Volume 32, Issue 3

Some inverse problems related to mathematical modelling of hydrophysical fields in water areas (seas and oceans) under the presence of ‘liquid’ (open) boundaries are studied and solved numerically in the paper. Numerical solution algorithms for these problems are based on procedures of variational data assimilation.

A Monte Carlo algorithm admitting parallelization is constructed for estimation of probability moments of the spectral radius of the operator of the integral equation describing transfer of particles with multiplication in a random medium. A randomized homogenization method is developed with the same aim on the base of the theory of small perturbations and diffusive approximation. Test calculations performed for a one-group spherically symmetric model system have shown a satisfactory concordance of results obtained from two models.

Special asymptotic approximations, namely, the first and second homogenized problems are proposed for the boundary value problem describing a stationary radiative-conductive heat transfer in a two-dimensional system of heat-conducting plates of thickness ɛ separated by vacuum layers. The unique solvability of homogenized problems is proved, the comparison theorem is established, and estimates of solutions are obtained. The first homogenized problem has the error estimate of order O(ε ) $ O(\sqrt{\varepsilon}) $ .

The mathematical model of hydrothermodynamics of the Baltic Sea is considered with the pole moved to a neighbourhood of St. Petersburg in order to improve the horizontal resolution in the Gulf of Finland. The problem of variational assimilation of mean daily data for sea surface temperature (SST) is formulated and studied for the given type of calculation grid of this model. A new algorithm of solution of the inverse problem for reconstruction of the heat flux on the interface of two media is proposed on the base of variational assimilation of satellite observation data. Results of numerical experiments are presented for reconstruction of heat fluxes in the problem of variational assimilation of mean daily observations of SST data.

A statistical modelling algorithm is constructed for solution of the nonlinear kinetic Boltzmann equation based on the projection method. Hermite polynomials are used as an orthonormalized basis. The algorithm was tested on calculations for the problem of one-dimensional relaxation of gas with a known solution.

In the present paper we propose a mathematical model and study numerically the functioning of the p53 tumors suppressor protein and the direct positive connection of microRNA molecules related to it. The adequacy of the model is confirmed by qualitative concordance of calculation results with the experimental data concerning transactivation of specific p53-dependent microRNAs. Estimates of stability of individual and p53-mediated properties of microRNA as factors in diagnosing cancer and neurodegenerative diseases are given within the framework of the accepted model.