Symmetry-based analysis and numerical solutions of coupled heat and wave equations in a cylindrical domain with dynamic boundary conditions

This comprehensive study explores the coupled three-dimensional heat and wave equations within a cylindrical domain, addressing dynamic boundary conditions prevalent in thermoelastic and thermal-acoustic systems. Leveraging Lie symmetry theory, we derive analytical solutions by identifying invariant transformations that simplify the coupled partial differential equations (PDEs). Two illustrative examples—a weakly coupled system with oscillatory Dirichlet conditions and a strongly coupled system with mixed boundary conditions—are solved analytically and validated numerically using finite difference methods. The analytical solutions employ Bessel functions and separation of variables, while numerical schemes ensure robustness for complex cases. Error analysis confirms second-order accuracy, and applications in aerospace, energy, and biomedical engineering are discussed. This work provides a robust framework for modelling coupled PDEs, advancing theoretical and practical understanding of dynamic systems in cylindrical geometries.