlighting their physical interpretations and potential applications. The methodology demonstrates the power of Lie symmetry analysis in simplifying complex PDEs and uncovering
physically meaningful solutions.