Vol. 57 No. 3 (262) (2023)

DOI: https://doi.org/10.46991/PYSUA.2023.57.3
Published: 2023-12-20

Mathematics

  • Mathematics

    THE SOLVABILITY OF AN INFINITE SYSTEM OF NONLINEAR ALGEBRAIC EQUATIONS WITH TOEPLITZ MATRIX

    Khachatur A. Khachatryan, Vachagan G. Dilanyan
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    Abstract

    The work is focused on studying of the existence, uniqueness, and various qualitative properties of the constructive solution of an infinite system of algebraic equations with a concave nonlinearity property, which are generated by Toeplitz matrices. In addition to its independent mathematical interest, such systems have a significant application in several branches of mathematical physics and mathematical biology. Those particularly appear in discrete problems within radiative transfer theory, kinetic theory of gases, dynamic theory of p-adic strings, and the mathematical theory of epidemic propagation. We establish the existence of a positive solution for the system in the class of bounded sequences, as well as provide an iterative method to approximate to the solution. We also study the asymptotic behavior of the solution at infinity and the uniqueness of the nontrivial solution with non-negative elements in the class of bounded sequences. The last section of the paper provides examples of applications of the corresponding Toeplitz matrix and the function that describes the nonlinearity.

    References

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    Vladimirov V.S., Volovich Ya.I. Nonlinear Dynamics Equation in p-Adic String Theory. Theoret. and Math. Phys. 138 (2004), 297-309.

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    Khachatryan Kh.A. Questions of Solvability of Some Nonlinear Integral and Integro-differential Equations with Non Compact Operators in Critical Case. Doctorial Thesis. Yerevan State University (2011), 235.

    Khachatryan Kh.A., Broyan M.F. One-parameter Family of Positive Solutions for a Class of Non-linear Infinite Algebraic Systems with Toeplitz-Hankel Type Matrices. Nazvanie jurnala 48 (2013), 209-220.

Mechanics

  • Mechanics

    ON ONE PROBLEM OF OPTIMAL CONTROL OF VIBRATIONS OF A PLATE-STRIP IN A TEMPERATURE FIELD

    Edmon R. Grigoryan
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    Abstract

    The problem of optimal control of elastic vibrations of an isotropic plate-strip under the influence of temperature and force fields is studied. The function of changing the external load on the plane of the plate is represented as a control function. Optimal control is also carried out by the distribution function of the temperature of the external field over the plate. The well-known classical hypotheses of thermo-elastic bending of the plate are accepted. The equations of transverse vibrations of the plate and heat conduction in the plate are solved under the boundary conditions of heat transfer and the stress state on the planes of the plate. The method of Fourier series, the method of representing moment relations, the well-known method of minimizing the functional are used.

    References

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    Jilavyan S.H., Grigoryan E.R., Khurshudyan As.Zh. Heating Control of Finite Rod with a Mobile Source. Archive of Control Sciences. Polish Academy of Sciences 31 (2021), 417-430. https://doi.org/10.24425/acs.2021.137425

    Xu B., Lu X., Bai Y. A Spatiotemporal Fuzzy Control Approach for Complex Distributed Parameter System with Application to Thermal Processes. J. Process Control 127 (2023). https://doi.org/10.1016/j.jprocont.2023.102993

    Wang Y.Z., Liu D., et al. Asymptotic Analysis of Thermoelastic Response in Functionally Graded Thin Plate Subjected to a Transient Thermal Shock. Compos. Struct. 139 (2016), 233-242. https://doi.org/10.1016/j.compstruct.2015.12.014

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  • Mechanics

    TRANSFER OF LOADS FROM THREE HETEROGENEOUS ELASTIC STRINGERS TO AN INFINITE SHEET THROUGH ADHESIVE LAYERS

    Aghasi V. Kerobyan, Karen P. Sahakyan
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    Abstract

    This paper considers the problem for an elastic infinite plate (sheet), which on parallel finite parts of its upper surface is strengthened by three finite stringers, two of which are located on the same line, having different elastic properties. The stringers are deformed under the action of horizontal forces. The interaction between infinite sheet and stringers takes place through thin elastic adhesive layers having other physical-mechanical properties and geometric configuration. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on three finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated. Further, for various values of changing characteristic parameters of the problem the multiple numerical results and its analysis are presented.

    References

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    Kerobyan A.V. Contact Problems for an Elastic Strip and the Infinite Plate with Two Finite Elastic Overlays in the Presence of Shear Interlayers. Proc. NAS RA: Mechanics 67 (2014), 22-34 (in Russian).

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    Grigoryan E.Kh., Kerobyan A.V., Shahinyan S.S. The Contact Problem for the Infinite Plate with Two Finite Stringers One from which is Glued, Other is Ideal Conducted. Proc. NAS RA: Mechanics 55 (2002), 14-23 (in Russian).

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    Kerobyan A.V., Sarkisyan V.S. The Solution of the Problem for an Anisotropic Half-plane on the Boundary of which Finite Length Stringer is Glued. Proc. of the Scientific Conference, Dedicated to the 60th Anniversary of the Pedagogical Institute of Gyumri 1 (1994). Gyumri, Vysshaya Shkola, 73-76 (in Russian).

    Grigoryan E.Kh. On Solution of Problem for an Elastic Infinite Plate, One the Surface of which Finite Length Stringer is Glued. Proc. NAS RA: Mechanics 53 (2000), 11-16 (in Russian).

    Sarkisyan V.S., Kerobyan A.V. On the Solution of the Problem for Anisotropic Half-plane on the Edge of which a Nonlinear Deformable Stringer of Finite Length is Glued. Proc. of NAS RA. Mechanics 50 (1997), 17-26 (in Russian).

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Physics

  • Physics

    THE RESONANT RESPONSE OF STRONGLY COUPLED NANORODS TO THE ELECTROMAGNETIC WAVE

    Davit A. Manukyan
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    Abstract

    The electromagnetic response of closely spaced nanorod dimers can be controlled via the modification of the nanoparticles interaction, caused by electron tunneling between them. The optical response to an intense external electromagnetic field of a system composed of two gold 230 nm long rods surrounded by air and separated by a gap of width 0.5 nm was analyzed. Using finite element method-based numerical simulations we show that the electron tunneling, taking place due to the extraordinary enhancement of the electromagnetic field inside the nanogap, results in the change of the nanoantenna coupling from the capacitive to the conductive one. As a result, the absorption-to-scattering ratio of the dimer significantly changes. Particularly, the scattering cross-section decreases by about three times, whereas the normalized absorption rises from about 12 to 18 with noticeably broadening of the spectral line.

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