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Depart ment of Differential Equations

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Arlen Mikhailovich Il'in

Scientific chief of the Department of Differential Equations,
Doctor of Science, Professor, Academician of RAS

ilam@go.ru

Scientific Interests

  • Application of asymptotic methods for solving nonlinear problems of mathematical physics

  • Asymptotics for the solutions of nonlinear equations

  • Catastrophe theory

Experience

  • Born 8 January 1932.
  • 1949 graduated from high-school № 59 with gold medal, Moscow.
  • 1949-1954 Student, Faculty of Mathematics and Mechanics, Moscow State University.
  • May 1957 PhD, supervisor Prof. O.A. Oleinik.
  • 1957-1963 work in the Department of Differential Equations, Moscow State University.
  • From 1963 work in the Sverdlovsk Division of V.A. Steklov Mathematical Institute (today Institute of Mathematics and Mechanics, Ural Branch of RAS).
  • 1973-1987 Head of the Sector of Differential Equations, Division of Physics and Mathematics, Bashkir Branch of the Academy of Sciences, then Head of the Department of Differential Equations, Institute of Mathematics, Ufa Scientific Center of RAS.
  • January 1988 Doctor's degree.
  • 1988-1994 Professor of the Ural Polytechnical Institute, Sverdlovsk.
  • 1990-1994 Leading Researcher, Institute of Mathematics and Mechanics, Ural Branch of RAS, Sverdlovsk.
  • March 1994 Corresponding member of the Russian Academy of Sciences.
  • 1994-2002 Head of the Department of Equations of Mathematical Physics, Institute of Mathematics and Mechanics, Ural Branch of RAS, Sverdlovsk.
  • 1995 I.G. Petrovskii prize of the Russian Academy of Sciences for a cycle of works "Asymptotic methods in mathematical physics" (together with O.A. Oleinik).
  • May 2000 Member of the Russian Academy of Sciences.
  • 2000 State prize for a cycle of works "Asymptotic methods for studying equations of mathematical physics" (together with V.S. Buslaev and M.V. Karasev).
  • 2002 - present Professor, Chelyabinsk State University.
  • Supervisor of 12 philosophy doctors. Scientific interests are concerned to the application of asymptotic methods for solving nonlinear problems of mathematical physics. Author of more than 60 publications.

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Conferences

  • May 1996 Int. conference "Complex analysis, differential equations and applications", Ufa, Russia
  • 3-8 August 1998 Int. conference "Exactly solvable models in mathematical physics", Chelyabinsk, Russia
  • 28 May - 1 June 2000 Int. conference "Complex analysis, differential equations and related problems", Ufa, Russia
  • 23-29 September 2000 Int. scientific seminar "Methods of functional analysis and function theory in different problems of mathematical physics", Seminar of Prof. R.S. Saks, Ufa, Russia
  • 26-30 May 2002 Int. conference "Asymptotics in differential equations", Ufa, Russia
  • May 2004 Int. conference "Differential Equations and Related Topics", Moscow, Russia

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Publications

  • Il'in A.M. On Dirichlet problem for an equation of elliptic type degenerating on some set of interior points of a region (Russian). Dokl.Akad.Nauk SSSR, 102, no.1 (1955), 9-12.
  • Il'in A.M., Oleinik O.A. Behavior of solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time (Russian). Dokl.Akad.Nauk SSSR, 120, no.1 (1958), 25-28.
  • Il'in A.M. Degenerate elliptic and parabolic equations (Russian). Mat.Sb., 50, no.4 (1960), 443-498.
  • Il'in A.M.,Oleinik O.A. Asymptotic behavior of the solutions of the Cauchy problem for some quasilinear equations for large values of the time (Russian). Mat.Sb., 51, no.2 (1960), 191-216.
  • Il'in A.M. On the behavior of the solutions of the Cauchy problem for a parabolic equation under unrestricted growth of time (Russian). Uspehi Mat. Nauk, 6, no.2 (1961), 115-121.
  • Il'in A.M., Kalashnikov A.S., Oleinik O.A. Second order linear equations of parabolic type (Russian). Uspehi Mat. Nauk, 7, no.3 (1962), 3-146.
  • Il'in A.M., Has'minskii R.Z. On an ergodic property of non-homogeneous diffusion processes (Russian). Dokl.Akad.Nauk USSR, 145, no. 5 (1962), 986-988.
  • Il'in A.M. On the fundamental solution of a parabolic equation (Russian). Dokl.Akad.Nauk SSSR, 147, no.4 (1962), 768-771.
  • Il'in A.M., Has'minskii R.Z. Asymptotic behavior of solutions of parabolic equations and an ergodic property of non-homogeneous diffusion processes (Russian). Mat.Sb., 60, no.3 (1963), 366-392.
  • Il'in A.M., Kamenkovich V.M. On influence friction on ocean streams. Dokl.Akad.Nauk USSR, v. 150, no 6 (1963), 1274-1277.
  • Il'in A.M., Has'minskii R.Z. On the equation of Brownion motion (Russian). Teor. Verojatnos. i Primen., 9, no. 3 (1964), 466-491.
  • Il'in A.M. On a class of ultraparabolic equations (Russian). Dokl.Akad.Nauk SSSR, 159, no. 61 (1964), 1214-1217.
  • Il'in A.M. On additional chains of numbers (Russian). Problemy kibernetiki, vyp.3 (1965), 245-248.
  • Il'in A.M. On eigenfunctions of elliptic operator in some unbounded domains (Russian). Dokl.Akad.Nauk SSSR, 161, no.4 (1965), 757-759.
  • Il'in A.M. Stability of difference scheme of Cauchy problem for sistems of partial differential equations (Russian). Dokl.Akad.Nauk SSSR, 164, no.3 (1965), 491-494.
  • Il'in A.M. On parabolic equations with continious coefficients (Russian). Dokl.Akad.Nauk SSSR, 171, no. 6 (1966), 1264-1268.
  • Il'in A.M. On parabolic equations with coefficients which do not satisfy Dini condition (Russian). Matem. zametki, 1, no.1 (1967), 71-80.
  • Il'in A.M., Emel'yanov K.V. On a number of arithmetic actions which is nesessary for approximate solving of Fredholm second type integral equation (Russian). Jurn. Vychisl. Mat. i Matem. Fiz., 7, no.4 (1967), 905-910.
  • Il'in A.M. Difference scheme for differential equation with a small parameter at higher derivatives (Russian). Matem. zametki, 6, no.2 (1969),237-248.
  • Il'in A.M., Shishkin G.I. The first boundary value problem for the equation of Brownian motion on the halfline (Russian). Ural. Gos. Univ. Mat. Zap., 7 (1969/1970), tetrad' 2, 59-75.
  • Il'in A.M. The asymptotic behavior of the solution of a certain boundary value problem (Russian). Matem. zametki, 8, no.3 (1970), 273-284.
  • Il'in A.M., Brainina H.Z., Yarunina G.V., Neiman N.Ya. Electrodissolving of the metals from the surface of the indifferent electrode. VI. Processes complicated by chemical reaction (Russian). Electrochemistry, v.7, No 6, 1971, 888-893.
  • Il'in A.M., Brainina H.Z., Yarunina G.V., Neiman N.Ya. Electrodissolving of the metals from the surface of the indifferent electrode. VII. Kinetics of oxidition of microphase of metal (Russian). Electrochemistry, v.12, No 12, 1972, 1790-1794.
  • Il'in A.M. The behavior of the solution of a certain boundary value problem as $t\to\infty$ (Russian). Mat.Sb., 87, no.4 (1972), 529-553.
  • Il'in A.M., Gor'kov Yu.P., Lelikova E.F. The method of matching asymptotic expansions (Russian). Dokl.Akad.Nauk SSSR, 217, no 5, (1974), 1033-1036.
  • Il'in A.M., Gor'kov Yu.P., Lelikova E.F. Asymptotic behavior of the solution of an elliptic equation with a small parameter multiplying the highest derivatives in the neighborhood of a singular characteristic of the limit equation (Russian). Trudy Sem. Petrovsk., vyp.1 (1975), 75-133.
  • Il'in A.M., Lelikova E.F. The method of matching asymptotic expansions for the equation $\varepsilon\Delta u-a(x,y)u_y=f(x,y)$ in a rectangle (Russian). Mat.Sb., 96, no.4 (1975), 568-583. English transl. in Math. USSR Sb., 25, no.4 (1975), 533-548.
  • Il'in A.M., Yemel'yanov K.V., Shishkin G.I., Kalyakin L.A., Solution of differential equations with a small parameter, Proc. of the seminar on the numerical methods, Novosibirsk, 1975 (in Russian).
  • Il'in A.M. The difference method for the solution of the Dirichlet problem for Laplace's equation in the case of discontinuous boundary conditions (Russian). Trudy Inst. Mat. i Mech., Ural Nauchn. Centr., vyp. 21 (1976), 44-51.
  • Il'in A.M., Shaigardanov Yu.Z. Boundary value problem for a differential equation with a small parameter at the highest derivative (Russian). Sbornik "Kraevye zadachi matematicheskoi phiziki i ih prilozheniya", Ufa, BFAN SSSR, (1976), 3-17.
  • Il'in A.M. A boundary value problem for elliptic second order equation in a domain with a narrow slit. 1. Two-dimensional case (Russian). Mat.Sb., 99, no.4 (1976), 514-537; English transl. in Math.USSR Sb., 28 (1976), 4, 459-480.
  • Il'in A.M. A boundary value problem for elliptic second order equation in a domain with a narrow slit. 2. Domain with a small hole (Russian). Mat.Sb., 103, no.2 (1977), 265-284; English transl. in Math.USSR Sb., 32(1977), 2, 227-244.
  • Il'in A.M. Degenerate partial differential equation (Russain). Mathematic encyclopedia, v.1 (1977), 810-811.
  • Il'in A.M., Nesterova T.N. Asymptotic behavior of the solution of the Cauchy problem for a quasilinear parabolic equation with a small parameter (Russian). Dokl.Akad.Nauk SSSR, 240, no.1 (1978), 11-13. English transl. in Soviet Math. Dokl., 19, no.3 (1978), 529-532.
  • Il'in A.M. Asymptotic behavior of a boundary value problem on the half-line for a parabolic equation (Russian). Application of the Method of Matched Asymptotic Expansions to Boundary Value Problems for Differential Equations, Ural Nauchn. Centr. AN SSSR, Sverdlovsk, 1979, 81-92.
  • Il'in A.M., Nasirov K.H. Matching method of the asymptotic expansions for elliptic boundary value problem with a small parameter (Russian). Sbornik " Differential equations with a small parameter. Ural Nauchn. Centr. AN SSSR, Sverdlovsk, 1980, 8-15.
  • Il'in A.M. Asymptotics of the solution of Poisson equation out of halfstrip (Russian). Sbornik "Matching method for asymptotic expansions in problems with singular perturbations", Ufa, BFAN SSSR, 1980, 115-134.
  • Il'in A.M. Study of the asymptotic behavior of the solution of an elliptic boundary value problem in a domain with a small hole (Russian). Trudy Sem. Petrovsk., vyp. 6 (1981), 57-82.
  • Il'in A.M. The structure of shokwave in presense of a small viscousity (Russian). Sbornik "Nonlinear waves", M., "Nauka", 1981, 234-237.
  • Il'in A.M. Asymptotics of the solution of a boundary value problem for a Poisson's equation in the exterior of a cylinder and a half-cylinder (Russian). Mat.Sb., 118, no.2 (1982), 184-202; English transl. in Math. USSR Sb., 6 (1983), 183-202.
  • Il'in A.M., Fedoryuk M.V. The method of a small parameter for partial differential equations (Russian). Mathematic encyclopedia, v.3, 1982.
  • Il'in A.M., Lelikova E.F. Asymptotics of solutions of some elliptic equations in unbounded domains (Russian). Mat.Sb., 119, no.3 (1982), 307-324; English transl. in Math. USSR Sb., 7, no.2 (1983), 295-313.
  • Il'in A.M., Suleimanov B.I. The asymptotics of the Green function for a second-order elliptic equation near the boundary of domain (Russian). Izv.Acad.Nauk SSSR, ser. mat., 47, no.6 (1983), 149-165; English transl. in Math. USSR Izvestiya, 23, no.3 (1984), 579-596.
  • Il'in A.M., Lelikova E.F. Asymptotics of solutions of some higher order elliptic equations in conical domains (Russian). Mat.Sb., 125, no.1 (1984), 88-116; English transl. in Math. USSR Sb., 53, no.1 (1986), 89-117.
  • Il'in A.M. On a sufficient condition of the stabilization of the solution of a parabolic equation (Russian). Matem. zametki, 37, no.6 (1985), 851-856.
  • Il'in A.M. Cauchy problem for a quasilinear parabolic equation with a small parameter (Russian). Dokl.Akad.Nauk SSSR, 283, no. 3 (1985), 530-534; English transl. in Soviet Math. Dokl., 32, no.1 (1985), 133-136.
  • Il'in A.M. Value problem for a Burger's equation without initial conditions (Russian). Uspehi Mat. Nauk, 41, no.5 (1986), 208-209.
  • Il'in A.M. Method of matched asymptoyic expansions. BAIL-IV (Novosibirsk, 1986), 98-109, Boole Press Conf. Ser., 8, Boole, Dun Laoghaire, 1986.
  • Il'in A.M. A boundary layer (Russian). Current problems in mathematics. Fundamental directions. vol.34, 175-213; Itogi Nauki i Tekhniki, Akad. Nauk SSSR, M., 1988.
  • Il'in A.M. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Transl. of Mathematical Monographs, AMS, v.102 (1992), 281 p.
  • Il'in A.M., Khudyaev S.I. On the asymptotics of astationary ignition wave in a condensed medium (Russian). Chim.Fiz., 8, no.4 (1989).
  • Il'in A.M. On the asymptotic behavior of the solution for a boundary value problem with a small parameter (Russian). Izv.Acad.Nauk SSSR, Ser. Mat., 53, no.2 (1989), 258-275; English transl.in Math. USSR Izvestiya, 34, no.2 (1990), 261-279.
  • Il'in A.M., Danilin A.R. Asymptotic of the Solution of the Optimal-Time Control Problem under Perturbed Initial Data (Russian). Izvestiya.Akad.Nauk RAN, Technicheskaya kibernetica, no. 2 (1994), 96-103.
  • Il'in A.M., Kalyakin L.A. Perturbation of N-soliton of the Korteveg-de Vries equation (Russian). Dokl.Akad.Nauk RAN, 336, no. 5 (1994), 595-598.
  • Il'in A.M., Danilin A.R. Asymptotic Behavior of the Solution of the Optimal-Time Control Problem under Perturbed Initial Data (Russian). Dokl.Akad.Nauk RAN, 350, no. 2 (1996), 155-157.
  • Il'in A.M., Neudachin D.I. On the influence of small dissipation on the solutions of a hyperbolic system with weak discontinuity. Dokl.Akad.Nauk RAN, 350, no. 5 (1996), 583-585.
  • Il'in A.M., Kalyakin L.A., Maslennikov S.I. The matching method for asymptotic solutions in chemical physics problem. Singular Perturbation Problems in Chemical Physics: Analytic and Computational Methods, in Advances in Chemical Physics Series, Vol. XCVII, John Wiley & Sons, 1997, 1-45.
  • Il'in A.M., Suleimanov B.I., On two special functions, concerned with the cusp singularity. Doklady RAN, 2002. V. 387, N 2. P. 156--158. (in Russian)
  • Il'in A.M., Suleimanov B.I., Coefficients of inner expansion in study of asymptotics for some singular boundary value problems. Dal'nevost. math. journal, 2003. V. 4, N 1. P. 78-85. (in Russian)
  • Ильин А.M., Сулейманов Б.И. Формирование контрастных структур типа ступеньки, связанное с катастрофой сборки. Тезисы конференции И.Г. Петровского. Москва, МГУ. 2004. С. 87.
  • Ильин А.М., Захаров С.В. Данилин А.Р. Применение метода согласования асимптотических разложений к решению краевых задач. //Современная математика и ее приложения. Т. 5 Асимптотические методы анализа.: Академия Наук Грузии, Институт Кибернетики: 2003, с.33-78.
  • Ильин А.М., Коврижных О.О. Асимптотика решения системы линейных уравнений с двумя малыми параметрами 2004 г., ДАН, т.396, №1, С. 23 - 24.
  • Ильин А.М., Нестерова В.В. Асимптотика решения вырождающегося дифференциального уравнения третьего порядка с малым параметром при старшей производной 2004 г., ЖВМиМФ, том 44, №9, С.1574 -1584
  • Ильин А.М., Сулейманов Б.И. Зарождение контрастных структур типа ступеньки, связанное с катастрофой сборки. Математический сборник. T. 195, № 12. 2004. С. 3-27.
  • Ильин А.M., Сулейманов Б.И. Зарождение кинкообразных асимптотических решений, связанное с катастрофой сборки. Тезисы докладов Всероссийской конференции «Новые математические модели в механике сплошных сред: построение и изучение». Новосибирск, ИГиЛ СО РАН, 2004. С. 64-65.
  • Ильин А.M., Сулейманов Б.И. Поведение решения эллиптического уравнения с малым параметром в окрестности внутренней особой точки. Алгоритмический анализ неустойчивых задач: Тезисы докладов Всероссийской конференции. Екатеринбург, Изд-во УрГУ. 2004, с.166-167.
  • Ильин А.М., Меленцов М.А.Асимптотика решений систем дифференциальных уравнений с малым параметром при больших значениях времени. Труды института математики и механики, т. 11 № 1,2005, стр 97-111.

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A.M.Il'in