Singular problems on the half-line

Irena Rachůnková; Jan Tomeček

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2009)

  • Volume: 48, Issue: 1, page 109-128
  • ISSN: 0231-9721

Abstract

top
The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form ( p ( t ) u ' ( t ) ) ' = p ( t ) f ( u ( t ) ) , u ' ( 0 ) = 0 , u ( ∞ ) = L . The existence of a strictly increasing solution (a homoclinic solution) of this problem is proved by the dynamical systems approach and the lower and upper functions method.

How to cite

top

Rachůnková, Irena, and Tomeček, Jan. "Singular problems on the half-line." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 48.1 (2009): 109-128. <http://eudml.org/doc/35191>.

@article{Rachůnková2009,
abstract = {The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form \[(p(t)u^\{\prime \}(t))^\{\prime \} = p(t)f(u(t)),\]\[u^\{\prime \}(0) = 0,\quad u(\infty ) = L.\] The existence of a strictly increasing solution (a homoclinic solution) of this problem is proved by the dynamical systems approach and the lower and upper functions method.},
author = {Rachůnková, Irena, Tomeček, Jan},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Singular ordinary differential equation of the second order; lower and upper functions; time singularities; unbounded domain; homoclinic solution; singular ordinary differential equation of the second order; lower and upper functions; time singularities; unbounded domain; homoclinic solutions},
language = {eng},
number = {1},
pages = {109-128},
publisher = {Palacký University Olomouc},
title = {Singular problems on the half-line},
url = {http://eudml.org/doc/35191},
volume = {48},
year = {2009},
}

TY - JOUR
AU - Rachůnková, Irena
AU - Tomeček, Jan
TI - Singular problems on the half-line
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2009
PB - Palacký University Olomouc
VL - 48
IS - 1
SP - 109
EP - 128
AB - The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form \[(p(t)u^{\prime }(t))^{\prime } = p(t)f(u(t)),\]\[u^{\prime }(0) = 0,\quad u(\infty ) = L.\] The existence of a strictly increasing solution (a homoclinic solution) of this problem is proved by the dynamical systems approach and the lower and upper functions method.
LA - eng
KW - Singular ordinary differential equation of the second order; lower and upper functions; time singularities; unbounded domain; homoclinic solution; singular ordinary differential equation of the second order; lower and upper functions; time singularities; unbounded domain; homoclinic solutions
UR - http://eudml.org/doc/35191
ER -

References

top
  1. Berestycki, H., Lions, P. L., Peletier, L. A., An ODE approach to the existence of positive solutions for semilinear problems in ℛ N , Indiana University Mathematics Journal 30, 1 (1981), 141–157. (1981) MR0600039
  2. Bonheure, D., Gomes, J. M., Sanchez, L., Positive solutions of a second–order singular ordinary differential equation, Nonlinear Analysis 61 (2005), 1383–1399. (2005) Zbl1109.34310MR2135816
  3. Dell’Isola, F., Gouin, H., Rotoli, G., Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations, Eur. J. Mech B/Fluids 15 (1996), 545–568. (1996) 
  4. Gouin, H., Rotoli, G., An analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids, Mech. Research Communic. 24 (1997), 255–260. (1997) Zbl0899.76064
  5. Kitzhofer, G., Koch, O., Lima, P., Weinmüller, E., Efficient numerical solution of the density profile equation in hydrodynamics, J. Sci. Comput. 32, 3 (2007), 411–424. (2007) Zbl1179.76062MR2335787
  6. Koch, O., Kofler, P., Weinmüller, E., Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind, Analysis 21 (2001), 373–389. (2001) Zbl1029.34002MR1867622
  7. Lima, P. M., Chemetov, N. V., Konyukhova, N. B., Sukov, A. I., Analytical–numerical investigation of bubble-type solutions of nonlinear singular problems, J. Comp. Appl. Math. 189 (2006), 260–273. (2006) Zbl1100.65066MR2202978
  8. Rachůnková, I., Koch, O., Pulverer, G., Weinmüller, E., On a singular boundary value problem arising in the theory of shallow membrane caps, J. Math. Anal. Appl. 332 (2007), 532–541. (2007) Zbl1118.34013MR2319681

NotesEmbed ?

top

You must be logged in to post comments.