Start Date: 2008-01-01 End Date: 2010-01-01
Mathematical modeling, investigation and numerical solution of applied problems are one of the actual problems of modern mathematics.
Electromagnetic field diffusion process in the substance the coefficient of electroconductivity of which essentially defends on the temperature belong to such problems. It is known, that such processes are accompanied with heat emission, which changes the conductivity of the substance and influences on the process of diffusion. The process of penetrating of electromagnetic field in the substance is described by the Maxwell system of nonlinear partial differential equations.
The above mentioned system is difficult for theoretic investigations as well as for studying concrete diffusion models, so, its simplified variants are often used. The important models of the process of diffusion are also the systems of partial differential equations describing the process of vein formation in meristematic tissues of young leaves, the mathematical model of the temperature conditions of the atmosphere's surface layer and system describing of the vortex fields in the atmosphere. Therefore, mathematical modeling, investigation and numerical solution of such problems of diffusion are tasks of importance.
The purpose of the project is investigation and numerical solution of some classes of nonlinear partial differential and integro-differential equations. The investigated models occur when describing above mentioned processes as well as other real processes. The main attention is paid to the penetration of the electromagnetic field into the substance.
Reduction of the system of Maxwell equations to the integro-differential models was performed in work Gordeziani D.G., Dzhangveladze T.A., Korshia T.K. Existence and Uniqueness of a Solution of Certain Nonlinear Parabolic Problems. Dfferential'nye Uravnenyia, 1983, V.19, N7, p.1197-1207 - which was followed by numerous scientific publications in Georgia as well as abroad. In these works the questions of existence, uniqueness, asymptotic behavior of the solution and approximate solution of the initial-boundary value problems are studied. The investigations are performed for some cases of nonlinearity.
The role of the authors of the project to solve these issues is essential and it is described in the numerous scientific works, which are published in famous journals.
The obtaining of comparison theorems for initial boundary value problems to different nonlinear models and there discrete analogous is important. In this direction scientific literature is rich too. These questions are only on the start position for the models described by us and one of the purposes of the project is to develop situation in this direction.
One must particularly note the scientific interest which is caused by so called nonlocal problems . In this regard the investigation and numerical solution of Bitsadze-Samarski type problems are important. These questions for linear ordinary equations as well as linear partial differential equations are still problematic. The consideration of possibility of variational formulation and investigation by decomposition methods of them is important.
It should be noted that, participants of the project have some experience and scientific potential in the studying of above mentioned problems too.