Existence theory for one-dimensional quasilinear coupled conductive–radiative flows

•We solve the conductive–radiative problem with a temperature dependent on the diffusion coefficient.•We use techniques of monotonicity in order to yield the solution of the problem.•We solve the problem with the Green Function Decomposition method.•We make the existence theory for quasilinear conductive–radiative problem.

The paper deals with the coupled conductive–radiative problem with a diffusion coefficient depending on the temperature. The technique of upper and lower solutions is used to generate a solution for this nonlinear problem in the space of Hölder continuous function by Pao (1992,2007) [1,2] together with certain integral representations given in Azevedo et al. (2011) [3]. We also produce numerical results using GFDNGFDN method, the Green Function Decomposition of the order N  , coupled with the Crank–Nicolson method and the Newton–Raphson method. The GFDNGFDN methodology arises from the integral representation involved and does not involve any a priori discretization on the angular variable μμ.