PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2013 | 11 | 1 | 112-132
Tytuł artykułu

Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t − au(t)/t 2 = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (−∞,−1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×ℝ. It is shown that S c = {u ∈ S: u′(T) = −c} is nonempty and compact for each c ≥ 0 and S = ∪c≥0 S c. The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S c,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.
Twórcy
Bibliografia
  • [1] Abraham F.F., Homogeneous Nucleation Theory, Academic Press, New York, 1974
  • [2] Agarwal R.P., O’Regan D., Singular Differential and Integral Equations with Applications, Kluwer, Dordrecht, 2003
  • [3] Agarwal R.P., O’Regan D., A survey of recent results for initial and boundary value problems singular in the dependent variable, In: Handbook of Differential Equations, Elsevier/North Holland, Amsterdam, 2004, 1–68
  • [4] Bongiorno V., Scriven L.E., Davis H.T., Molecular theory of fluid interfaces, J. Colloid Interface Sci., 1976, 57(3), 462–475 http://dx.doi.org/10.1016/0021-9797(76)90225-3
  • [5] Deimling K., Nonlinear Functional Analysis, Springer, Berlin, 1985 http://dx.doi.org/10.1007/978-3-662-00547-7
  • [6] Derrick G.H., Comments on nonlinear wave equations as models for elementary particles, J. Mathematical Phys., 1964, 5, 1252–1254 http://dx.doi.org/10.1063/1.1704233
  • [7] Fife P.C., Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomath., 28, Springer, Berlin- New York, 1979 http://dx.doi.org/10.1007/978-3-642-93111-6
  • [8] Fischer R.A., The wave of advance of advantegeous genes, Annals of Eugenics, 1937, 7, 355–369
  • [9] Hammerling R., Koch O., Simon C., Weinmüller E.B., Numerical solution of singular ODE eigenvalue problems in electronic structure computations, Comput. Phys. Comm., 2010, 181, 1557–1561 http://dx.doi.org/10.1016/j.cpc.2010.05.006
  • [10] Hamydy A., Existence and uniqueness of nonnegative solutions for a boundary blow-up problem, J. Math. Anal. Appl., 2010, 371(2), 534–545 http://dx.doi.org/10.1016/j.jmaa.2010.05.053
  • [11] Kalis H., Kangro I., Gedroics A., Numerical methods of solving some nonlinear heat transfer problems, Int. J. Pure Appl. Math., 2009, 57(4), 575–592
  • [12] Kiguradze I., Some Singular Boundary Value Problems for Ordinary Differential Equations, Izdat. Tbilis. Univ., Tbilisi, 1975 (in Russian)
  • [13] Kiguradze I.T., Shekhter B.L., Singular boundary value problems for second-order ordinary differential equations, J. Soviet Math., 1988, 43(2), 2340–2417 http://dx.doi.org/10.1007/BF01100361
  • [14] Kitzhofer G., Koch O., Lima P., Weinmüller E., Efficient numerical solution of the density profile equation in hydrodynamics, J. Sci. Comput., 2007, 32(3), 411–424 http://dx.doi.org/10.1007/s10915-007-9141-0
  • [15] Koleva M., Vulkov L., Blow-up of continuous and semidiscrete solutions to elliptic equations with semilinear dynamical boundary conditions of parabolic type, J. Comput. Appl. Math., 2007, 202(2), 414–434 http://dx.doi.org/10.1016/j.cam.2006.02.037
  • [16] Konyukhova N.B., Lima P.M., Morgado M.L., Soloviev M.B., Bubbles and droplets in nonlinear physics models: analysis and numerical simulation of singular nonlinear boundary value problems, Comput. Math. Math. Phys., 2008, 48(11), 2018–2058 http://dx.doi.org/10.1134/S0965542508110109
  • [17] Kubo A., Lohéac J.-P., Existence and non-existence of global solutions to initial boundary value problems for nonlinear evolution equations with strong dissipation, Nonlinear Anal., 2009, 71(12), e2797–e2806 http://dx.doi.org/10.1016/j.na.2009.06.053
  • [18] Linde A.D., Particle physics and inflationary cosmology, Proceedings of the Fourth Seminar on Quantum Gravity, Moscow, May 25–29, 1987, World Scientific, Teaneck, 1988
  • [19] O’Regan D., Theory of Singular Boundary Value Problems, World Scientific, River Edge, 1994 http://dx.doi.org/10.1142/2352
  • [20] Rachůnková I., Staněk S., Tvrdý M., Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations. III, In: Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2006, 607–722
  • [21] Rachůnková I., Staněk S., Tvrdý M., Solvability of Nonlinear Singular Problems for Ordinary Differential Equations, Contemp. Math. Appl., 5, Hindawi, New York, 2008
  • [22] Rottschäfer V., Kaper T.J., Blowup in the nonlinear Schrödinger equation near critical dimension, J. Math. Anal. Appl., 2002, 268(2), 517–549 http://dx.doi.org/10.1006/jmaa.2001.7814
  • [23] Sibley L., Armbruster D., Gouin H., Rotoli G., An analytical approximation of density profile and surface tension of microscopic bubbles for van der Waals fluids, Mech. Res. Comm., 1997, 24(3), 255–260 http://dx.doi.org/10.1016/S0093-6413(97)00022-0
  • [24] van der Waals J.D., Kohnstamm R., Lehrbuch der Thermodynamik. I, Maas & van Suchtelen, Leipzig-Amsterdam, 1908
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0047-1
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.