  This paper considers one-dimensional heat transfer in a media with
temperature-dependent thermal conductivity. To model the transient behavior of
the system, we solve numerically the one-dimensional unsteady heat conduction
equation with certain initial and boundary conditions. Contrary to the
traditional approach, when the equation is first discretized in space and then
in time, we first discretize the equation in time, whereby a sequence of
nonlinear two-point boundary value problems is obtained. To carry out the
time-discretization, we use the implicit Euler scheme. The second spatial
derivative of the temperature is a nonlinear function of the temperature and
the temperature gradient. We derive expressions for the partial derivatives of
this nonlinear function. They are needed for the implementation of the Newton
method. Then, we apply the finite difference method and solve the obtained
nonlinear systems by Newton method. The approach is tested on real physical
data for the dependence of the thermal conductivity on temperature in
semiconductors. A MATLAB code is presented.
