A comprehensive analysis of transient heat flow around fins

This study investigates transient heat transfer in longitudinal rectangular fins, where both thermal conductivity and the convective heat transfer coefficient are assumed to vary linearly with temperature. The governing nonlinear partial differential equation (PDE) is analyzed under two distinct boundary conditions: a step change in base temperature and a step change in base heat flux. Lie symmetry methods and conservation laws are used to derive invariant solutions and conserved vectors for various values of the nonlinearity exponent n . The analysis reveals that there are additional symmetries and nontrivial conservation laws for specific cases, notably $$n=-1, n=0$$ n = - 1 , n = 0 and $$n=1$$ n = 1 , while no conserved vectors are found for n ≥ 2 n ≥ 2 . Closed-form invariant solutions are obtained for the case $$n=-1$$ n = - 1 . To complement the analytical work, a numerical solution is carried out using the Method of Lines (MOL), with finite difference discretization in space, and the transient thermal behavior is visualized through surface and trajectory plots. Furthermore, the modified ( ZZ) Z ′ Z -expansion method is applied to construct exact traveling wave solutions for the physically relevant case $$n=2$$ n = 2 , yielding bright, periodic, kink, anti-kink, and rational soliton-type solutions depending on parameter choices. The combined analytical and numerical approaches provide a comprehensive understanding of the nonlinear thermal dynamics in fin systems, offering useful insights for engineering applications involving transient heat dissipation.