Heating of a semi-infinite Hooke chain
We investigate unsteady ballistic heat transport in a semi-infinite Hooke chain with a free end subjected to an arbitrary heat source. We derive an exact integral representation for the kinetic temperature, together with a weak-dissipation lattice approximation and two continuum-type descriptions. The continualization of the discrete solution for the kinetic temperature in the weakly dissipative limit is obtained in two representations. The first is a symmetrical continuum solution based on an even extension of the source with respect to the boundary. The second is a discrete-continuum solution derived by applying a large-time asymptotic approximation to the fundamental solution at the fronts of the incident and reflected thermal waves in an instantaneously perturbed conservative semi-infinite chain. This solution retains a contribution from the interaction of the incident and reflected waves, which is absent in the symmetrical continuum solution. Analysis of sudden point heat supply indicates that, even in the non-dissipative chain, the discrete solution for the kinetic temperature exhibits pronounced large-time slowing near the free end, consistent with a local nonequilibrium plateau, whereas the symmetrical continuum representation predicts unbounded logarithmic growth. The results are expected to provide a benchmark for continuum models of heat transport in lattices with a free boundary.