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Approximate solution of contact problem for some systems of elliptic equations in rectangular domain

Author: giorgi danelia

Mathematical modeling of variou practical problems requires the study of boundary and initial-boundary value problems posed on multi-structures, i.e. on the areas of various geometric characteristics. Mathematical modeling of multi-structural constructions and study of appropriate mathematical problems have both theoretical and practical importance. For practical application of special interest are the multi-structures composed of plates, shells, rods and other sub-structures. Two decades ago the researchers got interested in the construction and investigation of mathematical models for multi-structures. The study of mathematical models for multi-structures is associated with certain difficulties. We should mention that the difficulty is caused by the fact that contact conditions for constituent parts of multi-structures do not allow to produce direct splitting, and therefore, the use of an iterative method and its combination with decomposition method becomes necessary. It should be noted that the methods known in literature for construction of mathematical models for multi-structures and corresponding low-dimensional models can be used only in case of smallness compared to other dimensions of thickness and width of the sub-structures in the given multi-structure while the mentioned values may not be sufficiently small for the real bodies. Therefore, while studying various complex multi-structural constructions by mathematical methods and for approximate solution of the corresponding boundary-value problems, it is justified to use the method of successive approximation when the succession of approximate solutions of low-dimensional problem obtained at each step tends to exact solution of initial-value problem. The present master’s work deals with the boundary-contact problems for the elementary multi-structures. Particularly, approximate solution of the boundary-contact problem for the system of some elliptic equations in case of rectangular area and the boundary-contact problem for the "bridge-shaped" multi-structural flat body. The first section of §1 deals with the boundary-contact problem for the system of second order ordinary differential equations with common border for two united intervals. In the left end of the left interval and in the right end of the right interval Dirichlet boundary conditions are given and on the common point we have a smooth connection, i.e. solutions and their derivatives are equal. The solution of Dirichlet problem is written in each interval in the explicit form. In this case, it is assumed that the value of the solution in common point is yet unknown. It is explicitly determined with account of the contact conditions. In the second section of §1 the difference analogue of the continuous problem considered in the previous section is presented. For solution of difference system the factorization method is used. Here the essential fact is that on common point contact conditions known to continuous problem are transferred to the differential problem. The third section of §1 deals with the second order ordinary differential equations for the system boundary-contact problem in the middle of three common boundary points. As in the previous sections, there we also have in common points have continuous junction. For solution of this problem the approach is the same as in the first section (in case of two unified intervals). Solutions are written in explicit form in case of Dirichlet problem where boundary values in adhesion points are yet unknown values. With account of the contact conditions we obtain a system with two unknown equations which has only one solution. The second paragraph is devoted to the approximate solution of boundary-contact problem for the system of Poisson's equations with common border for two united rectangular areas. The first item of §2 considers the difference method for approximate solution of the given task. A standard scheme "cross" is written in each area. The system of the obtained difference equations is solved in a combination of factorization method with iterative method. Factorization is conducted along the x axis with account of the contact conditions, as we have it for one-dimensional case. In the second item of §2 discusses semi-discrete method for approximate solution of the problem posed in the previous paragraph. If discretization is performed according to y variable, then we get second order ordinary differential equations for the boundary-contact problem with common border for two united intervals. Solution of this problem is written in explicit form (in analogue with scalar equation the first item of § 1) by means of exponential matrix function. The inverse matrix entering this explicit representation the existence of functions is easily proved because the system matrix is symmetric and positively determined. The third paragraph studies boundary-contact problem for multi-structural flat body of “bridge shape”. Particularly, two membranes of rectangular shape connected with string. For membrane the classical linear boundary problems are regarded. The equation of the membrane bending represents Poisson equation and the string is described by nonlinear integro-differential equation of Kirkhoff type. Those boundaries of membrane on which a non-linear string is bounded, are free. Mathematically this means that the derivatives of the unknown functions in relation to x along these boundaries equal zero. For the given problem numerical solution algorithm is constructed. The program was created in the Matlab and numeric experiments were performed.

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