Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2014-04-11 End Date: 2016-04-11
It is known that the process of penetrating of electromagnetic field in the substance is described by the Maxwell complex system of nonlinear partial differential equations. The main features of these systems are expressed in fact that they contain equations of different kinds, which are strongly connected to each other.
Naturally arises question of approximate solution of these problems, which also are connected with serious complexities as well. Therefore, investigation and numerical solution of such problems of diffusion are tasks of importance.
Penetrating into a material, variable magnetic field induces variable electronic field in it, which causes the appearance of currents. The currents bring us to the heating of the material and rising of its temperature. For large oscillations of temperature the dependence has to be taken into consideration. One must note that this system of nonlinear partial differential equations can be reduced to the integro-differential model.
It should be noted that this and such type integro-differential models occur in mathematical simulation of many physical, chemical, technologic, aerodynamic, economic and other processes.
The Maxwell’s system and its equivalent integro-differential model is complex for the theoretical investigation and practical applications for solution of concrete diffusion problems. Therefore, there are also used its simplified variants.
Many authors study the convergence of finite difference schemes and Galerkin approximation for integro-differential models described in the project and problems similar to them. As it was mentioned, integro-differential model is complex and investigations mostly are made for one-dimensional and one-component magnetic field, but some researches for two-component magnetic field case and for multidimensional scalar case are done as well. Note that in these cases, nonlinearity is specific.
In the present project we will consider more wide range of nonlinearity as it was already studied. For the mentioned integro-differential systems will be studied solvability, uniqueness and asymptotic behavior as $$t \rightarrow \propto$$ of the solutions of initial-boundary value problems. Semi-discrete and discrete analogous will be constructed and studied. Galerkin and finite element methods for the investigated problems will be considered as well. Special attention will be paid to the construction of the finite difference schemes and proof of their convergence. On the basis of constructed algorithms the software pakages will be created. Numerous numerical experiments and their analysis will be carried out.
For the investigations, planned in the project, modern methods of numerical analysis, mathematical physics and nonlinear analysis will be used. The possibility of graphical visualization will be used as well. To develop software packages, OOP languages will be used.
The Laboratoire Jacques-Louis Lions, in which visits are planned, is world famous institution for its huge contributions to numerical methods for partial differential equations. There successfully realized applied investigations by using of finite differences, finite element, Galerkin methods and etc. Mentioned circumstances are giving possibility to investigate Maxwell systems considered in the project as well as great opportunity for Georgian and French teams’ future collaboration.
From abovementioned descriptions and from reference it is clear that to the study of the nonlinear integro-differential models considered in the project, together with other scientists are devoted the works of project director from Georgian part Prof. T. Jangveladze. Scientific cooperation between him and Prof. Z. Kiguradze has been begun many years ago. Already investigated cases of problems, foreseen in the project, turned out to be interesting for the some part of scientists.