zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Group classification of the generalized Emden-Fowler-type equation. (English) Zbl 1191.34045
This work is devoted to the [group-theoretical] analysis of the generalized Emden-Fowler equation $$xu^{\prime\prime}+nu^{\prime}+x^{\nu}F(u)=0.$$ In dependence of the function $F(u)$, the point symmetries of the equation are found, which gives eight possible cases. Up to the well known linearizable case, the number of these symmetries does not exceed three. These are then compared with the Noether symmetries in order to obtain first integrals. The symmetries are applied in some cases to reduce the corresponding equation.

MSC:
34C14Symmetries, invariants (ODE)
34A26Geometric methods in differential equations
34A05Methods of solution of ODE
WorldCat.org
Full Text: DOI
References:
[1] Thomson, W.: Collected papers. 5, 266 (1991)
[2] Emden, R.: Gaskugeln, anwendungen der mechanischen warmen-theorie auf kosmologie und meteorologische probleme. (1907) · Zbl 38.0952.02
[3] Fowler, R. H.: The form near infinity of real, continuous solutions of a certain differential equation of the second order. Q. J. Math. (Oxford) 45, 289-350 (1914) · Zbl 45.0479.01
[4] Fowler, R. H.: Further studies of Emden’s and similar differential equations. Q. J. Math. (Oxford) 2, 259-288 (1931) · Zbl 0003.23502
[5] Mellin, C. M.; Mahomed, F. M.; Leach, P. G. L.: Solution of generalized Emden--Fowler equations with two symmetries. Int. J. Nonlin. mech. 29, 529-538 (1994) · Zbl 0812.34001
[6] Meerson, E.; Megged, E.; Tajima, T.: On the quasi-hydrostatic flows of radiatively cooling self-gravitating gas clouds. Astrophys. J. 457, 321 (1996)
[7] Gnutzmann, S.; Ritschel, U.: Analytic solution of Emden--Fowler equation and critical adsorption in spherical geometry. Z. phys. B 96, 391 (1995)
[8] Bahcall, N. A.: The galaxy distribution in the cluster Abell 2199. Astrophys. J. 186, 1179 (1973)
[9] Bahcall, N. A.: Core radii and central densities of 15 rich clusters of galaxies. Astrophys. J. 198, 249 (1975)
[10] Horedt, G. P.: Seven-digit tables of Lane--Emden functions. Astronom. astrophys. 126, 357-408 (1986)
[11] Horedt, G. P.: Approximate analytical solutions of the Lane--Emden equation in N-dimensional space. Astronom. astrophys. 172, 359-367 (1987) · Zbl 0609.76082
[12] Bender, C. M.; Milton, K. A.; Pinsky, S. S.; Jr., L. M. Simmons: A new perturbative approach to nonlinear problems. J. math. Phys. 30, 1447-1455 (1989) · Zbl 0684.34008
[13] Lima, P. M.: Numerical methods and asymptotic error expansions for the Emden--Fowler equations. J. comput. Appl. math. 70, 245-266 (1996) · Zbl 0854.65067
[14] Lima, P. M.: Numerical solution of a singular boundary-value problem in non-Newtonian fluid mechanics. Appl. numer. Math. 30, 93-111 (1999) · Zbl 0929.65046
[15] Roxburgh, I. W.; Stockman, L. M.: Power series solutions of the polytrope equations. Mon. not. R. astron. Soc. 303, 466-470 (1999)
[16] Adomian, G.; Rach, R.; Shawagfen, N. T.: On the analytic solution of the Lane--Emden equation. Found. phys. Lett. 8, 161-181 (1995)
[17] Shawagfeh, N. T.: Nonperturbative approximate solution for Lane--Emden equation. J. math. Phys. 34, 4364-4369 (1993) · Zbl 0780.34007
[18] Burt, P. B.: Nonperturbative solution of nonlinear field equations. Nuovo cimento 100B, 43-52 (1987)
[19] Wazwaz, A. M.: A new algorithm for solving differential equations of Lane--Emden type. Appl. math. Comput. 118, 287-310 (2001) · Zbl 1023.65067
[20] Liao, S.: A new analytic algorithm of Lane--Emden type equations. Appl. math. Comput. 142, 1-16 (2003) · Zbl 1022.65078
[21] Chandrasekhar, S.: An introduction to the study of stellar structure. (1957) · Zbl 0079.23901
[22] Davis, H. T.: Introduction to nonlinear differential and integral equations. (1962) · Zbl 0106.28904
[23] Datta, B. K.: Analytic solution to the Lane--Emden equation. Nuovo cimento 111B, 1385-1388 (1996)
[24] Wrubel, M. H.: Stellar interiors. Encyclopedia of physics, 53 (1958)
[25] Momoniat, E.; Harley, C.: Approximate implicit solution of a Lane--Emden equation. New astron. 11, 520-526 (2006)
[26] Leach, P. G. L.: First integrals for the modified Emden equation q\ddot{}+${\alpha}(t)$q\dot{}+qn=0. J. math. Phys. 26, 2510-2514 (1985) · Zbl 0587.34004
[27] Bozhkov, Y.; Martins, A. C. G.: Lie point symmetries of the Lane--Emden systems. J. math. Anal. appl. 294, 334-344 (2004) · Zbl 1052.37059
[28] Bozhkov, Y.; Martins, A. C. G.: Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents. Nonlinear anal. 57, 773-793 (2004) · Zbl 1061.34030
[29] Euler, N.: Transformation properties of x\ddot{}+$f1(t)$x\dot{}+$f2(t)+f3(t)x+f3(t)$xn=0. J. nonlinear. Math. phys. 4, 310-337 (1997)
[30] Govinder, K. S.; Leach, P. G. L.: Integrability analysis of the Emden--Fowler equation. J. nonlinear. Math. phys. 14, No. 3, 435-453 (2007) · Zbl 1167.37032
[31] Kara, A. H.; Mahomed, F. M.: Equivalent Lagrangians and solutions of some classes of nonlinear equations q\ddot{}+$p(t)$q\dot{}+$r(t)q={\mu}$q\dot{}$2q-1+f(t)$qn. Int. J. Nonlinear mech. 27, 919-927 (1992) · Zbl 0760.34011
[32] Kara, A. H.; Mahomed, F. M.: A note on the solutions of the Emden--Fowler equation. Int. J. Nonlinear mech. 28, 379-384 (1993) · Zbl 0786.34001
[33] Wong, J. S. W.: On the generalized Emden--Fowler equation. SIAM rev. 17, 339-360 (1975) · Zbl 0295.34026
[34] Goenner, H.; Havas, P.: Exact solutions of the generalized Lane--Emden equation. J. math. Phys. 41, 7029-7042 (2000) · Zbl 1009.34002
[35] Goenner, H.: Symmetry transformations for the generalized Lane--Emden equation. Gen. relativity gravitation 33, 833-841 (2001) · Zbl 0989.83024
[36] Soh, C. Wafo; Mahomed, F. M.: Noether symmetries of y”=$f(x)$yn with applications to non-static spherically symmetric perfect fluid solutions. Classical quantum gravity 16, 3553-3566 (1999) · Zbl 0945.76101
[37] Ibragimov, N. H.: CRC handbook of Lie group analysis of differential equations. 1, 2 and 3 (1994--1996) · Zbl 0864.35001
[38] Lie, S.: Differentialgleichungen. (1967)
[39] Mahomed, F. M.; Kara, A. H.; Leach, P. G. L.: Lie and Noether counting theorems for one-dimensional systems. J. math. Anal. appl. 178, 116-129 (1993) · Zbl 0783.34002
[40] Khalique, C. M.; Ntsime, P.: Exact solutions of the Lane--Emden-type equation. New astron. 13, 476-480 (2008)
[41] Kara, A. H.; Mahomed, F. M.: Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear dynam. 45, 367-383 (2006) · Zbl 1121.70014
[42] Kara, A. H.; Mahomed, F. M.; Naeem, I.; Soh, C. Wafo: Partial Noether operators and first integrals via partial Lagrangians. Math. methods appl. Sci. 30, 2079-2089 (2007) · Zbl 1130.70012
[43] Olver, P. J.: Applications of Lie groups to differential equations. (1993) · Zbl 0785.58003