Steady state temperature fields in domains with temperature dependent heat conductivity and mixed boundary conditions involving a temperature dependent heat transfer coefficient and radiation were considered. The nonlinear heat conduction equation was transformed into Laplace's equation using Kirchhoff's transform. Due to this transform the non-linearity is transferred from the differential equation only to third kind boundary conditions. The remaining boundary conditions of first and second kind, became linear. Applying Green's theorem to transformed problem results in integral equation containing boundary integrals only. Discretization of this integral equation gives a system of algebraic equations with linear matrix and nonlinear right hand sides. Such set of equations can be solved iteratively. Numerical examples are included. © 1981 IPC Business Press Ltd.
Bialecki, R., & Nowak, A. J. (1981). Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions. Applied Mathematical Modelling, 5(6), 417–421. https://doi.org/10.1016/S0307-904X(81)80024-8