The application of mathematical simulation for the solution of linear algebraic and ordinary differential equations regarding to complex electrical circuits, systems of automatic control and regulation is discussed in the paper. MatLab software package of mathematical simulation and one of its built-in environments - Simulink were chosen to be used in the experiment. MatLab is the most commonly used application in engineering practice in comparison with the other similar programs and it has proved to be reliable for technical calculations. The calculation methods of complex electrical circuits by the solution of linear algebraic and differential equation systems were developed and tested. The equations of electric equilibrium were written for each element of an electric circuit on the basis of Ohm’s and Kirchhoff’s laws. The example of the calculation of branches currents and nodal potentials of electric circuit by the matrix method in MatLab was presented. The approximated polynomial function was calculated and its graph was plotted applying the developed technique. Thus, the coincidence with the original data was proved. The calculation of differential equations of successively connected inductive and resistive elements of electric circuit was made in Simulink environment. Finally, the dependences of voltage and current on time were found using the direct Laplace transformation. The graphs were plotted due to the application of the oscilloscope built into the program. It has been concluded that the application of mathematical simulation for engineering calculations in order to solve the problems of automatic control and power engineering is the most suitable and universal technique.

Standing and moving discrete breathers (or equally, intrinsic localized modes) in a square $\beta$-Fermi-Pasta-Ulam-Tsingou lattice are obtained by applying localizing functions to the delocalized nonlinear vibrational modes (DNVMs) found earlier by Ryabov and Chechin. The initial conditions used in our study do not correspond to exact spatially localized solutions, but make it possible to obtain long-lived quasibreathers. The approach employed in this work can easily be used to search for quasibreathers in three-dimensional crystal lattices, for which DNVMs with frequencies outside the phonon spectrum are known.

The problem of finding various discrete breathers (DBs) in the $\beta$-Fermi-Pasta-Ulam-Tsingou simple cubic lattice is addressed. DBs are obtained by imposing localizing functions on delocalized nonlinear vibrational modes (DNVMs) having frequencies above the phonon spectrum of the lattice. Among 27 DNVMs with the wave vector at the boundary of the first Brillouin zone there are three satisfying this condition. Seven robust DBs of different symmetries are found using this approach.

Self-diffusion in liquid alloys of Ni-Al and Ti-Al systems, depending on the concentration of the components, as well as in pure metals Al, Ni, and Ti, has been studied using the molecular dynamics method.