Volume 2, Issue 3

In this study, we examine one of the approaches to the investigation and solution of the Stokes equations: an original problem is regarded as an inverse problem and then reduced to a problem of optimal control. Iteration algorithms are suggested and results of numerical experiments are presented.

In this paper numerical methods for solving ’fuzzy partial differential equations’(FPDE) is considered. Fuzzy reachable set can be approximated by proposed methods with complete error analysis which is discussed in details. The methods are illustrated by solving some linear and nonlinear FPDE.

The distribution of electromagnetic fields, forces and temperatures induced by a three- phase axially-symmetric system of electric current in a conducting cylinder of finite length has been calculated. An original method was used to calculate the mean values of electromagnetic forces. The magnetohydrodynamic (MHD) flow of viscous incompressible liquid and distribution of temperature are obtained by the finite-difference method, using monotonic finite difference schemes.

To solve the boundary problems for singular ordinary differential equations of the second order the functional-discrete (FD-) method is proposed. The method gives the possibility of obtaining an approximate solution in the numerical-analytical form in the whole half-axis with a guaranteed accuracy. Sufficient conditions of convergence of the method at a rate of a geometrical progression are found. The two-sided approximations are established and an explicit evaluation of the two-sided estimation width is given. In conclusion the results of numerical experiments obtained by the system of computer algebra are presented.

The application of the dynamic adaptation method to the numerical solution of multidimensional Stefan problems with explicit tracking of interfaces is considered. The dynamic adaptation method is based on the idea of transition to any non-stationary coordinate system. The results of the numerical experiment on modelling the pulsed action of high-energy fluxes on metals are presented.

The elliptic equation under the nonhomogeneous Dirichlet boundary condition in 2D and 3D cases is solved. A rectangular nonuniform partition of a domain and polylinear finite elements are taken. For the interpolant of the exact solution u , a priori error estimates are proved provided that u possesses a weakened smoothness. Next error estimates are in terms of data. An estimate is established for the right–hand side f of the equation having a generalized smoothness. Error estimates are derived in the case of f which is not compatible with the boundary function. The proofs are based on some propositions from the theory of functions. The corresponding lower error estimates are also included; they justify the sharpness of the estimates without the logarithmic multipliers. Finally, we prove similar results in the case of 2D linear finite elements and a uniform partition.