Mathematical models of diffusion processes lead to nonstationary partial differential and integro-differential equations and systems of these equations with the corresponding initial-boundary conditions. Most of these models, as a rule are nonlinear and multi-dimensional.

In the project quantitative as well as qualitative characteristics of some nonlinear diffusion systems will be studied. These models arise at the mathematical modeling of process of electromagnetic field propagation in the medium. Investigated differential and integro-differential models are complex and investigations mostly are made for one-dimensional and one-component magnetic field. But some researches for two-component magnetic field case are done as well. Note that in already carried out investigations, nonlinearity is specific. Therefore, it is important to continue investigation of multi-dimensional as well as one-dimensional cases for these models. In the project, algorithms for approximate solution of these problems will be constructed and investigated. Special attention will be paid to construction of discrete analogs corresponding to the model, as well as to the construction and analysis of decomposition algorithms with respect to physical processes.

Uniqueness and long time asymptotic behavior of solutions of the different types initial-boundary value problems will be studied. Reduction of multi-dimensional models to one-dimensional problems by using the decomposition method. Construction and investigation of algorithms for approximate solutions will be carried out for these models too.

On the basis of approximate solution algorithms, constructed for problems discussed in the project, software packages will be created, by using of which numerical calculations for different modal problems will be fulfilled and analysis of the obtained numerical results will be carried out.

Project members:

• Mikheil Gagoshidze (Principal Investigator)