Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2013-04-14 End Date: 2016-04-14
In mathematical modeling of many natural phenomena and processes many nonlinear nonstationary equations are received. Often they can be described by the boundary and initial-boundary value problems posed for parabolic and hyperbolic differential and integro-differential equations. Their linearized variants are studied sufficiently, but they do not describe these phenomena with required exactness. Therefore, investigation of nonlinear mathematical models is actual both in practical and in theoretical aspects till today.
The initial and mixed problems posed for hyperbolic equations are studied well enough. As regards to the Cauchy-Goursat and Darboux boundary value problems posed in angular domains for nonlinear wave equation they are on the initial stage of study and so retain actuality. This especially concerns the study of conic body in supersonic flow and string oscillation in viscous liquid which can be reduced to the Darboux problem with the boundary conditions on non-characteristic curves starting from a common point.
One very important nonlinear nonstationary model is obtained in the mathematical modeling of processes of electromagnetic field propagation in the medium. In the quasistationary case, the corresponding system of Maxwell’s equations can be reduced to the equivalent integro-differential form.
It is important to study the questions of existence, uniqueness, asymptotic behavior of solutions and numerical resolution of initial-boundary value problems for such type models. In this direction some investigations are already performed and the goal of the project is to continue and deepen these researches. The mentioned problems mainly are conditioned by real physical problems, but part of them is the result of natural mathematical generalization and abstraction as well.
The study of one important nonlinear integro-differential model that arises in thermoelasticity quasistatic contact problems is planned as well.
The goal of project is to study the following questions to obtain some structural and qualitative properties of boundary and initial-boundary value problems for nonlinear hyperbolic type differential and parabolic type integro-differential equations: the influence of a nonlinearity nature on the solvability of boundary and initial-boundary value problems in view of the existence, non-existence (blow-up, gradient catastrophe, estimate of life-span) and uniqueness of solution; the establishment of the influence of nonlinear source and dissipative members on the correctness of problems; asymptotic behavior of solutions; construction and investigation of finite-difference schemes; carrying out numerical realization, analysis of the obtained results and comparison with theoretical conclusions.
The novelty of research consists in coalescence of methods of classical and modern theories of differential equations, nonlinear and numerical analysis.
One must note that the problems posed in the project are quite problematic and their solution is connected with certain scientific risks. But the knowledge and experience of the project participants in studying nonstationary nonlinear models make a premise that the objectives of the project will be successfully achieved, of course if the project will be funded accordingly.
The project aims at complete research for abovementioned problems, which will make the structural and qualitative theory and numerical methods for boundary and initial-boundary value problems for nonlinear nonstationary equations more precise.