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New analytic solution to the Lane-Emden equation of index 2. (English) Zbl 1264.65130
Summary: We present two new analytic methods that are used for solving initial value problems that model polytropic and stellar structures in astrophysics and mathematical physics. The applicability, effectiveness, and reliability of the methods are assessed on the Lane-Emden equation which is described by a second-order nonlinear differential equation. The results obtained in this work are also compared with numerical results of G. P. Horedt [“Seven-digit tables of Lane-Emden functions,” Astrophy. Space Sci. 126, No. 2, 357–408 (1986; doi:10.1007/BF00639386)] which are widely used as a benchmark for testing new methods of solution. Good agreement is observed between the present results and the numerical results. Comparison is also made between the proposed new methods and existing analytical methods and it is found that the new methods are more efficient and have several advantages over some of the existing analytical methods.

65L99 Numerical methods for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, New York, NY, USA, 1957. · Zbl 0079.23901
[2] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications Inc., New York, NY, USA, 1962. · Zbl 0106.28904
[3] A. Wazwaz, “A new method for solving singular initial value problems in the second-order ordinary differential equations,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 45-57, 2002. · Zbl 1030.34004 · doi:10.1016/S0096-3003(01)00021-2
[4] A. Wazwaz, “Adomian decomposition method for a reliable treatment of the Emden-Fowler equation,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 543-560, 2005. · Zbl 1061.65064 · doi:10.1016/j.amc.2003.12.048
[5] V. S. Ertürk, “Differential transformation method for solving differential equations of Lane-Emden type,” Mathematical & Computational Applications, vol. 12, no. 3, pp. 135-139, 2007. · Zbl 1175.34007
[6] A. Yildirim and T. Özi\cs, “Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method,” Physics Letters, Section A, vol. 369, no. 1-2, pp. 70-76, 2007. · Zbl 1209.65120 · doi:10.1016/j.physleta.2007.04.072
[7] H. Askari, Z. Saadatnia, D. Younesian, A. Yildirim, and M. Kalami-Yazdi, “Approximate periodic solutions for the Helmholtz Duffing equation,” Computers & Mathematics with Applications, vol. 62, pp. 3894-3901, 2011. · Zbl 1236.65103 · doi:10.1016/j.camwa.2011.09.042
[8] A. S. Bataineh, M. S. M Noorani, and I. Hashim, “Homotopy analysis method for singular IVPs of Emden-Fowler type,” Communications in Nonlinear Science andNumerical Simulation, vol. 14, pp. 1121-1131, 2009. · Zbl 1221.65197 · doi:10.1016/j.cnsns.2008.02.004
[9] S. Liao, “A new analytic algorithm of Lane-Emden type equations,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 1-16, 2003. · Zbl 1022.65078 · doi:10.1016/S0096-3003(02)00943-8
[10] A. Aslanov, “Determination of convergence intervals of the series solutions of Emden-Fowler equations using polytropes and isothermal spheres,” Physics Letters. A, vol. 372, no. 20, pp. 3555-3561, 2008. · Zbl 1220.35084 · doi:10.1016/j.physleta.2008.02.019
[11] C. Hunter, “Series solutions for polytropes and isothermal sphere,” Monthly Notices of the Royal Astronomical Society, vol. 303, pp. 466-470, 1999. · doi:10.1046/j.1365-8711.1999.02219.x
[12] C. Mohan and A. R. Al-Bayaty, “Power series solutions of the Lane-Emden equation,” Astrophysics and Space Science, vol. 100, pp. 447-449, 1984. · Zbl 0457.76039 · doi:10.1007/BF00651624
[13] I. W. Roxburgh and L. M. Stockman, “Power series solutions of the polytrope equation,” Monthly Notices of the Royal Astronomical Society, vol. 328, pp. 839-847, 2001. · doi:10.1046/j.1365-8711.2001.04914.x
[14] Z. F. Seidov, “The power series as solution of the Lane-Emden equation with index two,” Doklady Akademii Nauk AzSSSR, vol. 35, pp. 21-24, 1979.
[15] J. H. He, “Variational approach to the Lane-Emden equation,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 539-541, 2003. · Zbl 1022.65076 · doi:10.1016/S0096-3003(02)00382-X
[16] A. Yıldırım and T. Özi\cs, “Solutions of singular IVPs of Lane-Emden type by the variational iteration method,” Nonlinear Analysis, vol. 70, no. 6, pp. 2480-2484, 2009. · Zbl 1162.34005 · doi:10.1016/j.na.2008.03.012
[17] S. Liao, Beyond Perturbation, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. · Zbl 1051.76001
[18] K. Parand and M. Razzaghi, “Rational legendre approximation for solving some physical problems on semi-infinite intervals,” Physica Scripta, vol. 69, no. 5, pp. 353-357, 2004. · Zbl 1063.65146 · doi:10.1238/Physica.Regular.069a00353 · www.physica.org
[19] K. Parand and A. Pirkhedri, “Sinc-Collocation method for solving astrophysics equations,” New Astronomy, vol. 15, no. 6, pp. 533-537, 2010. · doi:10.1016/j.newast.2010.01.001
[20] K. Parand, A. R. Rezaei, and A. Taghavi, “Lagrangian method for solving LaneEmden type equation arising in astrophysics on semi-innite domains,” Acta Astronautica, vol. 67, pp. 673-680, 2010. · doi:10.1016/j.actaastro.2010.05.015
[21] S. S. Motsa and P. Sibanda, “A new algorithm for solving singular IVPs of Lane-Emden type,” in Proceedings of the 4th International Conference on Applied Mathematics, Simulation, Modelling, pp. 176-180, NAUN International Conferences, Corfu Island, Greece, 2010. · Zbl 1343.65087
[22] G. P. Horedt, “Seven-digit tables of Lane-Emden functions,” Astrophysics and Space Science, vol. 126, no. 2, pp. 357-408, 1986. · doi:10.1007/BF00639386
[23] F. G. Awad, P. Sibanda, S. S. Motsa, and O. D. Makinde, “Convection from an inverted cone in a porous medium with cross-diffusion effects,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1431-1441, 2011. · Zbl 1217.76071 · doi:10.1016/j.camwa.2011.01.015
[24] Z. G. Makukula, P. Sibanda, and S. S. Motsa, “A novel numerical technique for two-dimensional laminar flow between two moving porous walls,” Mathematical Problems in Engineering, vol. 2010, Article ID 528956, 15 pages, 2010. · Zbl 1195.76387 · doi:10.1155/2010/528956 · eudml:225106
[25] Z. G. Makukula, P. Sibanda, and S. S. Motsa, “A note on the solution of the von Kármán equations using series and Chebyshev spectral methods,” Boundary Value Problems, vol. 2010, Article ID 471793, 17 pages, 2010. · Zbl 1207.35248 · doi:10.1155/2010/471793 · eudml:226896
[26] Z. G. Makukula, P. Sibanda, and S. S. Motsa, “On new solutions for heat transfer in a visco-elastic fluid between parallel plates,” International Journal of Mathematical Models and Methods in Applied Sciences, vol. 4, no. 4, pp. 221-230, 2010. · Zbl 1341.80019
[27] Z. Makukula, S. Motsa, and P. Sibanda, “On a new solution for the viscoelastic squeezing flow between two parallel plates,” Journal of Advanced Research in Applied Mathematics, vol. 2, no. 4, pp. 31-38, 2010. · doi:10.5373/jaram.455.060310
[28] S. S. Motsa and S. Shateyi, “A new approach for the solution of three dimensional magnetohydrodynamic rotating flow over a shrinking sheet,” Mathematical Problems in Engineering, vol. 2010, Article ID 586340, 15 pages, 2010. · Zbl 1202.76157 · doi:10.1155/2010/586340 · eudml:222560
[29] S. S. Motsa and S. Shateyi , “Successive linearisation solution of free convection non-darcy flow with heat and mass transfer,” in Advanced Topics in Mass Transfer, M. El-Amin , Ed., pp. 425-438, InTech Open Access, 2011.
[30] S. Shateyi and S. S. Motsa, “Variable viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with Hall effect,” Boundary Value Problems, vol. 2010, Article ID 257568, 20 pages, 2010. · Zbl 1195.76438 · doi:10.1155/2010/257568 · eudml:225074
[31] M. M. Hosseini and H. Nasabzadeh, “On the convergence of Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 536-543, 2006. · Zbl 1111.65062 · doi:10.1016/j.amc.2006.04.015
[32] S. Karimi Vanani and A. Aminataei, “On the numerical solution of differential equations of Lane-Emden type,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2815-2820, 2010. · Zbl 1193.65151 · doi:10.1016/j.camwa.2010.01.052
[33] G. Adomian, “A review of the decomposition method and some recent results for nonlinear equations,” Mathematical and Computer Modelling, vol. 13, no. 7, pp. 17-43, 1990. · Zbl 0713.65051 · doi:10.1016/0895-7177(90)90125-7
[34] G. Adomian and R. Rach, “Noise terms in decomposition solution series,” Computers & Mathematics with Applications, vol. 24, no. 11, pp. 61-64, 1992. · Zbl 0777.35018 · doi:10.1016/0898-1221(92)90031-C
[35] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1994. · Zbl 0802.65122
[36] Y. Cherruault and G. Adomian, “Decomposition methods: a new proof of convergence,” Mathematical and Computer Modelling, vol. 18, no. 12, pp. 103-106, 1993. · Zbl 0805.65057 · doi:10.1016/0895-7177(93)90233-O
[37] J. R. Jabbar, “The boundary conditions for polytropic gas spheres,” Astrophysics and Space Science, vol. 100, no. 1-2, pp. 447-449, 1984. · doi:10.1007/BF00651624
[38] G. P. Horedt, Polytropes Applications in Astrophysics and Related Fields, Kluwer Academic Publishers, Dordrecht, The Netherland, 2004.
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