| Abstract [eng] |
In this dissertation, the method of dimension reduction is applied to some mathematical models. Due to it, the time required to find a solution to a differential equation is reduced a few times, however, some accuracy is lost this way. Two models of heat conduction are presented, where in some part of domain the dimension is reduced from 3 or 2 to 1. Then a FVM ADI numerical scheme is justified for obtained hybrid dimension models. Some important properties of this scheme are proved. Due to nonstandard additional conditions, the classical methods are modified. A unique existence of a numerical solution is proved. Also, a fourth order partial differential equation is investigated, which was derived from a model of viscous fluid flowing through an elastic vessel. Here the dimension is reduced to 1 in the whole domain. The specification of problem is discussed along with some numerical schemes and their properties. For the problems solved in this dissertation, provided numerical experiments agree well with theoretical estimates and justify the practical usage of constructed schemes. |
