Geodesic
Bubbles and Droplets in Nonlinear Physics Models: Analysis and Numerical Simulation of Singular Nonlinear Boundary Value Problem
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 11, pp. 2019-2023 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For a second-order nonlinear ordinary differential equation (ODE), a singular Boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble-type (droplet-type) solutions are sought. The equation, defined on a semi-infinite interval $0, possesses a regular singular point as $r\to0$ and an irregular one as $r\to\infty$. We give the restrictions to the parameters for a correct mathematical statement of the limit boundary conditions in singular points and their accurate transfer into the neighborhoods of these points using certain results for singular Cauchy problems and stable initial manifolds. The necessary and sufficient conditions for the existence of bubble-type (droplet-type) solutions are discussed (in the form of additional restrictions to the parameters) and some estimates are obtained. A priori detailed analysis of a singular nonlinear BVP leads to efficient shooting methods for solving it approximately. Some results of the numerical experiments are displayed and their physical interpretation is discussed. 
@article{ZVMMF_2008_48_11_a9,
     author = {N. B. Konyukhova and P. M. Lima and M. L. Morgado and M. B. Soloviev},
     title = {Bubbles and {Droplets} in {Nonlinear} {Physics} {Models:} {Analysis} and {Numerical} {Simulation} of {Singular} {Nonlinear} {Boundary} {Value} {Problem}},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2019--2023},
     year = {2008},
     volume = {48},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a9/}
}
TY  - JOUR
AU  - N. B. Konyukhova
AU  - P. M. Lima
AU  - M. L. Morgado
AU  - M. B. Soloviev
TI  - Bubbles and Droplets in Nonlinear Physics Models: Analysis and Numerical Simulation of Singular Nonlinear Boundary Value Problem
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 2019
EP  - 2023
VL  - 48
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a9/
LA  - en
ID  - ZVMMF_2008_48_11_a9
ER  - 
%0 Journal Article
%A N. B. Konyukhova
%A P. M. Lima
%A M. L. Morgado
%A M. B. Soloviev
%T Bubbles and Droplets in Nonlinear Physics Models: Analysis and Numerical Simulation of Singular Nonlinear Boundary Value Problem
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 2019-2023
%V 48
%N 11
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a9/
%G en
%F ZVMMF_2008_48_11_a9
N. B. Konyukhova; P. M. Lima; M. L. Morgado; M. B. Soloviev. Bubbles and Droplets in Nonlinear Physics Models: Analysis and Numerical Simulation of Singular Nonlinear Boundary Value Problem. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 11, pp. 2019-2023. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_11_a9/

[1] Dell'Isola F., Gouin H., Seppecher P., “Radius and surface tension of microscopic bubbles by second gradient theory”, C. r. Acad. Sci. Paris, 320 (1995), 211–216, Serie IIb

[2] Dell'Isola F., Gouin H., Rotoli G., “Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations”, Eur. J. Mech. B Fluids, 15:4 (1996), 545–568 | MR

[3] Rudakov V. A., Klassicheskie kalibrovochnye polya, Editorial URSS, M., 1999 | MR

[4] Linde A. D., Fizika elementarnykh chastits i inflyatsionnaya kosmologiya, Nauka, M., 1991 | MR

[5] Lima P. M., Konyukhova N. B., Sukov A. I., Chemetov N. V., “Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems”, J. Comput. Appl. Math., 189 (2006), 260–273 | DOI | MR | Zbl

[6] Lyapunov A. M., Obschaya zadacha ob ustoichivosti dvizheniya, Gostekhteorizdat, M., L., 1950 | Zbl

[7] Konyukhova N. B., “Singulyarnye zadachi Koshi dlya sistem obyknovennykh differentsialnykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 23:3 (1983), 629–645 | MR | Zbl

[8] Konyukhova N. B., “O povedenii reshenii vnutri i vne ustoichivogo mnogoobraziya nekotorykh dvumernykh nelineinykh sistem obyknovennykh differentsialnykh uravnenii”, Matem. zametki, 8:3 (1970), 285–295 | Zbl

[9] Konyukhova N. B., “O predelnom povedenii ogranichennogo resheniya sistemy evolyutsionnykh kvazilineinykh uravnenii s chastnymi proizvodnymi pervogo poryadka pri neogranichennom vozrastanii vremeni”, Differents. ur-niya, 28:9 (1992), 1561–1573 | MR | Zbl

[10] Konyukhova N. B., “Ob ustoichivykh mnogoobraziyakh Lyapunova dlya avtonomnykh sistem nelineinykh obyknovennykh differentsialnykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 34:10 (1994), 1358–1379 | MR | Zbl

[11] Derrick G. H., “Comments on nonlinear wave equations as models for elementary particles”, J. Math. Phys., 5:9 (1964), 1252–1254 | DOI | MR

[12] Gazzola F., Serrin J., Tang M., “Existence of ground states and free boundary problems for quasilinear elliptic operators”, Advances Different. Equat., 5:1–3 (2000), 1–30 | MR | Zbl