wave equation in nonhomogeneous media,
eiconal,
differential equation on network,
oscillation spectral theorems,
Green function of boundary value problem,
sign-regularity of discontinuous boundary value problem,
summarising of Fourie series in the order of coefficients decreasing,
constructive non-standard analysis.
Subject:
Theory of oscillation spectral properties. The analogue of Kellogg oscillation theorem is obtained for integral operators with discontinuous kernels and with integrating by measure is obtained. The effective conditions of sign-regularity (of Kalafaty type) of discontinuous boundary value problems are discovered. Theory of differential equations on networks. For forth order equations on the network the maximum principle is discovered. The system of determinating conditions for Green"s function of the forth order equation is found, the positivity if Green function proved. The method of the reduction of the problem on the network to a problem on an interval developed. Summarising of Fourie series. The formula of the maximal coeffitient in Fourie series and the formula of its number is obtained. Wave propagation in nonhomogeneous media. The formula of propagating waves for one-dimentional media is obtained. The group classification of eiconal equation for non-homogeneous and anisotropic medium fulfilled. Constructive non-standard analysis. The set of hyperreal values allowing the analysis of singularly perturbed equations is made constractively.
Biography
Graduated from Voronezh State University (Mathematical Department) in 1986. Candidate (Ph.D.) thesis "Boundary value problems with singularities" — 1990. About 20 referred publications.
Since 1996 — sci. secr. of Voronezh Spring Mathematical Schools and of Voronezh Winter Schools on the Function Theory.
Main publications:
Borovskikh A. V., Pokornyi Yu. V., “Sistemy Chebysheva–Khaara v teorii razryvnykh yader Kelloga”, Uspekhi matem. nauk, 49:3 (1994), 3–42
Borovskikh A.V., “Uravneniya eikonala dlya neodnorodnoi
i anizotropnoi sredy”, Uravneniya matematicheskoi fiziki, Sovremennaya matematika i ee prilozheniya, 64, 18–38
K. S. Platonova, A. V. Borovskikh, “Group analysis of the Boltzmann and Vlasov equations”, Theoret. and Math. Phys., 203:3 (2020), 794–823
2019
2.
A. V. Borovskikh, K. S. Platonova, “Group analysis of the one-dimensional Boltzmann equation: IV. Complete group classification in the general case”, Theoret. and Math. Phys., 201:2 (2019), 1612–1642 (cited: 1)
2018
3.
K. S. Platonova, A. V. Borovskikh, “Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful”, Theoret. and Math. Phys., 195:3 (2018), 886–915 (cited: 2)
2015
4.
I. V. Astashova, A. V. Borovskikh, V. V. Bykov, A. N. Vetokhin, A. Yu. Goritskii, N. A. Izobov, Yu. S. Ilyashenko, T. O. Kapustina, V. V. Kozlov, A. A. Kon'kov, I. V. Matrosov, V. V. Palin, N. Kh. Rozov, M. S. Romanov, I. N. Sergeev, E. V. Radkevich, O. S. Rozanova, I. V. Filimonova, A. V. Filinovskii, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova, “Scientific heritage of Vladimir Mikhailovich Millionshchikov”, J. Math. Sci. (N. Y.), 210:2 (2015), 115–134
2014
5.
A. V. Borovskikh, “Eikonal equation for anisotropic media”, J. Math. Sci. (N. Y.), 197:2 (2014), 248–289
Borovskikh A.V., “Group analysis of eikonal equation for non-homogeneous and anisotropic medium”, Abstr. of Intern. Conf. “Symmetry in non-linear physics” (Kiev, 21-27 iyunya 2009 g.), IM NAN Ukrainy, Kiev, 2009, po ssylke http://imath.kiev.ua/~appmath/
Borovskikh A.V., “Formulas of boundary control of an inhomogeneous string: I”, Differential Equations, 43:1 (2007), 69-95 (cited: 2) (cited: 2) (cited: 5)
34.
Borovskikh A.V., “Formulas of boundary control of an inhomogeneous string: II”, Differential Equations, 43:5 (2007), 656-666 (cited: 2) (cited: 2) (cited: 3)
35.
Borovskikh A.V., Rozov N.Kh., “O materialnoi i simvolnoi kulture v shkolnom obrazovanii”, Teoriya i praktika innovatsionnoi deyatelnosti v obrazovanii, MAKS Press, Moskva, 2007, 14-17
36.
Borovskikh A.V., Reikhani E., Rozov N.Kh., “Razvitie geometricheskogo myshleniya shkolnikov”, Teoriya i praktika innovatsionnoi deyatelnosti v obrazovanii, MAKS Press, Moskva, 2007, 28-46
37.
Borovskikh A.V., Reikhani E., Rozov N.Kh., “Uzly v shkole”, Teoriya i praktika innovatsionnoi deyatelnosti v obrazovanii, MAKS Press, Moskva, 2007, 47-62
A. V. Borovskikh, “The two-dimensional eikonal equation”, Siberian Math. J., 47:5 (2006), 813–834 (cited: 5) (cited: 9) (cited: 9) (cited: 18)
41.
Borovskikh A.V., “Equivalence groups of eikonal equation and classes of equivalent equations”, Bulletin of NSU, 2006, no. 4, 3-42
42.
Borovskikh A.V., Rasprostranenie voln v neodnorodnoi srede, Diss.…doktora fiz.-mat. nauk. 01.01.02 – Differentsialnye uravneniya., MGU, Moskva, 2006 , 300 pp.
43.
A. V. Borovskikh, “Equivalence groupes of eikonal equation and classes of equivalent equations”, Vestnik, Quart. J. of Novosibirsk State Univ., Series: Math., mech. and informatics, 6:4 (2006), 3–42
2005
44.
Borovskikh A.V., “An expression of the Riemann function of the wave equation in an Inhomogeneous medium via the transport coefficients”, Differential equations, 41:6 (2005), 895-896
45.
Borovskikh A.V., “Group analysis of the two-dimentional eikonal equation”, Differential equations, 41:6 (2005), 899
A. V. Borovskikh, “Group classification of the eikonal equation for a 3-dimensional inhomogeneous medium”, Sb. Math., 195:4 (2004), 479–520 (cited: 6) (cited: 3)
Borovskikh A.V., “Rasprostranenie voln v neodnorodnoi srede”, Sovremennye problemy teorii funktsii i ikh prilozheniya: Tez. dokl. 12-i Saratovskoi zimnei shkoly. (Saratov, 27 yanvarya – 3 fevralya 2004 g.), SGU, Saratov, 2004, 32-33
51.
Borovskikh A.V., “Investigation of Relaxation Oscillations with the Use of Constructive Nonstandard Analysis: I.”, Differential Equations, 40:3 (2004), 309-317 (cited: 1)
52.
Borovskikh A.V., “Investigation of Relaxation Oscillations with the Use of Constructive Nonstandard Analysis: II”, Differential Equations, 40:4 (2004), 491-501 (cited: 1)
Borovskikh A.V., “Uravnenie eikonala v neodnorodnoi srede”, Tez. dokl. Mezhdunar. konf. po differentsialnym uravneniyam i dinamicheskim sistemam (Suzdal, 5-10 iyulya 2004 g.), VGPU, Vladimir, 2004, 39-40
2005
57.
Borovskikh A.V., “Method of Propagating Waves for a One-Dimensional Inhomogeneous Medium”, Journal of Mathematical Sciences, 127:5 (2005), 2135-2158 (cited: 2)
2004
58.
Borovskikh A.V., “The Moving Source Phantom in Geometric Optics of Inhomogeneous Media”, Differential Equations, 40:7 (2004), 927-933
59.
Borovskikh A.V., “On a certain Luzin problem”, Analysis. Mathematica, 30:4 (2004), 259-287
2003
60.
A. V. Borovskikh, “Sign Regularity Conditions for Discontinuous Boundary-Value Problems”, Math. Notes, 74:5 (2003), 607–618
61.
Pokornyi Y.V., Pryadiev V.L., Borovskikh A.V., “The wave equation on a spatial network”, Doklady Mathematics, 67:1 (2003), 10-12
Borovskikh A.V., “Illusion of the moving source in geometric optics of non-homogeneous medium”, International Seminar “Days on Diffraction”. Abstracts. (S.-Peterburg, 24-27 iyunya 2003 g.), S.-Pb otdelenie MIRAN, S.-Peterburg, 2003, 17
65.
Borovskikh A.V., “The eiconal equation in an inhomogeneous medium”, Doklady Mathematics, 68:1 (2003), 103-106
2005
66.
Borovskikh A.V., “Group Classification of Eikonal Equations for the Wave Equation in Inhomogeneous Media”, Journal of Mathematical Sciences, 129:1 (2005), 3537-3549
2002
67.
Borovskikh A.V., “Summirovanie primera Feiera v poryadke ubyvaniya koeffitsientov”, Tez. dokl. shkoly “Sovremennye problemy teorii funktsii i ikh prilozheniya” (Saratov, 27 yanvarya – 3 fevralya 2002 g.), Saratovskii gosuniversitet, Saratov, 2002, 36
68.
Borovskikh A.V., Lazarev K.P., “Printsip maksimuma dlya differentsialnykh uravnenii chetvertogo poryadka na grafakh”, Tez. dokl. shkoly “Sovremennye problemy teorii funktsii i ikh prilozheniya” (Saratov, 27 yanvarya – 3 fevralya 2002 g.), Saratovskii gosuniversitet, Saratov, 2002, 37
Borovskikh A.V., Gruppovaya klassifikatsiya uravnenii eikonala dlya volnovogo uravneniya v neodnorodnoi srede, Dep. v VINITI 29.05.2002, N 953-V2002., VINITI, Moskva, 2002 , 143 pp.
73.
Borovskikh A.V., “A Traveling Wave Formula in a One-Dimensional Inhomogeneous Medium”, Differential Equations, 38:6 (2002), 796-807
74.
Borovskikh A.V., “Group classification of eikonal equation for an inhomogeneous medium”, Differential Equations, 38:11 (2002), 1669-1670
75.
Borovskikh A.V., “Geomerty of fronts and rays in a medium with linear and quadratic velocity function”, Differential Equations, 38:11 (2002), 1673
76.
Borovskikh A.V., “Application of constructive non-standard analysis to investigation of a system of connected Van-der-Pol equations”, Differential Equations, 38:11 (2002), 1678-1679
77.
Pokornyi Yu.V., Borovskikh A.V., “On defects of axiomatics of Green's function”, Doklady Mathematics, 65:3 (2002), 399-403
Pokornyi Yu.V., Borovskikh A.V., “Differential equations on networks (geometrical graphs)”, Journal of Mathematical Sciences, 119:6 (2004), 691-718
80.
Borovskikh A.V., Lazarev K.P., “Differential equations of the forth order on geometrical graphs”, Journal of Mathematical Sciences, 119:6 (2004), 719-738
81.
Pokornyi Yu.V., Borovskikh A.V., “Connection between the Green's function and the influence function for the non-classical problems”, Journal of Mathematical Sciences, 119:6 (2004), 739-768
2001
82.
Borovskikh A.V., “Rasprostranenie voln v odnomernoi neodnorodnoi srede”, Tez. dokl. shkoly “Sovremennye metody teorii funktsii i smezhnye problemy” (Voronezh, 27 yanvarya – 4 fevralya 2001 g.), VGU, Voronezh, 2001, 50-51
83.
Borovskikh A.V., “Konstruktivnyi nestandartnyi analiz”, Analiticheskie i chislennye metody v matematike i mekhanike. Tr. XXII konf. molodykh uchenykh mekh.-mat. f-ta MGU. (Moskva, 17-22 aprelya 2000 g.), izd-voMGU, Moskva, 2001, 28-31
84.
Borovskikh A.V., “Volny v odnomernoi neodnorodnoi srede”, Sovremennye metody v teorii kraevykh zadach. “Pontryaginskie chteniya - XII”. Tez. dokl. shkoly. (Voronezh, 3-9 maya 2001 g.), VGU, Voronezh, 2001, 34-35
A. V. Borovskikh, K. P. Lazarev, Yu. V. Pokornyi, “On the Kellogg kernels in discontinuous problems”, Proc. Steklov Inst. Math., 211 (1995), 92–108
105.
Borovskikh A.V., “Ob ekstremumakh razryvnykh sistem Chebysheva i Markova”, Tez. dokl. shkoly “Sovremennye metody teorii funktsii i smezhnye problemy prikladnoi matematiki i mekhaniki” (Voronezh, 25 yanvarya – 1 fevralya 1995 g.), VGU, Voronezh, 1995, 48
106.
Borovskikh A.V., Kolmykov V.A., “Integrirovanie po diskretnym meram na grafakh”, Tez. dokl. shkoly “Sovremennye metody teorii funktsii i smezhnye problemy prikladnoi matematiki i mekhaniki” (Voronezh, 25 yanvarya – 1 fevralya 1995 g.), VGU, Voronezh, 1995, 49
Borovskikh A.V., Burlutskaya M.Sh., “O differentsirovanii i integrirovanii po mere na abstraktnykh lineinykh kontinuumakh i o kvazichislakh”, Tez.dokl. II mezhdun. konferentsii “Algebraicheskie, veroyatnostnye, geometricheskie, kombinatornye i funktsionalnye metody v teorii chisel” (Voronezh, 25-30 sentyabrya 1995 g.), VGU, Voronezh, 1995, 25
113.
Borovskikh A.V., Mustafokulov R., Lazarev K.P., Pokornyi Yu.V., “A class of fourth-order differential equations on a spatial net”, Doklady Mathematics, 52:3 (1995), 433-435
1994
114.
A. V. Borovskikh, Yu. V. Pokornyi, “Chebyshev–Haar systems in the theory of discontinuous Kellogg kernels”, Russian Math. Surveys, 49:3 (1994), 1–42 (cited: 4) (cited: 3)
115.
Borovskikh A.V., Lazarev K.P., “Matematicheskaya model sterzhnevoi sistemy”, Tez.dokl. shkoly “Sovremennye problemy mekhaniki i matematicheskoi fiziki” (Voronezh, 21-28 yanvarya 1994 g..), VGU, Voronezh, 1994, 17