A Study of the One-Dimensional Heat-Conduction Equation with Radiation
Mihai Halic
TL;DR
The paper tackles the steady-state one-dimensional heat-conduction equation with radiation, modeled by u_heat'' = b^2(u_heat^4 − t^4) with boundary data u_heat(0) = 1 and u_heat(1) = t. It develops an analytic envelope framework built on auxiliary transcendental equations and a residue-based comparison to obtain explicit upper and lower bounds for the exact solution, addressing its strong parameter sensitivity and boundary-layer behavior. The contributions include global and partial envelopes for y_heat (and hence u_heat), a deformation-based residue analysis, and a MAPLE-implemented numerical methodology that achieves high-precision estimates with errors shrinking like O(t e^{−3B}) as the boundary layer intensifies. By bridging transcendental-analysis with explicit ODE solutions, the work yields practical bounds and insights into the boundary-layer structure, providing a robust toolkit for both theoretical insight and computational verification of radiative heat-conduction models. The results have potential impact on accurately predicting temperature profiles in materials where radiation dominates, with a scalable approach to parameter-driven envelope construction and shooting-method stabilization.
Abstract
We consider a boundary value problem (BVP) modelling one-dimensional heat-conduction with radiation, which is derived from the Stefan-Boltzmann law. The problem strongly depends on the parameters, making difficult to estimate the solution. We use an analytical approach to determine upper and lower bounds to the exact solution of the BVP, which allows estimating the latter. Finally, we support our theoretical arguments with numerical data, by implementing them into the MAPLE computer program.