Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics
https://periodicals.karazin.ua/mech_math
<p><span style="color: #009900;"><strong>Indexed/Abstracted</strong></span> in <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=se:00002693" target="_blank" rel="noopener"><strong>Zentralblatt MATH</strong></a> ( since <strong>1999</strong>; indexed more than <strong>360</strong> documents). <br> Zentralblatt MATH (<a href="https://zbmath.org/about/" target="_blank" rel="noopener"><strong>zbMATH</strong></a>) is the world’s most comprehensive and longest-running abstracting and reviewing service in pure and applied mathematics.</p>V. N. Karazin Kharkiv National Universityen-USVisnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics2221-5646<p>The copyright holder is the <strong>author</strong>.</p> <p>Authors who publish with this journal agree to the following terms:</p> <p>1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a <strong>Creative Commons Attribution License</strong> that allows others to share the work with an acknowledgement of the work's authorship and <strong>initial publication in this journal</strong>. (Attribution-Noncommercial-No Derivative Works licence). </p> <p>2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.</p> <p>3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (see The Effect of Open Access).</p>Discrete mathematical model of the scattering process of E-polarized wave on a periodic impedance grating
https://periodicals.karazin.ua/mech_math/article/view/14946
<p>The method of numerical modeling of wave scattering by periodic impedance grating is considered.<br>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.<br>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.<br>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.<br>Also, the difference between the total and incident fields satisfies the Sommerfeld radiation condition.<br>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.<br>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.<br>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.<br>The method of parametric representations of integral operators makes it possible to obtain systems of integral equations of other types.<br>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.<br>A numerical experiment was conducted for cases of different location of tapes.<br>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.</p>Vladimir D. DushkinStanislav ZhuchenkoOleksii V. Kostenko
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2019-12-232019-12-239042510.26565/2221-5646-2019-90-01Boundary value problems for systems of non-degenerate difference-algebraic equations
https://periodicals.karazin.ua/mech_math/article/view/14947
<p>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.<br>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.<br>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.</p>Sergey ChuikoYaroslav KalinichenkoMykyta Popov
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2019-12-232019-12-2390264110.26565/2221-5646-2019-90-02Mathematical modeling of particle aggregation and sedimentation in the inclined tubes
https://periodicals.karazin.ua/mech_math/article/view/14948
<p>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.</p>Vitaliia BaranetsNatalya Kizilova
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2019-12-232019-12-2390425910.26565/2221-5646-2019-90-03On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case
https://periodicals.karazin.ua/mech_math/article/view/14949
<p>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.</p> <p>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.</p> <p>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.</p>Sergey ChuikoOlga V. Nesmelova
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2019-12-232019-12-2390607210.26565/2221-5646-2019-90-04