On the 1D heat equation for laser-medium interaction including convection and radiation at the boundaries

• Semi-analytical solution for problems with convective and radiative boundaries. • Application of spectral method and perturbation theory to laser-medium interaction. • Generalization of solving the heat equation with nonlinear boundary conditions. • Alternative, slightly more intuitive approach to the Homotopy Analysis Method. The interaction of laser radiation and a non-perfectly reflecting medium leads to the conversion of electromagnetic energy into heat at the medium boundary. Heat partially dissipates through convection and radiation at the medium boundaries. Evaluation of this problem requires a solution to the heat equation with nonlinear boundary conditions. This paper demonstrates through application of the spectral method and perturbation theory an analytically approximated temperature solution that is globally smooth and stabilizes for time t → ∞ . This behavior is verified by comparison with the previously reported analogy of convective and radiative cooling of a sphere. The method is generalized in order to solve the heat equation with nonlinear boundary conditions in terms of nth degree polynomial functionals.