2025, Vol. 6, Issue 2, Part C

This paper explores the (3+2)-dimensional heat and wave equations using Lie symmetry theory to address complex dynamic boundary conditions in a general domain. Lie point symmetries are derived to reduce the partial differential equations (PDEs) to ordinary differential equations (ODEs), enabling exact solutions. Two fully solved examples for the heat equation and two for the wave equation demonstrate the method’s efficacy, incorporating time-dependent and nonlinear boundary conditions. Conservation laws, derived via Ibragimov’s method, ensure physical consistency. The results are significant for applications in heat transfer, wave propagation, and fluid dynamics, providing analytical tools for high-dimensional systems with dynamic boundaries.