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Semiclassical approximation formalism is developed for the multidimensional Fokker–Planck–Kolmogorov equation with non-local and nonlinear drift vector with respect to a small diffusion coefficient D, D→0, in the class of trajectory concentrated functions. The Einstein−Ehrenfest system of (0, M)-type is obtained. A family of semiclassical solutions localized around a point driven by the Einstein−Ehrenfest system accurate to O(D^(M+1)/2) is found.

We developed an algorithm for fast simulation of VLSI CMOS (Very Large Scale Integration with Complementary Metal-Oxide-Semiconductors) with an accuracy control. The algorithm provides an ability of parallel numerical experiments in multiprocessor computational environment. There is computation speed up by means of block-matrix and structural (DCCC) decompositions application. A feature of the approach is both in a choice of moments and ways of parameters synchronization and application of multi-rate integration methods. Due to this fact we have ability to estimate and control error of given characteristics.

Methods of the solution of a problem of focal approximation — approach on point-by-point given smooth closed empirical curve by multifocal lemniscates are investigated. Criteria and convergence of the developed approached methods with use of the description, both in real, and in complex variables are analyzed. Topological equivalence of the used criteria is proved.

Necessary and sufficient conditions for the existence of solutions of nonlinear nonautonomous periodic problem for Hill’s equation in the case of parametric resonance. A characteristic feature of the task is the need of finding, as desired solution, and the corresponding eigenfunction, which ensures solvability of the periodic problem for Hill’s equation in the case of parametric resonance. To construct solutions of the periodic problem for Hill’s equation and the corresponding eigenfunction in the case of parametric resonance proposed iterative scheme, based on the method of simple iterations with used list-square technics.

Modification of the lattice Boltzmann method for computation of viscous incompressible fluid flows is proposed. The method is based on the splitting of differential operator in Navier–Stokes equation and on the idea of instantaneous Maxwellisation of distribution function. The method is based on explicit schemes and didn’t have any problems with parallelization of computations. The stability of the method is demonstrated using von Neumann method in a wide range of input parameter values. The efficiency of the method proposed is demonstrated on the solution of the problem of 2D lid-driven cavity flow.

The paper provides a solution of a task of calculating the parameters of a Rician distributed signal on the basis of the maximum likelihood principle in limiting cases of large and small values of the signal-tonoise ratio. The analytical formulas are obtained for the solution of the maximum likelihood equations’ system for the required signal and noise parameters for both the one-parameter approximation, when only one parameter is being calculated on the assumption that the second one is known a-priori, and for the two-parameter task, when both parameters are a-priori unknown. The direct calculation of required signal and noise parameters by formulas allows escaping the necessity of time resource consuming numerical solving the nonlinear equations’ s system and thus optimizing the duration of computer processing of signals and images. There are presented the results of computer simulation of a task confirming the theoretical conclusions. The task is meaningful for the purposes of Rician data processing, in particular, magnetic-resonance visualization.

The paper provides a solution of the two-parameter task of joint signal and noise estimation at data analysis within the conditions of the Rice distribution by the techniques of mathematical statistics: the maximum likelihood method and the variants of the method of moments. The considered variants of the method of moments include the following techniques: the joint signal and noise estimation on the basis of measuring the 2-nd and the 4-th moments (MM24) and on the basis of measuring the 1-st and the 2-nd moments (MM12). For each of the elaborated methods the explicit equations’ systems have been obtained for required parameters of the signal and noise. An important mathematical result of the investigation consists in the fact that the solution of the system of two nonlinear equations with two variables — the sought for signal and noise parameters — has been reduced to the solution of just one equation with one unknown quantity what is important from the view point of both the theoretical investigation of the proposed technique and its practical application, providing the possibility of essential decreasing the calculating resources required for the technique’s realization. The implemented theoretical analysis has resulted in an important practical conclusion: solving the two-parameter task does not lead to the increase of required numerical resources if compared with the one-parameter approximation. The task is meaningful for the purposes of the rician data processing, in particular — the image processing in the systems of magnetic-resonance visualization. The theoretical conclusions have been confirmed by the results of the numerical experiment.

Equations of motion in Newtonian and Hamiltonian forms are used for classical molecular dynamics simulation of particle system time evolution. When Newton equations of motion are used for finding of particle coordinates and velocities in $N$-particle system it takes to solve $3N$ ordinary differential equations of second order at every time step. Traditionally numerical schemes of Verlet method are used for solving Newtonian equations of motion of molecular dynamics. A step of integration is necessary to decrease for Verlet numerical schemes steadiness conservation on sufficiently large time intervals. It leads to a significant increase of the volume of calculations. Numerical schemes of Verlet method with Hamiltonian conservation control (the energy of the system) at every time moment are used in the most software packages of molecular dynamics for numerical integration of equations of motion. It can be used two complement each other approaches to decrease of computational time in molecular dynamics calculations. The first of these approaches is based on enhancement and software optimization of existing software packages of molecular dynamics by using of vectorization, parallelization and special processor construction. The second one is based on the elaboration of efficient methods for numerical integration for equations of motion. A procedure for constructing of explicit, implicit and symmetric symplectic numerical schemes with given approximation accuracy in relation to integration step for solving of molecular dynamic equations of motion in Hamiltonian form is proposed in this work. The approach for construction of proposed in this work procedure is based on the following points: Hamiltonian formulation of equations of motion; usage of Taylor expansion of exact solution; usage of generating functions, for geometrical properties of exact solution conservation, in derivation of numerical schemes. Numerical experiments show that obtained in this work symmetric symplectic third-order accuracy scheme conserves basic properties of the exact solution in the approximate solution. It is more stable for approximation step and conserves Hamiltonian of the system with more accuracy at a large integration interval then second order Verlet numerical schemes.

The paper has methodical character; it is devoted to three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Special attention was payed to the matter conservation law and the offensive effect of excessive hexagonal symmetry.

It has been shown that in contrary to finite-difference approach, in spite of terminological equivalence of CA local transition function to the pattern of computing double layer explicit method, CA approach contains the replacement of matrix technique by iterative ones (for instance, sweep method for three diagonal matrixes). This suggests that discretization of boundary conditions for CA-cells needs more rigid conditions.

The correct local transition function (LTF) of the boundary cells, which is valid at least for the boundaries of the rectangular and circular shapes have been firstly proposed and empirically given for the hexagonal grid and the conservative boundary conditions. The idea of LTF separation into «internal», «boundary» and «postfix» have been proposed. By the example of this problem the value of the Courant-Levy constant was re-evaluated as the CA convergence speed ratio to the solution, which is given at a fixed time, and to the rate of the solution change over time.

A set of implicit difference schemes on the five-pointwise stensil is under construction. The analysis of properties of difference schemes is carried out in a space of undetermined coefficients. The spaces were introduced for the first time by A. S. Kholodov. Usually for properties of difference schemes investigation the problem of the linear programming was constructed. The coefficient at the main term of a discrepancy was considered as the target function. The optimization task with inequalities type restrictions was considered for construction of the monotonic difference schemes. The limitation of such an approach becomes clear taking into account that approximation of the difference scheme is defined only on the classical (smooth) solutions of partial differential equations.

The functional which minimum will be found put in compliance to the difference scheme. The functional must be the linear on the difference schemes coefficients. It is possible that the functional depends on net function – the solution of a difference task or a grid projection of the differential problem solution. If the initial terms of the functional expansion in a Taylor series on grid parameters are equal to conditions of classical approximation, we will call that the functional will be the generalized condition of approximation. It is shown that such functionals exist. For the simple linear partial differential equation with constant coefficients construction of the functional is possible also for the generalized (non-smooth) solution of a differential problem.

Families of functionals both for smooth solutions of an initial differential problem and for the generalized solution are constructed. The new difference schemes based on the analysis of the functionals by linear programming methods are constructed. At the same time the research of couple of self-dual problems of the linear programming is used. The optimum monotonic difference scheme possessing the first order of approximation on the smooth solution of differential problem is found. The possibility of application of the new schemes for creation of hybrid difference methods of the raised approximation order on smooth solutions is discussed.

The example of numerical implementation of the simplest difference scheme with the generalized approximation is given.