Volume 22, Issue 3

A method for obtaining an approximate response matrix for a discrete stochastic dynamical system in the case of an external action from a subspace is considered. The method is based on the calculation of approximate covariation matrices. The dependence of errors on the parameters and the dimension of the problem is known for this method. The main goal of the paper is experimental verification of this dependence. In addition, some problems of software implementation of the method, the choice of subspaces, its connection to an effective number of degrees of freedom are discussed.

This paper is devoted to the construction of a mathematical model of the endocrine system regulatory network. The model describes interdependence between the concentrations of forty-five hormones, carrier proteins, their complexes, and several physiological parameters. The integrated network including six endocrine subsystems or axes (somatotropic, gonadotropic, thyroid, hypothalamicpituitary-adrenal axis (HPA axis), mineral-corticoid and the axis of calcium homeostasis) is described. The model can be applied to the investigation of development, aging, long-term adaptation and approaches to correction of unfavorable dynamics of these complex physiological processes.

A grid approximation of the Dirichlet problem is considered on a segment for a parabolic convection-diffusion equation; the high derivative of the equation contains a parameter ε taking arbitrary values from the half-interval (0,1]. A difference scheme on a posteriori adaptive grids is constructed for the boundary value problem. The classic approximation of the equation on uniform grids in the main domain and also in domains refined to improve the accuracy of the grid solution are used. Such subdomains are determined based on majorant functions for singular components of the solutions to the boundary value problem and the difference scheme. Special schemes on a posteriori piecewise-uniform grids are constructed, these schemes permit to obtain approximations convergent in the whole grid domain ' almost e-uniformly ', namely, with an error weakly depending on the value of the parameter ε; the scheme converges ε-uniformly with the first order of accuracy (up to logarithmic cofactors) outside a sufficiently small neighborhood of the 'outlet' part of the boundary through which the characteristics of the limit equation leave the domain.

A sixth-order accurate composite grid method for solving a mixed boundary value problem for Laplace's equation on staircase polygons (the polygons may have polygonal cuts and be multiply connected) is constructed and justified. The O( h 6 ) order of accuracy for the number of nodes O( h –2 ln h –1 ) is obtained by using 9-point scheme on exponentially compressed polar and square grids, as well as constructing the sixth-order matching operator connecting the subsystems. This estimate is obtained for requirements on the functions specifying the boundary conditions which cannot be essentially lowered in C k,λ . Finally, we illustrate the high accuracy of the method in solving the well known Motz problem which has singularity due to abrupt changes in the type of boundary conditions.