**Full Volume**/ Весь том (UA) [ (PDF) ]**Cover**/ Титул (UA) [ (PDF) ]**Editorial Board**/ Редакційна колегія (UA).*P. 2.*[ (PDF) ]**Contents**/ Зміст (UA).*P.3.*[ (PDF) ]**V.D. Dushkin, S. Zhuchenko, O.V. Kostenko**. Discrete mathematical model of the scattering process of E-polarized wave on a periodic impedance grating.*P. 4-25.*DOI: 10.26565/2221-5646-2019-90-01. [ Abstract] [ Full-text available (PDF)]**S.M. Chuiko, Ya.V. Kalinichenko, M.V. Popov.**Boundary value problems for systems of non-degenerate difference-algebraic equations.*P. 26-41.*DOI: 10.26565/2221-5646-2019-90-02. [ Abstract ] [ Full-text available (PDF) ]**V. Baranets, N. Kizilova.**Mathematical modeling of particle aggregation and sedimentation in the inclined tubes.*P. 42-59.*DOI: 10.26565/2221-5646-2019-90-03. [ Abstract ] [ Full-text available (PDF) ]**S.M. Chuiko, O.V. Nesmelova.**On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case*P. 60-72.*DOI: 10.26565/2221-5646-2019-90-04. [ Abstract ] [ Full-text available (PDF) ]

*Short abstract:*
It is considered the discrete mathematical models which describe the interaction process of the E-polarized wave and periodic system of impedance tapes. It is shown that the discrete model for various values of the discretization parameter is equivalent to the system of singular integral equations. Calculations were performed for the proposed model and for the model based on the hypersingular equations.
The obtaining results showed the closeness of the field characteristics.

*Extended abstract:*
The method of numerical modeling of wave scattering by periodic impedance grating is considered.
In the case of a harmonic dependence of the field on time and the uniformity of the structure along a certain axis, the three-dimensional problem reduces to considering of two 2D problems for the components of the E-polarized and H-polarized waves.
The signle nonzero component of the electric field created by the incident E-polarized wave is the solution of the boundary value problem for the Helmholtz equation with Robin boundary conditions.
It follows from the physical formulation of the problem that its solutions satisfy the Floquet quasiperiodicity condition, the condition of finiteness of energy in any bounded region of the plane.
Also, the difference between the total and incident fields satisfies the Sommerfeld radiation condition.
Following the ideas of the works of Yu.V. Gandel, using the method of parametric representations of integral operators, the boundary-value problem reduces to two systems of integral equations.
The first one is the system of singular equations of the first kind with additional integral conditions. The second system consists of the Fredholm boundary integral equations of the second kind with a logarithmic singularity in the integrand.
A discrete model for various values of the discretization parameter is equivalent to systems of singular integral equations. By solving these equations, approximate values of the main field characteristics are determined.
The method of parametric representations of integral operators makes it possible to obtain systems of integral equations of other types.
In particular, the initial boundary-value problem reduces to a system consisting of hypersingular integral equations of the second kind and the Fredholm integral equation of the second kind.
A numerical experiment was conducted for cases of different location of tapes.
Calculations were performed for the proposed model and the model based on hypersingular equations. They showed the closeness of the obtained results in a wide range of parameters studied.

*Keywords:* mathematical model; impedance structures; numerical experiment.

2010 Mathematics Subject Classification: 41A55. [ Full-text available (PDF) ] Top of the page.

*Short abstract:*
The conditions of existence and the construction of solutions of Cauchy problem for difference-algebraic
system are determined. The conditions of existence and the construction of solutions of a linear Noetherian difference-algebraic boundary-value problem are determined. An original classification and a single scheme of construction of the solutions of difference-algebraic equations are proposed.

*Extended abstract:*
The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N.N. Luzin and
F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell,
Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis.
The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems
in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability.
At the same time, the study of differential algebraic boundary value problems is closely related to the study of boundary
value problems for difference equations, initiated in A.A. Markov, S.N. Bernstein, Ya.S. Besikovich, A.O. Gelfond, S.L. Sobolev,
V.S. Ryaben'kii, V.B. Demidovich, A. Halanay, G.I. Marchuk, A.A. Samarskii, Yu.A. Mitropolsky, D.I. Martynyuk, G.M. Vayniko,
A.M. Samoilenko, O.A. Boichuk and O.M. Stanzhitsky. Study of nonlinear singularly perturbed boundary value problems for
difference equations in partial differences is devoted to the work of V.P. Anosov, L.S. Frank, P.E. Sobolevskii, A.L. Skubachevskii
and A. Asheraliev.
Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A.M. Samoilenko and
O.A. Boichuk on linear boundary value problems for difference-algebraic equations, in particular finding the necessary and
sufficient conditions for the existence of the desired solutions, and also the construction of the Green's operator of
the Cauchy problem and the generalized Green operator of a linear boundary value problem for a difference-algebraic equation.
The solvability conditions are found in the paper, as well as the construction of a generalized Green operator for
the Cauchy problem for a difference-algebraic system. The solvability conditions are found, as well as the construction
of a generalized Green operator for a linear Noetherian difference-algebraic boundary value problem. An original
classification of critical and noncritical cases for linear difference-algebraic boundary value problems is proposed.

*Keywords:* boundary-value problems; difference-algebraic equations; pseudoinverse matrices.

2010 Mathematics Subject Classification: 15A24; 34B15; 34C25. [ Full-text available (PDF) ] Top of the page.

*Short abstract:* Sedimentation of the aggregating particles of different technical suspensions,
blood and nanofluids in the gravity is investigated. The dependence of the sedimentation rate on the angle of inclination
is considered. The two phase model of the aggregating particles is generalized to the inclined tubes. In the suggestion
of small angles of inclination the equations are averaged over the transverse coordinate and the obtained hyperbolic
system of equations is solved by the method of characteristics.

*Extended abstract:*
Sedimentation of the aggregating particles in the gravity field is widely used as an easy and cheap test of the suspension
stability of different technical suspensions, blood and nanofluids. It was established the tube inclination makes the test
much faster that is known as the Boycott effect. It is especially important for the very slow aggregating and sedimenting
blood samples in medical diagnostics or checking the ageing of the nanofluids. The dependence of the sedimentation rate on
the angle of inclination is complex and poorly understood yet. In this paper the two phase model of the aggregating particles
is generalized to the inclined tubes. The problem is formulated in the two-dimensional case that corresponds to the narrow
rectangle vessels or gaps of the viscosimeters of the cone-cone type. In the suggestion of small angles of inclination
the equations are averaged over the transverse coordinate and the obtained hyperbolic system of equations for is solved by
the method of characteristics. During the sedimentation the upper region (I) of the fluid free of particles, the bottom
region (III) of the compactly located aggregates without fluid, and the intermediate region of the sedimenting aggregates (II)
appear. The interface between I and II can be registered by any optic sensor and its trajectory is the sedimentation curve.
Numerical computations revealed the increase in the initial concentration of the particles, their aggregation rate,
external uniform force and inclination angle accelerate the sedimentation while any increase in the fluid viscosity
decelerates it that is physically relevant. Anyway, the behaviors of the acceleration are different. For the elevated force
the interfaces I-II and II-III shifts uniformly, while for the elevated concentration or aggregation rate the interface I-II
or II-III moves faster. Small increase of the inclination angle accelerates the sedimentation while at some critical
angles is starts to decelerate due to higher shear drag in the very viscous mass of the compactly located aggregates.
Based on the results, a novel method of estimation of the suspension stability is proposed.

*Keywords:* Boycott effect; suspension; aggregation; sedimentation; medical diagnostics.

2010 Mathematics Subject Classification: 76T20; 76Zxx; 83C55. [ Full-text available (PDF) ] Top of the page.

*Short abstract:*
We construct necessary and sufficient conditions for the existence of solution and iterative scheme for the approximate
solutions of nonlinear Noetherian differential-algebraic boundary value problem in critical case.

*Extended abstract:*
The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of nonlinear boundary-value problems for ordinary differential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk.
The study of the nonlinear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the nonlinear boundary value problems for the differential algebraic equations, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the nonlinear differential algebraic boundary value problems.
In this article we found the conditions of the existence and constructed the iterative scheme for finding the solutions of the weakly nonlinear Noetherian differential-algebraic boundary value problem. The proposed scheme of the research of the nonlinear differential-algebraic boundary value problems in the article can be transferred to the nonlinear matrix differential-algebraic boundary value problems. On the other hand, the proposed scheme of the research of the nonlinear Noetherian differential-algebraic boundary value problems in the critical case in this article can be transferred to the autonomous seminonlinear differential-algebraic boundary value problems.

*Keywords:* boundary-value problems; differential-algebraic equations; noncritical case; pseudoinverse matrices.

2010 Mathematics Subject Classification: 34B15. [ Full-text available (PDF) ] Top of the page.

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