** Number 1, pp.1-92 Number 2, pp.93-198 Number 3, pp.199-293 Number 4, pp.295-394**

**Averina T.A. Statistic algorithm of simulation of dynamic systems with random structure **(

The paper considers dynamic systems with random structure. The statistic algorithm was constructed for probabilistic analysis of two types of systems with random structure. It based on numerical methods for solving the Stochastic Differential Equations (SDEs). There are given results of numerical experiments.

**
Voytishek A.V., Golovko N.G., Shkarupa E.V. Error estimation for
multi-dimensional analogue of the polygon of frequencies method **
(

Decomposition to three components for L_2-error of multi-dimensional analogue of the polygon of frequencies method is obtained. For every component an upper bound is constructed. The statement about the finiteness of maximum of variances of stochastic estimates in grid nodes is derived. Upper bounds for displacements of estimates in nodes for C-approach and L_2-approach are obtained. On that basis it is shown that the application of smooth approximations of the solution for the polygon of frequencies method is inexpedient.

**V.L. Leont'ev
About convergence of a mixed variational-grid method **(

The variational-grid method connected to mixed variational principle and to the approximation of the exact solution by orthogonal finite functions is considered. Such functions differ from other orthogonal finite functions by simplicity of structure and symmetry and consequently simplify the algorithm of a method. The convergence of the approximate solutions in a task of mathematical physics and in a plane task of the theory of elasticity is investigated. The estimations of speed of their convergence are established. The orthogonal finite basic functions define the structure of the system of the grid equations of the variational-grid method, which supposes the exception of a part of unknown nodal quantities. It makes possible the use of classical technique of the research of the convergence of difference schemes and eliminates the basic lack of the mixed variational-grid methods, which is connected to the increased number of nodal unknown quantities in comparison with methods based on variational principles for convex functionals. Thus all advantages of the mixed methods, caused by independent approximation of the unknown functions and their derivatives are kept.

**
V.V. Smelov
A correct version of S_n-method in transport radiation theory **(

The rise of the local instability of the well-known *S_n*-method is
explained. Another absolute stability version of this method is proposed.

**
S.A. Uhinov, D.I. Yurkov
Monte Carlo estimates for parametric derivatives of polarized radiation **(

In the given paper, the results of theoretical and numerical research of parametric derivatives of double local estimate of the Monte Carlo method for the vectorial transfer equation dealing with polarization in the spherical atmosphere of the planet are presented.

**
J.H. He
Approximate analytical solution for certain strongly nonlinear
oscillations by the variational iteration method** (

In this paper, a new kind of analytical method of nonlinear problem solving called variational iteration method is described and used to give approximate solutions for certain strongly oscillations. In this method, a correction functional is constructed via a general Lagrange multiplier which can be identified optimally via the variational theory. The proposed technique does not depend on the small parameter assumption and therefore can overcome the disadvantages and limitations of the perturbation techniques. Some examples reveal that even the first-order approximates are of high accuracy, and are uniformly valid not only for weakly nonlinear systems, but also for strongly ones.

**
G.I. Shishkin** (

Grid approximations with an improved rate of convergence for singularly perturbed elliptic equations in domains with characteristic boundaries

On a rectangle, we consider the Dirichlet problem for
singularly perturbed elliptic equations with convective terms
in the case of characteristics of the reduced equations which are
parallel to the sides. For such convection-diffusion problems
the uniform (with respect to the perturbation parameter \eps)
convergence rate of the well-known special schemes on piecewise
uniform meshes is of order not higher than one (in the uniform L_{\infty}-norm). For the above problem, based on asymptotic
expansions of the solutions, we construct schemes that converge \eps-uniformly with the rate
\Oh{N^{-2}\lnēN}, where * N*
defines the number of mesh points with respect to each variable.
For not too small values of the parameter we apply classical
finite difference approximations on piecewise uniform meshes condensing
in boundary layers; for small values of the parameter we use
approximations of auxiliary problems, which describe the main terms of
asymptotic representation of the solution in a neighbourhood of
the boundary layer and outside of it. Note that the computation
of solutions of the constructed difference scheme is simplified
for sufficiently small values of the parameter \eps.

**Artemiev S.S., Yakunin M.A.**

**Stochastic wave models of prices of various financial instruments ** (

We consider various mathematical models of prices of stocks, currencies, and financial futures in the form of the stochastic differential equations, with allowance for the wave nature of the dynamics of prices on the stock, the exchange, and the futures markets. The way of calculation of estimates of parameters of one model and an example of calculation by the real price observations are presented.

**Bakushinskii A.B., Kokurin M.Yu., Yusupova N.A. **

We propose and study a class of iterative methods of the Newton type for approximate solution of nonlinear ill-posed operator equations without the regularity property. A possible

The equilibrium and the non-equilibrium behaviours of the uniform tandem with jump-like service rate in each queue are examined. The uniform tandem represents a specific multiplicative queueing network and involves a sequence of queues being identical with respect to customer's services. It is a suitable mathematical model providing analysis of the effect of the jump-like service rate on the probability-temporal measures related, in particular, to transient processes.

For the equilibrium behaviour, the state space structure is determined, and the Laplace-Stieltjes transform of the cycle times distribution is obtained. For the non-equilibrium behaviour, the recurrence solution of the Kolmogorov differential equations is developed, the transient process time is evaluated, the integral and the phase trajectories for the respective Markov process are investigated.

The paper deals with the three-dimensional Dirichlet problem for a second order elliptic equation in a domain with a smooth curvilinear boundary. To construct a finite element scheme, the embedded subspaces of basic functions are used without strict embeddedness of a sequence of spatial triangulations. It is proved that the discretization error is of the same order as in the case of the standard piecewise linear elements on a polyhedron. For solving the obtained system of linear algebraic equations on a sequence of grids, the cascadic organization of two iterative processes is applied providing a simple version of a multigrid method without any preconditioning or restriction to coarser grids. The cascadic algorithm starts on the coarsest grid where the grid problem is directly solved. On finer grids, approximate solutions are obtained by an iterative process, where interpolation of an approximate solution from the previous coarser grid is taken as an initial guess. It is proved that the convergence rate of this algorithm does not depend on the number of unknown values as well as on the number of grids.

A three-point difference scheme with an infinite number of grid nodes is considered. The method of reduction of such scheme to the scheme with a finite number of grid nodes is investigated. This method is based on the extraction of a set of solutions satisfying the limit conditions on infinity. The results of numerical experiments are discussed.

The paper deals with a new class of explicit schemes for a solution to parabolic problems describing multi-scale processes. The method is based on a compound scheme of the first order of accuracy and predictor-corrector explicit scheme of the second order. The stability theorems are proven, and a numerical example is considered.

Monte Carlo methods offer the possibility to estimate derivatives of a solution to a boundary value problem even without estimating the solution to the problem making them attractive in various practical applications. In this paper, Monte Carlo methods are applied to estimate derivatives of a solution to the third boundary value problem of the diffusion equation with a constant complex parameter with respect to the parameter as well as with respect to a spatial variable. Moreover, the estimation of the spatial derivative of non-centric Green's function for the diffusion operator in a ball is used. All the derivatives estimators are obtained by the method of "walking on spheres" with boundary reflection. Based on the derivatives estimators of the solution to the diffusion equation problem, derivatives of a solution of the heat equation boundary value problem with respect to the time variable, the spatial variable, and a constant real parameter are obtained.

An extensive literature is devoted to the ill-posed problems connected with a nonlinear operator and differential-operator equations. A regularization method is usually constructed by using the "operator" approach and special properties of the problem operator (for instance, monotonicity). In this paper, stable approximate solutions of an ill-posed differential problem are constructed by a method of the quasi-inversion type. The convergence of the constructed approximate solutions to the exact solution of the initial problem is investigated.

In the present work, the stability in the space *l_p*, *1<p\leq\infty*,
for a wide class of difference analogs of kinetic transport as well as for the Carleman nonlinear system in the Baltazar
equation theory has been proved. The stability in the norm of the space *l_p*
results, as a particular case, in the stability in *l_2*, which coincides with the stability in the energy space, and
at *p=\infty*, with the norm in the space *C*. In this case, the result is gained in the manner
similar to the methods of obtaining * a priori* estimations in the norm of the space *L_p* for differential problems by themselves.

**Amelkin V.A. **

Sets of binary and

**Danaev N.T., Smagulov Sh.S., Tukenova L.M.
On one class of iterative schemes for solving the Navier-Stokes equations **(

A class of numerical iterative schemes for
incompressible Navier--Stokes equations is considered.
The iterative algorithms are proposed and problems of stability
and convergence of these algorithms are investigated using the method of *a
priori* estimates.
The results of numerical experiments are given.

**
Derevtsov E.Yu., Kashina I.G.
Numerical solution to the vector tomography problem using polynomial basis**
(

The problem of a reconstruction of a solenoidal part of a vector field in the circle is considered if its ray transform is known. Two variants of numerical solution of the problem are developed. In the first of them, a polynomial approximation of the vector field that was obtained by means of the least squares method contains an potential part. Thus a further step of solving the problem is to separate from the approximation a potential vector field by finding a solution for a homogeneous boundary value problem for the Poisson equation. Investigation of the structure of finite-dimensional subspaces of solenoidal and potential vector fields of the polynomial type allows to state a problem of determining of coefficients of the polynomial approximation of the potential part as the problem of step-by-step solving of a set of systems of linear equations of increasing dimensions. The second way consists in constructing subspaces of the basis polynomial solenoidal fields. In this case, the least squares method immediately gives a polynomial approximation of a solenoidal part of the vector field. Efficiency of the constructed algorithms is verified by the numerical simulation. The results of comparative test of the algorithms show that the accuracy of both algorithms is good and similar to one another.

**
Kovaleva I.M.
Recovery and integration of functions from the
Korobovs anisotropic class** (

The problem of approximate recovery of functions from the class *E^(r_1,...,r)_s* by means of an operator in the form of an algebraic
polynomial is considered. The algorithm based on application of the
theory of divisors in cyclotomic fields of the algebraic integers
is applied to determination of optimum factors of an operator.
The problem of approximate integration of functions from the class *E^{r_1,...,r_s}* in the domain distinct from *[0,1]^s* is considered.

**
Noskov M.V., Schmid H.J.
Minimal cubature formulae of an even degree for the 2-torus**

(

In this paper, we derive the minimal even degree formulas for the 2-torus in the trigonometric case. All such formulas are obtained by solving several matrix equations. As far as we know, this is the first approach to determine all formulae of this type. Computational results by using a Computer Algebra System are presented. They verify that up to degree 30 there is only one minimal formula of even degree (and its dual) if one node is fixed. In all the cases computed, it turned out that the known lattice rules of rank 1 are the only minimal formulas.

**
Shadimetov Kh.M.
Construction of weight lattice optimal
quadrature formulas in the space L_2^m(0,N) **
(

In this paper, the optimal
coefficients of weight quadrature formulas are explicitly found
in the space * L_2^m(0,N)* for any *m\geq 1* using the
algorithm proposed by S.L. Sobolev.

Necessary conditions for a given convergence rate of iterative methods for solution of linear ill-posed operator equations in a Banach space

We study the rate of convergence of iterative methods for solution of linear ill-posed equations with sectorial operators in the Banach space. It is stated that the power sourcewise representation condition on the initial discrepancy with an arbitrary positive exponent which is sufficient for the power estimate of the convergence rate with the same exponent is actually close to be necessary and thus cannot be essentially relaxed.

**
Malyshkin G.N.
Mathematical model of disperse medium to simulate attenuation of
non-scattered radiation
**
(

Mathematical model of a two-component random disperse medium was developed to simulate the attenuation of non-scattered radiation by disperse media. The model is based on the description of the distribution of disperse medium components along the line in the form of an alternating renewal process. Using this model, attenuation of the non-scattered radiation flux caused by a flat layer of mono-disperse sample medium with cubic and spherical grains was computed and compared to the results obtained with the effective cross-section method.

**
Makhotkin O.A. **
(

Simulation of points uniformly distributed in polygons

An algorithm for the simulation of random points uniformly distributed in polygons is considered. This algorithm uses the decomposition of polygons on triangles. The correctness of the proposed algorithm is proved, its efficiency is demonstrated on different concrete examples. The problems of computer realization for the methods of decomposition and simulation are discussed. The effective schemes of simulation algorithms are listed using a pseudo-Pascal language.

**
Namm R.V., Sachkoff S.A. **
(

On a stable duality scheme method for solution of the Mosolov and the Miasnikov problem with boundary friction

The paper deals with the construction of a stable method for solution of the Mosolov and the Miasnikov variational problem with boundary friction and can be considered as the sequel of the investigations started in [1]. An approximate solution is carried out on the basis of the iterative prox-regularization method. In this case, each auxiliary problem is coming to the search of the Lagrange functional saddle point. The error estimation is defined for the numerical solution of the problem in the case of realization of this algorithm using the finite element method on a sequence of triangulations. The results of numerical computations are presented.

**
Popov A.S. **
(

The search for the sphere of the best cubature formulae invariant under octahedral group of rotations

A new optimality criterion of the cubature formula, invariant under any symmetry group for a sphere is proposed. An essential difference of this criterion from others consists in using the main term of the cubature formula error. The work of the new criterion is demonstrated on an example of the cubature formulae invariant under the octahedral group of rotations. The table which contains the main characteristics of all the best today cubature formulae of the octahedral group of rotations up to the 35th algebraic order of accuracy is given. The weights and the coordinates of the new cubature formulae of the 26th and the 27th orders of accuracy are given to 16 significant digits.

**
Racheva M.R., Andreev A.B. **
(

Variational aspects of one-dimensional fourth-order problems with eigenvalue parameter in the boundary conditions

We study a general type of eigenvalue problems for one-dimensional fourth-order operators. The case where the spectral parameter linearly appears in the boundary conditions is discussed. It is well-known that the Galerkin methods depend on the variational formulations of the given boundary problem. The conditions, when the variational bilinear forms are symmetric and the eigenfunctions belong to an appropriate Hilbert space, are presented. Our investigation has a direct application to the eigenoscillations of the mechanical systems. The effect of the theoretical results are illustrated by some examples.

**
Shkarupa E.V. **
(

The use of quantum computer for global the integral estimation depending on a parameter

Some aspects of application of the quantum algorithms for estimation of integrals are considered. The new quantum algorithms for the global estimation of the integral which is dependent on a parameter are presented. The upper bounds of errors of the presented algorithms are obtained in C-metrics. The optimal relations between parameters of these algorithms are obtained. The comparison of computational costs of the quantum functional algorithms and Monte-Carlo functional algorithms is made.