International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

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.

Numerical modeling of electronic state evolution due to external electric field in the structure metal-insulator-semiconductor with solitary donor center is carried out. Considering a nanometer disc-shaped gate as a source of the electric field, the problem for the Laplace equation in infinite multilayered medium is solved to determine the gate potential. The energy spectrum of a bound electron is calculated from the problem for the stationary Schrödinger equation. Finite difference schemes are constructed to solve both the problems. Difference scheme for the Schrödinger equation takes into account cusp condition for the wave function at the donor location. To solve the problem for the Laplace equation, asymptotic boundary conditions for approximating the potential at large distances from the gate are proposed. On the basis of calculation results, a controlling parameter is suggested, which allows to determine the localization of electron wave function regardless of insulator thickness and permittivity.