Start Date: 2017-11-23 End Date: 2017-11-25
In the mathematical modeling of many natural processes systems of nonlinear partial differential equations arise very often. Investigation and approximate solution of the initial-boundary value problems posed for these systems are the actual sphere of contemporary mathematical physics and numerical analysis.
An economical finite-difference scheme for one system of nonlinear multi-dimensional partial differential equations is constructed. In a particular case, the model can be used as a mathematical simulation of the process of vein formation in meristematic tissues of young leaves. The stability and convergence of the developed scheme are proven. Numerical experiments verifying theoretical findings for the three-dimensional case are carried out. The appropriate graphical illustrations are given.
The second important model is obtained at the mathematical modeling of processes of electromagnetic field penetration in the substance. In the quasi-stationary approximation, this diffusion process, taking into account Joule law is described by a nonlinear system of Maxwell equations. For a more thorough description of electromagnetic field propagation in the medium, it is desirable to take into consideration different physical effects, first of all – the heat conductivity of the medium. Special attention is paid to the construction of discrete analogs, corresponding to one-dimensional models as well as to the construction and analysis of decomposition algorithms with respect to physical processes. The above-mentioned decomposition is defined by splitting this model into two parts: in the first part the Joule heat release is taken into account and in the second – part the heat conductivity of the medium is considered.