Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions

Steady-state temperature fields in domains with temperature-dependent heat conductivity and mixed boundary conditions involving a temperature-dependent heat transfer coefficient and radiation are considered. The nonlinear heat conduction equation is transformed into Laplace's equation using Kirchhoff's transform. Because of this transform, the nonlinearity is transferred from the differential equation only to boundary conditions of the third kind. The remaining boundary conditions (of the first and second kind) become linear. Applying Green's theorem to the transformed problem yields an integral equation containing only boundary integrals. Discretization of this integral equation produces a system of algebraic equations with a linear matrix and nonlinear right hand sides. A set of equations of this type can be solved iteratively. Numerical examples are provided.