A perturbation method for nonlinear, one-dimensional conduction with heat generation

The Fourier conduction equation is nonlinear when the thermal conductivity and bulk heat generation rate are temperature-dependent. For the particular case involving Joulean heating with constant voltage gradient, the latter is inversely proportional to the temperature. This paper describes a simplified, closed form solution for the one-dimensional slab insulated on one wall with a fixed temperature on the other. The Kirchoff transformation is used to eliminate the nonlinearity due to temperature-dependent thermal conductivity and the transformed temperature is then asymptotically expanded in the thermal coefficient of electrical resistivity. The resulting equation through the first order terms is examined and the requirements to obtain 0.1 percent accuracy determined.