**
Number 1, pp. 1-99
Number 2, pp. 101-216
Number 3, pp. 217-313
Number 4, pp. 315-427**

**
Aida-zade
Kamil Rajab, Abdullaev Vagif Maarif**

**
On the
numerical solution to loaded systems of ordinary differential equations with
non-separated multipoint and integral conditions
**

We propose a
numerical method of solving systems of linear non-autonomous ordinary loaded
differential equations with non-separated multipoint and integral conditions.
This method is based on the operation of convolution of integral conditions to
local conditions. This approach allows reducing the solution to the original
problem to solving the Cauchy problem with respect to a system of ordinary
differential equations and to linear algebraic equations. Numerous computational
experiments on several test problems with application of the formulas and
schemes of the numerical solution have been carried out. The results of the
experiments have shown a sufficiently high efficiency of the approach described.

**
Key words:
**ordinary
loaded differential equations system, non-separated conditions, integral
conditions, non-local multipoint conditions, sequential shift operation.

**
Aleksandrov
Vladimir
**

**
**

A new method
of solving time-optimal control problems in real time has been developed. The
method is based on the following: 1) approximating the attainability sets with a
family of hyperplanes; 2) subdividing the whole computational process into the
computations performed beforehand and those that are carried out while the
control takes place; 3) integrating differential equations only over the
displacement intervals of the control completion moment and the switching
moments. The labor-intensive characteristic of the method is evaluated.
Characteristics of calculating the optimal control of a linear system with
retarded control in real time are considered. The results of modeling and
numerical estimations are presented.

**
Key words:
**optimal
control, speed, switching moment, retardation, adjoint system, phase trajectory.

==============================================

**
Behl
Ramandeep, Kanwar V., Sharma Kapil K.**

**
New modified
optimal families of King's and Traub-Ostrowski's method
**

**
**
Based on
quadratically convergent Schröder's method, we derive many new interesting
families of fourth-order multipoint iterative methods without memory for
obtaining simple roots of nonlinear equations by using the weight function
approach. The classical King's family of fourth-order methods and
Traub-Ostrowski's method are obtained as special cases. According to the Kung-Traub
conjecture, these methods have the maximal efficiency index because only three
functional values are needed per step. Therefore, the fourth-order family of
King's method and Traub-Ostrowski's method are the main findings of the present
work. The performance of proposed multipoint methods is compared with their
closest competitors, namely, King's family, Traub-Ostrowski's method, and
Jarratt's method in a series of numerical experiments. All the methods
considered here are found to be effective and comparable to the similar robust
methods available in the literature.

**
Key words:
**nonlinear
equations, Newton's method, King's family, Traub-Ostrowski's method, Jarratt's
method, optimal order of convergence, efficiency index.

**
==============================================**

**
Vikhtenko
Ellina Mihailovna, Maksimova Nadezhda Nikolaevna, Namm Robert Viktorovich**

**
A
sensitivity functionals in variational inequalities of mechanics and their
application to duality schemes**

Characteristic properties of a sensitivity functional in the variational
inequalities mechanics on an example of a scalar Signorini problem are
investigated. Applications of sensitivity functionals in duality schemes are
considered.

**
Key words:
**scalar
Signorini problem, duality scheme, modified Lagrangian functional, sensitivity
functional.

**
Kokurin
Mikhail Yurjevich, Kozlov Alexander Ivanovich**

**
On a
posteriori approximation of a set of solutions to a system of quadratic
equations with the use of the Newton method
**

For
quadratic systems of algebraic equations we propose an algorithm generating a
posteriori estimates of a convex hull of a set of solutions using the results of
a step of the Newton method. Results of numerical tests are given.

**
Key words:
**quadratic
operator, the Newton method, a posteriori estimate, numerical range, convex
hull.

==============================================

**
Okuonghae
R.I., Ikhile M.N.O.**

**
A family of
highly stable second derivative block methods for stiff IVPs in ODEs
**

This paper
considers a class of highly stable block methods for the numerical solution of
initial value problems (IVPs) in ordinary differential equations (ODEs). The
boundary locus of the proposed parallel one-block, *r*-output point algorithms shows that
the new schemes are *A*-stable for
output points *r*=2(2)8 and
*A*(α)-stable
for output points *r*=10(2)20, where
*r* is the number of processors in a
particular block method in the family. Numerical results of the block methods
are compared with a second derivative linear multistep method in [8].

**
Key words:
**block
methods, continuous methods, collocation and interpolation, boundary locus,

**
==============================================**

**
Prashanth
M., Gupta D.K., Singh S.**

**
Semilocal
convergence for the Super-Halley's method
**

The semilocal convergence of Super-Halley's method for solving nonlinear equations
in Banach spaces is established under the assumption that the second Frёchet
derivative satisfies the ω-continuity
condition. This condition is milder than the well known Lipschitz and Hölder
continuity conditions. The importance of our work lies in the fact that
numerical examples can be given to show that our approach is successful even in
cases where the Lipschitz and Hölder continuity conditions fail. Difficult
computation of the second Frёchet derivative is also avoided by replacing it
with a divided difference containing only the first Frёchet derivatives. A
number of recurrence relations based on two parameters are derived. A
convergence theorem is established to estimate a priori error bounds along with
the domains of existence and uniqueness of the solutions. The
*R*-order of convergence of the method
is shown to be at least three. Two numerical examples are worked out to
demonstrate the efficiency of our method. It is observed that in both examples
the existence and uniqueness regions of solution are improved when compared with
those obtained in [7].

**
Key words:**
nonlinear operator equations, ω-continuity
condition, recurrence relations,

**
To the 7Oth Birthday of Academician B.G. Mikhailenko
(**

**
To the 80th Anniversary of Gennadii Alekseevich Mikhailov
(**

**
Abdikalykov A.K., Ikramov Kh.D., Chugunov V.N.**

**
On eigenvalues of ( T+H)-circulants
and (T+H)-skew-circulants**

Explicit formulas for calculating eigenvalues of the Hankel circulants, Hankel
skew-circulants, (*T+H*)-circulants, and
(*T+H*)-skew-circulants are obtained. It
is shown that if
ϕ
≠ ±1, then the set of matrices that can be represented as sums of a Toeplitz ϕ-circulant
and a Hankel ϕ-circulant is not an algebra.

**
Key words:**
Toeplitz matrix, Hankel matrix, circulant, skew-circulant, eigenvalues.

**
Burmistrov A.V., Korotchenko M.A.**

**
Weight Monte Carlo algorithms for estimation and parametric analysis of
the solution to the kinetic coagulation equation
**

**
**

The Smoluchowski equation with linear coagulation coefficients depending on two
parameters is considered. We construct weight algorithms for estimating various
linear functionals in an ensemble, which is governed by the equation under
study. The algorithms constructed allow us to estimate the functionals for
various parameters as well as parametric derivatives using the same set of
trajectories. Moreover, we construct the value algorithms and analyze their
efficiency for estimating the total monomer concentration as well as the total
monomer and dimer concentration in the ensemble. A considerable gain in
computational costs is achieved via the approximate value simulation of the time
between interactions combined with the value simulation of the interacting pair
number.

**
Key words:**
statistical modeling, evolution of many-particle system, Smoluchowski equation,
value function, parametric derivative, multiplicative weight, computational
cost.

**
Imomnazarov Kh.Kh., Mikhailov A.A.**

**
Application of a spectral method for numerical modeling of propagation of
seismic
waves in porous media for dissipative case
**

**
**

This paper presents the algorithm, based on the application of the spectral
Laguerre method for approximation of temporal derivatives as applied to the
problem of seismic wave propagation in the porous media with dissipation of
energy. The initial system of equations is written down as the first order
hyperbolic system in terms of velocities, stresses and pore pressure. For the
numerical solution of the problem in question, the method of a combination of
the analytical Laguerre transformation and a finite difference method is used.
The proposed method of the solution can be considered to be an analog to the
known spectral method based on the Fourier transform. However, unlike the
Fourier transform, application of the integral Laguerre transform with respect
to time allows us to reduce the initial problem to solving a system of equations
in which the parameter of division is present only in the right-hand side of
equations and has a recurrent dependence. As compared to the time-domain method,
with the help of an analytical transformation in the spectral method it is
possible to reduce an original problem to solving a system of differential
equations, in which there are only derivatives with respect to spatial
coordinates. This allows us to apply a known stable difference scheme for
recurrent solutions to similar systems. Such an approach is effective when
solving dynamic problems for porous media. Thus, because of the presence of the
second longitudinal wave with a low velocity, the use of difference schemes in
all coordinates for stable solutions requires a consistent small step both with
respect to time and space, which inevitably results in an increase in computer
costs.

**
Key words: **
Laguerre transform, porous media, numerical modeling, wave field, difference
scheme.

**
Mikhailenko B.G., Mikhailov A.A.**

**
Numerical modeling of acoustic-gravity waves propagation in a
heterogeneous «Earth-Atmosphere» model with a wind in the atmosphere
**

**
**

A numerical-analytical solution for seismic and acoustic-gravitational waves propagation is applied to a heterogeneous «Earth-Atmosphere» model. Seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. Propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations with the a wind. The proposed algorithm is based on the integral Laguerre transform with respect to time, the finite integral Fourier transform along the spatial coordinate with the finite difference solution of the reduced problem.

**
Key words: **seismic
waves, acoustic-gravitational waves, Navier-Stokes equations, Laguerre
transform, finite difference method.

**
Mikhailenko B.G., Fatyanov A.G.**

**
Numerical-analytical modeling of wave fields for complex subsurface
geometries and structures
**

**
**

In this paper we propose an analytical method of modeling seismic wave fields
for a wide range of geophysical media: elastic, non-elastic, anisotropic,
anisotropic-non-elastic, porous, random-inhomogeneous, etc. for super-remote
(far) distances. As finite difference approximations are not used, there is no
grid dispersion when computing wave fields for arbitrary media models and
observation points. The analytical solution representation in the spectral
domain makes possible to carry out the analysis of a wave field in parts,
specifically, to obtain the primary waves. Based on the developed program of
computing the wave fields, we have carried out the simulation of water waves and
seismic «ringing» on the Moon. The monotone displacement resonant to the lower
frequency area with increasing the recording distance has been explained. Such a
displacement was detected in experiments with a seismic vibrator.

**
Key words:**
mathematical modeling, analytical solution, full wave fields, primary waves,
elastic, non-elastic, anisotropic-non-elastic, porous, random-inhomogeneous
media.

**
Mikhailov G.A.**

**
About efficient algorithms of numerically-statistical simulation
**

**
**

A set of numerical algorithms for the simulation of random variables and
functions as well for the parametrical numerically-statistical analysis are
considered. Important specifications and explanations of the algorithms
formulations and substantiation, which are effective from the standpoint of
practice, are given.

**
Key words: **
base random
number, probability density function, discrete superposition method, branching
of trajectories, similar trajectory method, random field, histogram.

**
Pchelintsev A.N.**

**
Numerical and physical modeling of the Lorenz system dynamics
**

**
**

This paper describes a modification of a power series for the construction of
approximate solutions of the Lorenz system. The results of the computer-aided
simulation are presented. Also, the physical modeling of the dynamics of the
Lorenz system of the processes occurring in the circuit are considered.

**
Key words: **Lorenz
system,
analog multiplier, integrator, method of power series,
radius of convergence, free convection, Lorenz attractor.

Smirnov S.V.

**
On calculation of seiche oscillations of the middle part of Peter the
Great Gulf
( in Russian),
pp. **

Characteristics of barotropic seiche oscillations of the middle part of Peter
the Great Gulf are considered with the use of spectral-finite difference model.
The model is based on the linearized system of shallow water equations.
Difference approximation is carried out on an irregular triangular spatial mesh.
The numerical method involves the solution of the eigenvalue problem and is able
to directly obtain a set of frequencies and the corresponding forms of seiche
oscillations. The grid computational domain covers the Amurskiy gulf and the
Ussuriyskiy gulf. The Zolotoy Rog bay and the Alekseev bay are described in more
detail on the grid. The calculated and presented spatial-temporal
characteristics of a number of seiche oscillations corresponding to well-defined
peaks of the energy spectrum of the sea level data from the station
«Vladivostok» of the Russian of tsunami warning service. The results of
calculations for the Alekseev bay are compared to the data of natural
measurements and the solutions of the Cauchy problem.

Key words:
seiches,
harbour oscillations.

**
Afanasyev I.V.**

**
A cellular automata model of three organisms populations in lake Baikal **

**
**

A cellular automata model of population dynamics of three organisms in
Lake Baikal is proposed and investigated. Each species is divided into age
groups. There are eight groups all together. The model allows one to take into
account a spatial organisms distribution, a seasonal dependency of birth rates,
a possible habitat pollution and water streams. A computational experiment was
carried out for the case of pollution that is in the south area of lake Baikal.
It demonstrates that the population dynamics tends to the oscillating process
with a period equal to 1 year. The assessment of the critical pollution
intensity which leads to the total extinction is presented. The model was
verified within production-to-biomass and frequency of occurrence ratios.

**
Key words:**
cellular
automata, discrete modeling, populations dynamics, lake Baikal, prey-predator
systems.

**
Lutsenko N.A., Tarasov G.V., Gyrnik K.A.**

**
An OpenMP version of the parallel algorithm for calculation of unsteady
gas flow through porous objects with energy sources: Analysis and Application****( in Russian),
pp. **

**
**

The gas flows in the gravity field through the porous objects with energy
sources, which may originate from the natural or man-caused disasters, have been
investigated. An OpenMP version of the parallel algorithm has been developed for
the calculation of unsteady 2D gas flows through porous self-heating media of
complex subsurface geometries. The structure of the sequential algorithm and the
transition from it to the OpenMP version have been described, the performance
and efficiency of parallelization have been analyzed. The unsteady gas flows
through axisymmetric porous self-heating objects with a partial closure of the
object outlet (with a top cover) have been investigated by means of the
developed parallel algorithm. The influence of the partial closure of the object
outlet on the cooling process of the porous objects with a non-uniform
distribution of heat sources has been analyzed.

**
Key words:**
parallel
algorithms, numerical modeling, porous objects, gas cooling, heat release.

**
Orlov A.V., Malyshev A.V.**

**
The test problem generation for quadratic-linear pessimistic bilevel
optimization****( in Russian),
pp. **

**
**

The generation method of quadratic-linear bilevel optimization test
problems in a pessimistic formulation is proposed and justified. The
propositions about the exact form and the number of local and global pessimistic
solutions in generated problems are proved.

**
Key words: **
test problem generation, bilevel optimization, guaranteed (pessimistic)
solution, kernel problems.

Romankov A.S., Romenski E.I.

**
The Runge-Kutta/WENO method for solving equations for small-amplitude
wave propagation in a saturated porous medium
**

**
**

A high-accuracy Runge-Kutta/WENO method up to fourth order with respect
to time and fifth order with respect to space is developed for the numerical
modeling of the small-amplitude wave propagation in a steady fluid-saturated
porous medium. The system of governing equations is derived from the general
thermodynamically compatible model of a compressible fluid flow through a
saturated elastic porous medium, which is described by the hyperbolic system of
conservation laws with allowance for finite deformations of the medium. The
results of numerical solution of one- and two-dimensional wavefields demonstrate
efficiency of the method developed.

**
Key words:**
high-accuracy methods, hyperbolic system of conservation laws, saturated elastic
porous media, wave propagation.

**
Tripathy M., Sinha Rajen Kumar**

**
Convergence of H^{1}-Galerkin
mixed finite element method for parabolic problems with reduced regularity of
initial datas**

**
**

We study the convergence of an *H
*^{1}-Galerkin mixed finite element method for parabolic problems in
one space dimension. Both semi-discrete and fully discrete schemes are analyzed
assuming reduced regularity of the initial data. More precisely, for a spatially
discrete scheme error estimates of order \mathcal{O}(*
h*^{2}*t *^{-1/2})
for positive time are established assuming the initial function
*p*_{0 }
϶
*H *
^{2(}Ω)
∩ *H*_{0}^{1}(Ω). Further, we use an energy technique together with a
parabolic duality argument to derive error estimates of order \mathcal{O}(*
h*^{2}*t *^{-1}) when *p*_{0
}is only in *H*_{0}^{1}(Ω). A discrete-in-time backward Euler method is analyzed
and almost optimal order error bounds are established.

**
Key words:**
parabolic
problems, *H*^{1}-Galerkin mixed finite element method, semi-discrete
scheme, backward Euler method, error estimates.

**
Shary S.P.**

**
On the full rank interval matrices****
( in Russian),
pp. **

**
**

For interval matrices, the paper considers the problem of determining
whether a matrix has a full rank. We propose the full rank criterion that relies
on the search for diagonal dominance as well as criteria based on
pseudoinversion of the midpoint matrix and comparison of the midpoint and the
radius matrices for the interval matrix under study.

**
Key words:**
interval
matrix, full rank, full rank criteria.

**Shlychkov V.A., Krylova A.I.**

**
A numerical model of density currents in estuaries of the Siberian
rivers**

**
**

A numerical model for studying the dynamic mixing of the sea and the
river waters in the estuarial area is proposed. Computations are based on the
two-dimensional longitudinal vertical stratified fluid mechanics equations and
the equation of transport of salt. The model focuses on the reproduction of the
local density currents at the mouth of arms branched deltas of the rivers of
Siberia. The results of numerical experiments are given, the dynamic structure
of the flow and salinity profiles are compared to the observational data.

**
Key words:**
numerical simulation, turbulence,
gradient-density flows, river flow, sea area, transport of salt.
**
**

Vitvitsky A.A.

Cellular automata with a dynamic structure for simulating
the biological tissues growth*in Russian*),
pp.**315-327**

*L*-system (Lindenmayer
system). Furthermore, the proposed DCA-model allows one to simulate growth of
individual biological cells and to visualize the substances dynamics in these
cells (decay, synthesis and diffusion).

**Key words:**
computer simulation, cellular automata, cellular automata with dynamic
structure, orphogenesis,
apical meristem escape, Arabidopsis Thaliana.

**Kotel'nikov E.A.**** **

**Minimization of a quadratic function on the sphere***in Russian*),
pp.**329-338**

In this paper, a sequential algorithm for solving the problem of minimization of
a quadratic function on a sphere is proposed. At each iteration of the scheme, a
two-dimensional problem of minimization is solved. Numerical comparisons with
other methods are presented.

**Key words:**
quadratic
optimization on sphere, Cholesky decomposition, trust region, step trajectory,
quadratic model.

**
Leonov A.S.**

**
Which of inverse problems can have a priori approximate solution accuracy
estimates comparable in order with the data accuracy
(in Russian),
pp.
339-348 **

It is proved that a priori global accuracy estimate for approximate solutions to
linear inverse problems with perturbed data can be of the same order as
approximate data errors for well-posed in the sense of Tikhonov problems only. A
method for assessing the quality of selected sets of correctness is proposed.
The use of the generalized residual method on a set of correctness allows us to
solve the inverse problem and to obtain a posteriori accuracy estimate for
approximate solutions, which is comparable with the accuracy of the problem
data. The approach proposed is illustrated by a numerical example.

**Key words:**
linear inverse problems, correctness in the sense of Tikhonov, a
priori and a posteriori accuracy estimate.

**Litvenko K.V., Prigarin S.M.**

**Numerical stochastic models of the sea surface undulation and extreme ocean
waves***in Russian*),
pp.**349-361**

The paper deals with simulation of the time-space stochastic structure of the sea surface undulation and extreme ocean waves. Numerical algorithms are constructed on the basis of conditional spectral models and models of time series adapting data of observations. Estimates for frequencies of extreme waves appearance are studied on the basis of the theory of random processes and fields.

**Key words:**
simulation of random fields, conditional spectral models, time series, sea
surface undulation, extreme ocean waves (freak-waves, rogue waves).

**Moskalensky E.D.**

**On finding exact solutions of the two-dimensional eikonal equation when the
front of the wave propagating in a medium is a circle***in Russian*),
pp.**363-372**

Wave propagation in a two-dimensional medium is considered in the case when the
front of the wave is a circle with the center (*a*(*t*),0) and radius
*r*(*t*).
A question is posed: what is the distribution of velocity in the medium? Common
characteristics and examples of such media are given.

**Key words:**

** **

Okuonghae R.I., Ikhile M.N.O.

**
L(α)-stable
variable order implicit second derivative Runge Kutta methods
**

This paper considers the extension of the popular Runge Kutta methods (RKMs) to
second derivative Runge Kutta methods (SDRKMs) for the direct solution of stiff
initial value problems (IVPs) of ordinary differential equations (ODEs). The
methods are based on using collocation and interpolation techniques. The last
stage of the input approximation is identical to the output method. The SDRKMs
are *
L*(α)-stable
for the methods examined. Numerical experiments are given comparing one of these
methods with a two derivative Runge Kutta method (TDRKM) and a second derivative
linear multistep method (SDLMM).

**Key words:** second
derivative, Runge Kutta method, collocation, interpolation.

**
Salov G.I.**

**A new three-sample non-parametric statistical test with its special case
equivalent to the Whitney test** *in Russian*),
pp.**389-397**

In this paper, we propose a new non-parametric statistical test for the problem
of homogeneity of three samples. We consider an alternative for which one sample
values tend to be stochastically larger than every one from the two other
samples values. The Whitney test is equivalent to special (linear) case of this
test. Some comparisons are made for the cases with samples from exponential and
uniform distributions.

**Key words: **three
samples, homogeneity test, nonparametric statistical test.

**
Smelov V.V.**

**
**
**
A network version of the non-standard trigonometric basis and its advantages
with respect to a similar polynomial basis**

A trigonometry-based functional basis as a network version is proposed. It is
aimed at the approximation with high orders of accuracy of smooth and
piecewise-smooth functions. A comparative analysis of the features of the basis
proposed and a polynomial one is made. The trigonometric version offers
considerable advantages over the polynomial bases.

**Key words: **
functional basis, elliptic operator, energy scalar product, functional,
generalized solution, conjugation condition.**
**
**
**

**Tarakanov V.I., Lysenkova S.A., Nesterenko M.V.**

**
Iterative scheme of finding a spectrum of the product of two non-commutative
operators
**

We consider spectral features and an iterative scheme of finding a spectrum of
the product of two non-commutative partially symmetric operators in the Hilbert
space *H*. In this case it is assumed
that one of operators is compact, the second not necessarily being compact and
even restricted in *H*. Numerical
implementation of the iterative scheme for finding the operator spectrum of the
problem of eigen-oscillations of the Rayleigh beam is presented.

**
Key words:
**
operator,
spectrum, iterative algorithm.