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Open Access Open Badges Research Article

Terminal value problem for singular ordinary differential equations: Theoretical analysis and numerical simulations of ground states

Alex P Palamides1* and Theodoros G Yannopoulos2

Author Affiliations

1 Department of Telecommunications Science and Technology, University of Peloponesse, Tripolis 22100, Greece

2 Department of Mathematics, Technological Educational Institute (TEI) of Athens, Egaleo 12210, Greece

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Boundary Value Problems 2006, 2006:28719  doi:10.1155/BVP/2006/28719

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2006/1/28719

Received:18 October 2005
Revisions received:26 July 2006
Accepted:13 August 2006
Published:5 November 2006

© 2006 Palamides and Yannopoulos

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A singular boundary value problem (BVP) for a second-order nonlinear differential equation is studied. This BVP is a model in hydrodynamics as well as in nonlinear field theory and especially in the study of the symmetric bubble-type solutions (shell-like theory). The obtained solutions (ground states) can describe the relationship between surface tension, the surface mass density, and the radius of the spherical interfaces between the fluid phases of the same substance. An interval of the parameter, in which there is a strictly increasing and positive solution defined on the half-line, with certain asymptotic behavior is derived. Some numerical results are given to illustrate and verify our results. Furthermore, a full investigation for all other types of solutions is exhibited. The approach is based on the continuum property (connectedness and compactness) of the solutions funnel (Knesser's theorem), combined with the corresponding vector field's ones.


  1. Agarwal, RP, Kiguradze, I: Two-point boundary value problems for higher-order linear differential equations with strong singularities. Boundary Value Problems. 2006, 32 pages (2006)

  2. Baxley, JV: Boundary value problems on infinite intervals. In: Henderson J (ed.) Boundary Value Problems for Functional-Differential Equations, pp. 49–62. World Scientific, New Jersey (1995)

  3. Berestycki, H, Lions, P-L, Peletier, LA: An ODE approach to the existence of positive solutions for semilinear problems in . Indiana University Mathematics Journal. 30(1), 141–157 (1981). Publisher Full Text OpenURL

  4. Bonheure, D, Gomes, JM, Sanchez, L: Positive solutions of a second-order singular ordinary differential equation. Nonlinear Analysis. 61(8), 1383–1399 (2005). Publisher Full Text OpenURL

  5. Copel, WA: Stability and Asymptotic Behavior of Differential Equations, Heath, Massachusetts (1965)

  6. Dell'Isola, F, Gouin, H, Rotoli, G: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. European Journal of Mechanics. B. Fluids. 15(4), 545–568 (1996)

  7. Gazzola, F, Serrin, J, Tang, M: Existence of ground states and free boundary problems for quasilinear elliptic operators. Advances in Differential Equations. 5(1–3), 1–30 (2000)

  8. Kuratowski, K: Topology II, Academic Press, New York (1968)

  9. Palamides, PK: Singular points of the consequent mapping. Annali di Matematica Pura ed Applicata. Serie Quarta. 129, 383–395 (1982) (1981). Publisher Full Text OpenURL

  10. Palamides, PK: Boundary-value problems for shallow elastic membrane caps. IMA Journal of Applied Mathematics. 67(3), 281–299 (2002). Publisher Full Text OpenURL

  11. Palamides, PK, Galanis, GN: Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line. Nonlinear Analysis. 57(3), 401–419 (2004). Publisher Full Text OpenURL

  12. Rocard, Y: Thermodynamique, Masson, Paris (1967)

  13. Walter, W: Ordinary Differential Equations, Graduate Texts in Mathematics,p. xii+380. Springer, New York (1998)

  14. Wasov, W: Asymptotic Expressions for Ordinary Differential Equations, John Wiley & Sons, New York (1965)