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.
Author: Filipov S.
CONSTRAINED SIMILARITY OF 2-D TRAJECTORIES BY MINIMIZING THE H1 SEMI-NORM OF THE TRAJECTORY DIFFERENCE
This paper defines constrained functional similarity between 2-D trajectories via minimizing the H1 semi-norm of the difference between the trajectories. An exact general solution is obtained for the case wherein the components of the trajectories are mesh- functions defined on a uniform mesh and the imposed constraints are linear. Various examples are presented, one of which features application to mechanics and two-point boundary value problems. A MATLAB code is given for the solution of one of the examples. The code could easily be adjusted to other cases.
THEORETICAL FOUNDATIONS AND SPECIFICITY OF MATHEMATICAL MODELLING
LAPLACIAN PRESERVING TRANSFORMATION OF SURFACES AND APPLICATION TO BOUNDARY VALUE PROBLEMS FOR LAPLACE’S AND POISSON’S EQUATIONS
This paper shows that the constrained similarity transformation of surfaces under boundary constraints is a Laplacian preserving transformation. First, a general proof is presented and then the result is verified for mesh-functions through particular examples. The fact that the constrained similarity transformation, subject to boundary constraints, is a Laplacian preserving transformation is used to construct a method for solving boundary value problems for the Laplace’s and the Poisson’s equations. Given any solution to these equations we apply the constrained similarity transformation to get the particular solution that satisfies the given boundary conditions.
TECHNOLOGICAL BASIS OF “INDUSTRY 4.0”
CONSTRAINED SIMILARITY OF SURFACES BY MINIMIZING THE L2 NORM OF THE GRADIENT OF THE SURFACE DIFFERENCE
This paper defines constrained similarity between surfaces via minimizing the L2 norm of the gradient of the difference between the surfaces. An exact general solution is obtained for the case wherein the surfaces are given as mesh-functions defined on a uniform mesh and the imposed constraints are linear. Various examples are presented as well as a MATLAB code for the solution of one of the examples. The code could be adjusted to other cases.