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

Abstract

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.

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References

  1. 1.

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

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Baxley JV: Boundary value problems on infinite intervals. In Boundary Value Problems for Functional-Differential Equations. Edited by: Henderson J. World Scientific, New Jersey; 1995:49-62.

    Google Scholar 

  3. 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 1981,30(1):141-157. 10.1512/iumj.1981.30.30012

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bonheure D, Gomes JM, Sanchez L: Positive solutions of a second-order singular ordinary differential equation. Nonlinear Analysis 2005,61(8):1383-1399. 10.1016/j.na.2005.02.029

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Copel WA: Stability and Asymptotic Behavior of Differential Equations. Heath, Massachusetts; 1965.

    Google Scholar 

  6. 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 1996,15(4):545-568.

    MATH  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Kuratowski K: Topology II. Academic Press, New York; 1968.

    Google Scholar 

  9. 9.

    Palamides PK: Singular points of the consequent mapping. Annali di Matematica Pura ed Applicata. Serie Quarta 1981, 129: 383-395 (1982). 10.1007/BF01762151

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Palamides PK: Boundary-value problems for shallow elastic membrane caps. IMA Journal of Applied Mathematics 2002,67(3):281-299. 10.1093/imamat/67.3.281

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Palamides PK, Galanis GN: Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line. Nonlinear Analysis 2004,57(3):401-419. 10.1016/j.na.2004.02.021

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Rocard Y: Thermodynamique. Masson, Paris; 1967. chapter 5

    Google Scholar 

  13. 13.

    Walter W: Ordinary Differential Equations, Graduate Texts in Mathematics. Volume 182. Springer, New York; 1998:xii+380.

    Google Scholar 

  14. 14.

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

    Google Scholar 

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Correspondence to Alex P Palamides.

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Palamides, A.P., Yannopoulos, T.G. Terminal value problem for singular ordinary differential equations: Theoretical analysis and numerical simulations of ground states. Bound Value Probl 2006, 28719 (2006). https://doi.org/10.1155/BVP/2006/28719

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Keywords

  • Surface Tension
  • Ordinary Differential Equation
  • Vector Field
  • Asymptotic Behavior
  • Functional Equation
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