ASYMPTOTICS OF a PARTICLES TRANSPORT PROBLEM

Vestnik MGSU 11/2017 Volume 12
  • Kuzmina Ludmila Ivanovna - National Research University Higher School of Economics Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, National Research University Higher School of Economics, 20 Myasnitskaya st., Moscow, 101000, Russian Federation.
  • Osipov Yuri Viktorovich - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.

Pages 1278-1283

Subject: a groundwater filtration affects the strength and stability of underground and hydro-technical constructions. Research objectives: the study of one-dimensional problem of displacement of suspension by the flow of pure water in a porous medium. Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck in the pores. It is assumed that size distributions of the solid particles and the pores overlap. In this case, the main mechanism of particle retention is a size-exclusion: the particles pass freely through the large pores and get stuck at the inlet of the tiny pores that are smaller than the particle diameter. The concentrations of suspended and retained particles satisfy two quasi-linear differential equations of the first order. To solve the filtration problem, methods of nonlinear asymptotic analysis are used. Results: in a mathematical model of filtration of suspensions, which takes into account the dependence of the porosity and permeability of the porous medium on concentration of retained particles, the boundary between two phases is moving with variable velocity. The asymptotic solution to the problem is constructed for a small filtration coefficient. The theorem of existence of the asymptotics is proved. Analytical expressions for the principal asymptotic terms are presented for the case of linear coefficients and initial conditions. The asymptotics of the boundary of two phases is given in explicit form. Conclusions: the filtration problem under study can be solved analytically.

DOI: 10.22227/1997-0935.2017.11.1278-1283

Download

Asymptotic expansion of the integral with two parameters

Vestnik MGSU 7/2014
  • Kuzmina Lyudmila Ivanovna - National Research Institute “Higher School of Economics” Candidate of Physico-Mathematical Sciences, Professor Assistant, Department of Higher Mathematics of Moscow Institute of Electronics and Mathematics, National Research Institute “Higher School of Economics”, 20 ulitsa Myasnitskaya, Moscow, 101000, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Osipov Yuriy Viktorovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physico-Mathematical Sciences, Professor, Department of Information Sciences and Applied Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 34-40

In our work we study a classic pursuit problem in which two material points - a Pursuer and a Pursued - move in a plane at constant velocities. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at a higher speed, it will overtake the Pursued for any initial angle between velocity vectors. For example, a crane system simultaneously producing three movements: rotation, extension/retraction and luffing, may seize the cargo moving in a straight line while the crane is standing motionless. In the coordinate system of the Pursuer, the path length of the Pursued is given by an integral, depending on two parameters: the ratio of the initial velocities of two points and the initial angle between them. The theorem on the asymptotic integral expansion is formulated and proved considering the speed of the Pursuer is much greater than the speed of the Pursued. The first two nonzero terms of the asymptotic expansion provide fast convergence to the exact value of the integral because of the absence of the first- and the third-order asymptotic elements. The third nonzero element of the fifth order allows to determine the difference of path lengths corresponding to the adjacent initial angles between the velocities of the points.

DOI: 10.22227/1997-0935.2014.7.34-40

References
  1. Nahin Paul J. Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, 2007, 270 pp.
  2. Mungan C.E. A Classic Chase Problem Solved from a Physics Perspective. European Journal of Physics. 2005, vol. 26, pp. 985—990.
  3. Simoson A.J. Pursuit Curves for the Man in the Moone. The College Mathematics Journal. Washington, 2007, vol. 38, no. 5, pp. 330—338.
  4. Rikhsiev B.B. Differentsial'nye igry s prostym dvizheniem [Differential Games with Simple Motion]. Tashkent, Fan Publ., 1989, 232 p.
  5. Bernhart A. Curves of Pursuit. Scripta Mathematica. 1954, vol. 20, pp. 125—141.
  6. Krasovskiy N.N. Igrovye zadachi o vstreche dvizheniy [Game Problems on the Meeting of Movements], Moscow, Nauka Publ., 1970, 420 p.
  7. Azamov A.A., Kuchkarov A.Sh., Samatov B.O. O svyazi mezhdu zadachami presledovaniya, upravlyaemosti i ustoychivosti v tselom v lineynykh sistemakh s raznotipnymi ogranicheniyami [The Relation between the Chase, Controllability and Overall Stability Problems in Linear Systems with Heterogeneous Constraints]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 2007, vol. 71, no. 2, pp. 259—263.
  8. Barton J.C., Eliezer C.J. On Pursuit Curves. The Journal of the Australian Mathematical Society. Ser. B41, 2000, pp. 358—371.
  9. Kuzmina L.I., Osipov Yu.V. Raschet dliny traektorii dlya zadachi presledovanya [Path Length Calculation in the Pursuit Problem]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 12, pp. 20—26.
  10. Silagadze Z.K., Chashchina O.I. Zadacha presledovaniya zaytsa volkom kak uprazhnenie elementarnoy kinematiki [The Dog-and-rabbit Purcuit Problem as an Exercise in Introductory Kinematics]. Vestnik NGU, Seriya Fizika [Bulletin of Novosibirsk State University, Issue Physics]. 2010, vol. 5, no. 2, pp. 111—115.
  11. Silagadze Z.K., Tarantsev G.I. Comment on ‘Note on the Dog-and-rabbit Chase Problem in Introductory Kinematics’. European Journal of Physics. 2010, vol. 31, pp. 37—38.
  12. Kuzmina L.I., Osipov Yu.V. Calculation of the Pursuit Curve Length. Journal for Computational Civil and Structural Engineering. Moscow, ASV Publ., 2013, vol. 9, no. 3, pp. 31—39.
  13. Kuzmina L.I., Osipov Yu.V. Asimptotika dliny traektorii v zadache presledovaniya [Path Length Asymptotics in the Pursuit Problem]. Voprosy prikladnoy matematiki i vychislitel'noy mekhaniki [Problems of Applied Mathematics and Computational Mechanics]. Moscow, MGSU Publ., 2013, vol. 16, pp. 238—249.
  14. Maslov V.P. Asimptoticheskie metody i teoriya vozmushcheniy [Asymptotic Methods and Perturbation Theory]. Moscow, Nauka Publ., 1988, 310 p.
  15. Olver F. Introduction to Asymptotics and Special Functions. New York, Academic Press, 1974, 375 p.

Download

Calculation of the path lengthin the pursuit problem

Vestnik MGSU 12/2013
  • Kuzmina Lyudmila Ivanovna - National Research Institute “Higher School of Economics” Candidate of Physico-Mathematical Sciences, Professor Assistant, Department of Higher Mathematics of Moscow Institute of Electronics and Mathematics, National Research Institute “Higher School of Economics”, 20 ulitsa Myasnitskaya, Moscow, 101000, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Osipov Yuriy Viktorovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physico-Mathematical Sciences, Professor, Department of Information Sciences and Applied Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 20-26

Classic pursuit problem is studied with two material points — a Pursuer and an Evader, who move in plane at constant speeds. The velocity vector of the Evader does not change its direction and the velocity vector of the Pursuer turns and is always aimed at the Evader. If the Pursuer moves at a higher speed, he will overtake the Pursued for any initial angle between velocity vectors.The mechanical path geometry is established. The path line rotates around the origin of coordinates so that at the final meeting point the line tangent to the motion trajectory always coincides with the velocity vector of the Evader. The two-parameter integral for the length of the pursuit curve is considered, its asymptotics up to quartic is calculated on the assumption that the speed of the Pursuer is much higher than the speed of the Evader. Rapid convergence of the asymptotics to the integral for the path length is provided by the absence of the first and third members of the asymptotic expansion. Numerical computation of the path length is compared to the asymptotic formulas. Calculations show that the resulting asymptotics is a good approximation of the integral for the path length, and the quartic in the asymptotic formulas significantly improves the approximation.

DOI: 10.22227/1997-0935.2013.12.20-26

References
  1. Simoson A.J. Pursuit Curves for the Man in the Moone. The College Mathematics Journal. Washington, 2007, vol. 38, no. 5, pp. 330—338.
  2. Nahin P.J. Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, 2007, 270 p.
  3. Krasovskiy N.N. Igrovye zadachi o vstreche dvizheniy [Game Problems on the Meeting of Movements]. Moscow, Nauka Publ., 1970, 420 p.
  4. Rikhsiev B.B. Differentsial'nye igry s prostym dvizheniem [Differential Games with Simple Motion]. Tashkent, Fan Publ., 1989, 232 p.
  5. Azamov A.A., Kuchkarov A.Sh., Samatov B.O. O svyazi mezhdu zadachami presledovaniya, upravlyaemosti i ustoychivosti v tselom v lineynykh sistemakh s raznotipnymi ogranicheniyami [The Relation between the Pursuit, Handling and Overall Stability Problems in Linear Systems with Heterogeneous Constraints]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 2007, vol. 71, no. 2, pp. 259—263.
  6. Sigaladze Z.K., Chashchina O.I. Zadacha presledovaniya zaytsa volkom kak uprazhnenie elementarnoy kinematiki [The Dog-and-rabbit Chase Problem as an Exercise in Introductory Kinematics]. Vestnik NGU. Seriya Fizika [Bulletin of the Novosibirsk State University. Physics Series]. 2010, vol. 5, no. 2, pp. 111—115.
  7. Bernhart A. Curves of Pursuit. Scripta Mathematica. 1954, vol. 20, pp. 125—141.
  8. Barton J.C., Eliezer C.J. On Pursuit Curves. The Journal of the Australian Mathematical Society, ser. B41. 2000, pp. 358—371.
  9. Petrosyan L.A. Differential Games of Pursuit. World Scientific. Singapore, 1993, 326 p.
  10. Alekseev E.R., Chesnokova O.V. Reshenie zadach vychislitel'noy matematiki v paketakh Mathcad 12, MATLAB 7, Maple 9 [Solving Computational Mathematics Problems Using Packages Mathcad 12, MATLAB 7, Maple 9. Moscow, NT Press Publ., 2006, 492 p.
  11. Kuzmina L.I., Osipov Yu.V. Calculation of the Pursuit Curve Length. Journal for Computational Civil and Structural Engineering. Moscow, ASV Publ., 2013, vol. 9, no. 3, pp. 31—39.
  12. Kuz'mina L.I., Osipov Yu.V. Asimptotika dliny traektorii v zadache presledovaniya [Asymptotics of the PathLength in the Pursuit Problem]. Voprosy prikladnoy matematiki i vychislitel'noy mekhaniki [Problems of Applied Mathematics and Computational Mechanics]. 2013, ¹ 16, pp. 238—249.
  13. Maslov V.P. Asimptoticheskie metody i teoriya vozmushcheniy [Asymptotic Methods and Perturbation Theory]. Moscow, Nauka Publ., 1988, 310 p.
  14. Olver F. Introduction to Asymptotics and Special Functions. New York, Academic Press, 1974, 375 pp.

Download

Results 1 - 3 of 3