Applications of Mathematics, first online, pp. 1-9
Exact solutions of generalized Lane-Emden equations of the second kind
Kısmet Kasapoğlu
Received December 20, 2023. Published online September 13, 2024.
Abstract: Contact and Lie point symmetries of a certain class of second order differential equations using the Lie symmetry theory are obtained. Generators of these symmetries are used to obtain first integrals and exact solutions of the equations. This class of equations is transformed into the so-called generalized Lane-Emden equations of the second kind y"(x)+\frac{k}xy'(x)+ g(x)e^{ny}=0. Then we consider two types of functions $g(x)$ and present first integrals and exact solutions of the Lane-Emden equation for them. One of the considered cases is new.
Keywords: Lie point symmetry; contact symmetry; first integral; Lane-Emden differential equation
References: [1] G. W. Bluman, S. Kumei: Symmetries and Differential Equations. Applied Mathematical Sciences 81. Springer, Berlin (1989). DOI 10.1007/978-1-4757-4307-4 | MR 1006433 | Zbl 0698.35001
[2] Y. Bozhkov, A. C. Gilli Martins: Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents. Nonlinear Anal., Theory Methods Appl., Ser. A 57 (2004), 773-793. DOI 10.1016/j.na.2004.03.016 | MR 2067733 | Zbl 1061.34030
[3] P. L. Chambré: On the solution of the Poisson-Boltzmann equation with application to the theory of thermal explosions. J. Chem. Phys. 20 (1952), 1795-1797. DOI 10.1063/1.1700291
[4] R. Emden: Gaskugeln. Anwendung der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme. B. G. Teubner, Leipzig (1907). (In German.) JFM 38.0952.02
[5] D. A. Frank-Kamenetskii: Diffusion and Heat Transfer in Chemical Kinetics. Plenum Press, New York (1969).
[6] H. Goenner: Symmetry transformations for the generalized Lane-Emden equation. Gen. Relativ. Gravitation 33 (2001), 833-841. DOI 10.1023/A:1010255807935 | MR 1839009 | Zbl 0989.83024
[7] C. Harley, E. Momoniat: Steady state solutions for a thermal explosion in a cylindrical vessel. Mod. Phys. Lett. B 21 (2007), 831-841. DOI 10.1142/S0217984907013250 | Zbl 1115.80005
[8] C. Harley, E. Momoniat: Alternate derivation of the critical value of the Frank-Kamenetskii parameter in cylindrical geometry. J. Nonlinear Math. Phys., Suppl. 1 15 (2008), 69-76. DOI 10.2991/jnmp.2008.15.s1.6 | MR 2434706
[9] C. Harley, E. Momoniat: Instability of invariant boundary conditions of a generalized Lane-Emden equation of the second-kind. Appl. Math. Comput. 198 (2008), 621-633. DOI 10.1016/j.amc.2007.08.077 | MR 2405965 | Zbl 1146.34031
[10] P. E. Hydon: Symmetry Methods for Differential Equations: A Beginner's Guide. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2000). DOI 10.1017/CBO9780511623967 | MR 1741548 | Zbl 0951.34001
[11] N. H. Ibragimov: Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley Series in Mathematical Methods in Practice 4. Willey, Chichester (1999). MR 1679646 | Zbl 1047.34001
[12] C. M. Khalique, F. M. Mahomed, B. Muatjetjeja: Langrangian formulation of a generalized Lane-Emden equation and double reduction. J. Nonlinear Math. Phys. 15 (2008), 152-161. DOI 10.2991/jnmp.2008.15.2.3 | MR 2430716 | Zbl 1169.34033
[13] C. M. Khalique, P. Ntsime: Exact solutions of the Lane-Emden-type equation. New Astronomy 13 (2008), 476-480. DOI 10.1016/j.newast.2008.01.002
[14] B. Muatjetjeja, C. M. Khalique: Exact solutions of the generalized Lane-Emden equations of the first and second kind. Pramana - J. Phys. 77 (2011), 545-554. DOI 10.1007/S12043-011-0174-4
[15] F. Oliveri: Lie symmetries of differential equations: Classical results and recent contributions. Symmetry 2 (2010), 658-706. DOI 10.3390/sym2020658 | MR 2804858 | Zbl 1284.22014
[16] P. J. Olver: Application of Lie Groups to Differential Equations. Graduate Texts in Mathematics 107. Springer, New York (1993). DOI 10.1007/978-1-4612-4350-2 | MR 1240056 | Zbl 0785.58003
[17] H. Stephani: Differential Equations: Their Solution Using Symmetries. Cambridge University Press, Cambridge (1989). DOI 10.1017/CBO9780511599941 | MR 1041800 | Zbl 0704.34001
Affiliations: Kısmet Kasapoğlu, Department of Mathematics, Faculty of Science, Trakya University, Koca Sinan Mahallesi, 22030 Edirne, Turkey, e-mail: kismetkasapoglu@trakya.edu.tr
