phmathВестник Государственного университета просвещения. Серия: Физика-МатематикаBulletin of Federal State University of Education. Series: Physics and Mathematics2949-50832949-5067Federal State University of Educationphmath-307Research ArticleМАТЕМАТИКАMATHEMATICSАНАЛИТИЧЕСКОЕ РЕШЕНИЕ ДИСПЕРСИОННОГО УРАВНЕНИЯ ИЗ ВТОРОЙ ЗАДАЧИ СТОКСАTHE ANALYTICAL SOLUTION OF THE DISPERSION EQUATION FROM SECOND STOKES PROBLEMАкимоваВ. А.AkimovaV. .fakul-fm@mgou.ruБугримовАнатолий ЛьвовичBugrimovA. .al.bugrimov@mgou.ruЛатышевАнатолий ВасильевичLatyshevA. .fakul-fm@mgou.ruЮшкановАлександр АлексеевичYushkanovA. .kaf-tfiz@mgou.ruМосковский государственный областной университет (МГОУ)РоссияMoscow State Regional University (MSRU)Russian Federation20130801202301320Copyright © Акимова В.А., Бугримов А.Л., Латышев А.В., Юшканов А.А., 20232023Акимова В.А., Бугримов А.Л., Латышев А.В., Юшканов А.А.Akimova V..., Bugrimov A..., Latyshev A..., Yushkanov A...Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.physmathmgou.ru/jour/article/view/307

Исследуются нули дисперсионной функции из второй задачи Стокса. Вторая задача Стокса - задача о поведении разреженного газа, заполняющего полупространство. Плоскость, ограничивающая полупространство, совершает гармонические колебания в своей плоскости. Используется линеаризованное кинетическое эллипсоидально статистическое уравнение с параметром. Доказывается формула факторизации дисперсионной функции. С помощью формулы факторизации в явном виде находятся нули дисперсионной функции и проводится их исследование в зависимости от величины безразмерной частоты ограничивающей газ плоскости и параметра уравнения.

Zero of dispersion function from second Stokes problem are investigated. Second problem Стокса is a problem about behavior of the rarefied gas filling half-space. The plane limiting half-space, makes harmonious oscillations. It is used linear kinetic ellipsoidal statistical equation with parameter. The factorization formula to dispersion function is proved. By means of the factorization formula are found zero of dispersive function in an explicit form and their research depending on quantity of dimensionless frequency of a plane limiting gas and equation parameter is carried out.

вторая задача Стоксакинетическое эллипсоидально статистическое уравнениеразделение переменныхнули дисперсионной функциисобственные решениянепрерывный и дискретный спектрыфакторизация дисперсионной функцииsecond Stokes problemkinetic ellipsoidal statistical equationseparation of variableszero of dispersion functioneigen solutionscontinuous and discrete spectrafactorization of dispersion functionReferencesStokes G. G. On the effect of internal friction of fluids on the motion of pendulums. Trans. Cambr. Phil., IX, 1851, 8-106.Stokes G. G. On the effect of internal friction of fluids on the motion of pendulums. Trans. Cambr. Phil., IX, 1851, 8-106.Akimova V. A., Latyshev A. V., Yushkanov A. A. Analytical solution of the second Stokes problem on behaviour of gas over oscillation surface. Part I: eigenvalues and eigensolutions// arXiv: 1111.3429v1 [math-ph] 15 Nov 2011, 27 pp.Akimova V. A., Latyshev A. V., Yushkanov A. A. Analytical solution of the second Stokes problem on behaviour of gas over oscillation surface. Part I: eigenvalues and eigensolutions// arXiv: 1111.3429v1 [math-ph] 15 Nov 2011, 27 pp.Akimova V. A., Latyshev A. V., Yushkanov A. A. Analytical solution of the second Stokes problem on behaviour of gas over oscillation surface. Part II: mathematical apparatus of solving of problem// arXiv: 1111.5182v1 [math-ph] 22 Nov 2011, 26 pp.Akimova V. A., Latyshev A. V., Yushkanov A. A. Analytical solution of the second Stokes problem on behaviour of gas over oscillation surface. Part II: mathematical apparatus of solving of problem// arXiv: 1111.5182v1 [math-ph] 22 Nov 2011, 26 pp.Akimova V. A., Latyshev A. V., Yushkanov A. A. Analytical Solution of Second Stokes Problem on Behavior of Gas over Oscillation Surface. Part III: Solving of Problem and Applications// arXiv: 1112.1283v1 [math-ph] 6 Dec 2011, 40 pp.Akimova V. A., Latyshev A. V., Yushkanov A. A. Analytical Solution of Second Stokes Problem on Behavior of Gas over Oscillation Surface. Part III: Solving of Problem and Applications// arXiv: 1112.1283v1 [math-ph] 6 Dec 2011, 40 pp.Akimova V. A., Latyshev A. V., Yushkanov A. A. Zero of the dispersion equations from the second Stokes problem about behaviour of gas over the fluctuating surface. Vestnik MGOU (Moscow state regional univ.). Ser. Physics-Mathematics, 2012. №. 2, 3-14.[in russian].Akimova V. A., Latyshev A. V., Yushkanov A. A. Zero of the dispersion equations from the second Stokes problem about behaviour of gas over the fluctuating surface. Vestnik MGOU (Moscow state regional univ.). Ser. Physics-Mathematics, 2012. №. 2, 3-14.[in russian].Sharipov F. and Kalempa D. Gas flow around a longitudinally oscillating plate at arbitrary ratio of collision frequency to oscillation frequency // Rarefied Gas Dynamics: 25-th International Symposium, ed. by M.S.Ivanov and A.K.Rebrov. Novosibirsk, 2007, 1140-1145.Sharipov F. and Kalempa D. Gas flow around a longitudinally oscillating plate at arbitrary ratio of collision frequency to oscillation frequency // Rarefied Gas Dynamics: 25-th International Symposium, ed. by M.S.Ivanov and A.K.Rebrov. Novosibirsk, 2007, 1140-1145.Karabacak D. M., Yakhot V., and Ekinci K. L. High-Frequency Nanofluidics: An Experimental Study using Nanomechanical Resonators, Phys. Rev. Lett., 98 (2007), 254505.Karabacak D. M., Yakhot V., and Ekinci K. L. High-Frequency Nanofluidics: An Experimental Study using Nanomechanical Resonators, Phys. Rev. Lett., 98 (2007), 254505.Yakhot V. and Colosqui C. Stokes’ second flow problem in a highfrequency limit: application to nanomechanical resonators // J. Fluid Mechanics. V. 586. Sept. 2007, pp. 249-258.Yakhot V. and Colosqui C. Stokes’ second flow problem in a highfrequency limit: application to nanomechanical resonators // J. Fluid Mechanics. V. 586. Sept. 2007, pp. 249-258.Dudko V. V., Yushkanov A. A., Yalamov Yu. I. Influence of properties of a surface on characteristics of shift waves. J. tech. phys.,75:4 (2005), 134-135. [in russian].Dudko V. V., Yushkanov A. A., Yalamov Yu. I. Influence of properties of a surface on characteristics of shift waves. J. tech. phys.,75:4 (2005), 134-135. [in russian].Dudko V. V., Yushkanov A. A., Yalamov Yu. I. Generation by a fluctuating surface of shift waves in gas, High Temperature, 47:2 (2009), 262-268. [in russian].Dudko V. V., Yushkanov A. A., Yalamov Yu. I. Generation by a fluctuating surface of shift waves in gas, High Temperature, 47:2 (2009), 262-268. [in russian].Dudko V. V. Sliding of rarefied gas lengthways motionless and fluctuating surfaces. Thesis, Moscow, 2010. [in russian].Dudko V. V. Sliding of rarefied gas lengthways motionless and fluctuating surfaces. Thesis, Moscow, 2010. [in russian].Latyshev A. V., Yushkanov A. A. Analytical solution of second Stokes problem about behaviour of gas over fluctuating surface by means of ellipsoidal statistical equation// arXiv:1210.4135v1 [mathph] 15 Oct 2012, 24 pp.Latyshev A. V., Yushkanov A. A. Analytical solution of second Stokes problem about behaviour of gas over fluctuating surface by means of ellipsoidal statistical equation// arXiv:1210.4135v1 [mathph] 15 Oct 2012, 24 pp.Burniston E. E. and Siewert C. E. The Use of Riemann-Hilbert Problem in Solving a Class of Transcendental Equations // Proc. of the Cambridge Phil. Soc.,73 (1973).Pp. 111-118.Burniston E. E. and Siewert C. E. The Use of Riemann-Hilbert Problem in Solving a Class of Transcendental Equations // Proc. of the Cambridge Phil. Soc.,73 (1973).Pp. 111-118.Siewert C.E. and Thomas J. R., Jr. Half-Space Problems in the Kinetic Theory of Gases // The Physics of Fluids, 16 (1973). Pp.1557-1559.Siewert C.E. and Thomas J. R., Jr. Half-Space Problems in the Kinetic Theory of Gases // The Physics of Fluids, 16 (1973). Pp.1557-1559.Siewert C. E. Explicit Results for the Quantum-Mechanical Energy States Basic to a Finite Square-Well Potential// Journal of Mathematical Physics, 19 (1978) 434-435.Siewert C. E. Explicit Results for the Quantum-Mechanical Energy States Basic to a Finite Square-Well Potential// Journal of Mathematical Physics, 19 (1978) 434-435.Siewert C. E. On Computing Eigenvalues in Radiative Transfer// Journal of Mathematical Physics, 21 (1980). Pp. 2468-2470.Siewert C. E. On Computing Eigenvalues in Radiative Transfer// Journal of Mathematical Physics, 21 (1980). Pp. 2468-2470.Anastasselou E. G. and Ioakamidis N. I. Application of the Cauchy theorem to the location of zeros of sectionally analytic functions// J. Appl. Math. Phys. (ZAMP). Vol. 35, №5, 1984. Pp. 705-711.Anastasselou E. G. and Ioakamidis N. I. Application of the Cauchy theorem to the location of zeros of sectionally analytic functions// J. Appl. Math. Phys. (ZAMP). Vol. 35, №5, 1984. Pp. 705-711.Ioakamidis N. I. Application of the generalized Siewert-Burniston method to locating zeros and poles of meromorphic functions// J. Appl. Math. Phys. (ZAMP). Vol. 36, №5, 1984. Pp. 733-742.Ioakamidis N. I. Application of the generalized Siewert-Burniston method to locating zeros and poles of meromorphic functions// J. Appl. Math. Phys. (ZAMP). Vol. 36, №5, 1984. Pp. 733-742.Anastasselou E. G. and Ioakamidis N. I. A generation of the Siewert-Burniston method for the determination of zeros of analytic functions// J. Math. Phys. Vol. 25, №8, 1984. 2422.Anastasselou E. G. and Ioakamidis N. I. A generation of the Siewert-Burniston method for the determination of zeros of analytic functions// J. Math. Phys. Vol. 25, №8, 1984. 2422.Ioakamidis N. I. A unified Riemann-Hilbert problem to the analytical determination of zeros of sectionally analytic functions// J. Math. Anal. and Appl. Vol. 129, №1, 1988. Pp. 134-141.Ioakamidis N. I. A unified Riemann-Hilbert problem to the analytical determination of zeros of sectionally analytic functions// J. Math. Anal. and Appl. Vol. 129, №1, 1988. Pp. 134-141.Cercignani C. Theory and application of the Boltzmann equation, Scottish Academic Press, Edinburgh and London, 1978.Cercignani C. Theory and application of the Boltzmann equation, Scottish Academic Press, Edinburgh and London, 1978.Zharinov V. V., Vladimirov V. S. Equations of mathematical physics, Fizmatlit, M., 2000. [in russian].Zharinov V. V., Vladimirov V. S. Equations of mathematical physics, Fizmatlit, M., 2000. [in russian].Gakhov F. D. Boundary value problems, Dover Publications, 1990.Gakhov F. D. Boundary value problems, Dover Publications, 1990.

The authors declare that there are no conflicts of interest present.