A two-point boundary value problem associated with nonlinear one-dimensional conduction in radiating heat shields and other applications is solved by perturbation and numerical methods. An exact numerical solution is compared with asymptotic results consisting of a previously developed weak conduction solution and its strong conduction counterpart, which is obtained in the present analysis. It is found that the strong conduction asymptotic applies over a much wider range of the radiation conduction parameter than the weak one. Typical calculations show that the maximum temperature of the heat shield is reduced by only nine percent, with a disproportionate increase in end temperature of 32 percent in increasing the radiation conduction parameter from zero to infinity. However, an important structural benefit is obtained by significant reduction in the temperature differences along the shield, with moderate increases in the radiation conduction parameter.