| Title:
|
Singular problems on the half-line (English) |
| Author:
|
Rachůnková, Irena |
| Author:
|
Tomeček, Jan |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2009 |
| Pages:
|
109-128 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Singular ordinary differential equation of the second order |
| Keyword:
|
lower and upper functions |
| Keyword:
|
time singularities |
| Keyword:
|
unbounded domain |
| Keyword:
|
homoclinic solution |
| MSC:
|
34B16 |
| MSC:
|
34B40 |
| idZBL:
|
Zbl 1209.34037 |
| idMR:
|
MR2641952 |
| . |
| Date available:
|
2010-02-11T13:57:59Z |
| Last updated:
|
2012-05-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137518 |
| . |
| Reference:
|
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| Reference:
|
[2] Bonheure, D., Gomes, J. M., Sanchez, L.: Positive solutions of a second–order singular ordinary differential equation.Nonlinear Analysis 61 (2005), 1383–1399. Zbl 1109.34310, MR 2135816 |
| Reference:
|
[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. |
| Reference:
|
[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. Zbl 0899.76064 |
| Reference:
|
[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. Zbl 1179.76062, MR 2335787 |
| Reference:
|
[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. Zbl 1029.34002, MR 1867622 |
| Reference:
|
[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. Zbl 1100.65066, MR 2202978 |
| Reference:
|
[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. Zbl 1118.34013, MR 2319681 |
| . |
