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A criterion for oscillations in the solutions of the polytropic Lane-Emden equations. (English) Zbl 1343.34080

Summary: We have previously formulated a simple criterion for deducing the intervals of oscillations in the solutions of second-order linear homogeneous differential equations. In this work, we extend analytically the same criterion to superlinear Lane-Emden equations with integer polytropic indices \(n > 1\). We illustrate the validity of the analytical results by solving numerically both the cylindrical and the spherical Lane-Emden equations subject to the usual astrophysical boundary conditions for self-gravitating fluids.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A34 Nonlinear ordinary differential equations and systems
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[1] Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972) · Zbl 0543.33001
[2] Whittaker, ET, Watson, GN: A Course of Modern Analysis, 3rd edn. Cambridge University Press, Cambridge (1920) · JFM 45.0433.02
[3] Hartman, P: Ordinary Differential Equations. Wiley, New York (1964) · Zbl 0125.32102
[4] Agarwal, RP, Grace, SR, O’Regan, D: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht (2002) · Zbl 1073.34002 · doi:10.1007/978-94-017-2515-6
[5] Wong, JSW: On second-order nonlinear oscillation. Funkc. Ekvacioj 11, 207-234 (1968) · Zbl 0184.12202
[6] Christodoulou, DM, Graham-Eagle, J, Katatbeh, QD: A program for predicting the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. Adv. Differ. Equ. (2016). doi:10.1186/s13662-016-0774-x · Zbl 1419.34059 · doi:10.1186/s13662-016-0774-x
[7] Katatbeh, QD, Christodoulou, DM, Graham-Eagle, J: The intervals of oscillations in the solutions of the radial Schrödinger differential equation. Adv. Differ. Equ. (2016). doi:10.1186/s13662-016-0777-7 · Zbl 1419.34061 · doi:10.1186/s13662-016-0777-7
[8] Lane, JH: On the theoretical temperature of the Sun. Am. J. Sci. Arts 50, 57-74 (1870) · doi:10.2475/ajs.s2-50.148.57
[9] Emden, R: Gaskugeln. Teubner, Leipzig (1907) · JFM 38.0935.01
[10] Chandrasekhar, S: An Introduction to the Study of Stellar Structure. University of Chicago Press, Chicago (1939) · JFM 65.1543.02
[11] Horedt, GP: Polytropes. Kluwer Academic, Dordrecht (2004) · Zbl 0629.76117
[12] Benguria, RD: The Lane-Emden equation revisited. Contemp. Math. 327, 11-19 (2003) · Zbl 1039.35032 · doi:10.1090/conm/327/05801
[13] Farina, A: On the classification of solutions of the Lane-Emden equation on unbounded domains of RN \(\mathbb{R}^N\). J. Math. Pures Appl. 87, 537-561 (2007) · Zbl 1143.35041 · doi:10.1016/j.matpur.2007.03.001
[14] Van Gorder, RA, Vajravelu, K: Analytic and numerical solutions to the Lane-Emden equation. Phys. Lett. A 372, 6060-6065 (2008) · Zbl 1223.85004 · doi:10.1016/j.physleta.2008.08.002
[15] Parand, K, Dehghan, M, Rezaei, AR, Ghaderi, SM: An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method. Comput. Phys. Commun. 181, 1096-1108 (2010) · Zbl 1216.65098 · doi:10.1016/j.cpc.2010.02.018
[16] Parand, K, Rezaei, AR, Taghavi, A: Lagrangian method for solving Lane-Emden type equation arising in astrophysics on semi-infinite domains. Acta Astronaut. 67, 673-680 (2010) · doi:10.1016/j.actaastro.2010.05.015
[17] Kumar, N, Pandey, RK, Cattani, C: Solution of the Lane-Emden equation using the Bernstein operational matrix of integration. Int. Sch. Res. Not. Astron. Astrophys. 2011, Article ID 351747 (2011)
[18] Robe, H: Équilibre et oscillations des cylindres polytropiques compressibles en rotation. Ann. Astrophys. 31, 549-558 (1968)
[19] Bailey, PB, Billingham, J, Cooper, RJ, Everitt, WN, King, AC, Kong, Q, Wu, H, Zettl, A: On some eigenvalue problems in fuel-cell dynamics. Proc. R. Soc. Lond. A 459, 241-261 (2003) · Zbl 1056.34091 · doi:10.1098/rspa.2002.1058
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