**Commenced**in January 2007

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**Edition:**International

**Paper Count:**30184

##### An Efficient Computational Algorithm for Solving the Nonlinear Lane-Emden Type Equations

**Authors:**
Gholamreza Hojjati,
Kourosh Parand

**Abstract:**

In this paper we propose a class of second derivative multistep methods for solving some well-known classes of Lane- Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. These methods, which have good stability and accuracy properties, are useful in deal with stiff ODEs. We show superiority of these methods by applying them on the some famous Lane-Emden type equations.

**Keywords:**
Lane-Emden type equations,
nonlinear ODE,
stiff problems,
multistep methods,
astrophysics.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1076576

**References:**

[1] A. Aslanov, Determination of convergence intervals of the series solutions of Emden-Fowler equations using polytropes and isothermal spheres. Phys. Lett. A., 372 (2008) 3555-3561.

[2] A.S Bataineh, M.S.M. Noorani, I. Hashim, Homotopy analysis method for singular IVPs of Emden-Fowler type, Commun. Nonlinear. Sci. Numer. Simul., 14 (2009) 1121-1131.

[3] C.M. Bender, K.A. Milton, S.S. Pinsky, Jr.L.M. Simmons, A new perturbative approach to nonlinear problems, J. Math. Phys., 30 (1989) 1447- 1455.

[4] J.P. Boyd, C. Rangan, P.H. Bucksbaum, Pseudospectral methods on a semi-infinite interval with application to the Hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions, J. Comput. Phys., 188 (2003) 56-74.

[5] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Second Edition, Dover, New York, 2000.

[6] J.P. Boyd, Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys., 69(1987) 112-142.

[7] J.P. Boyd, Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys., 70(1987) 63-88.

[8] J.R. Cash, Second Derivative Extended Backward Differentiation Formulas for the Numerical Integration of Stiff Systems, SIAM J. Numerical Anal., 18(1981) 21-36.

[9] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, 1967.

[10] M.S.H. Chowdhury, I. Hashim, Solutions of Emden-Fowler equations by homotopy perturbation method, Nonlinear. Anal. Real. World. Appl., 10(2009) 104-115.

[11] CI. Christov, A complete orthogonal system of functions in L2(−∞,∞) space, SIAM J. Appl. Math., 42(1982) 1337-1344.

[12] O. Coulaud, D. Funaro, O. Kavian, Laguerre spectral approximation of elliptic problems in exterior domains, Comput. Method. Appl. Mech. Eng., 80(1990) 451-458.

[13] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962.

[14] M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Eq., 26(2010) 448-479.

[15] M. Dehghan, M. Shakourifar, A. Hamidi, The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique, Chaos, Solitons and Fractals, 39(2009) 2509-2521.

[16] M. Dehghan, F. Shakeri, Use of He-s homotpy perturbation method for solving a partial differential equation arising in modeling of flow in porous media, Journal of Porous Media, 11(2008) 765-778.

[17] M. Dehghan, A. Hamidi, M. Shakourifar, The solution of coupled Burgers equations using Adomian-Pade technique, Appl. Math. Comput., 189(2007) 1034-1047.

[18] M. Dehghan, F. Shakeri, The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Physica Scripta, 78(2008) 1-11, Article No. 065004.

[19] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy, 13(2008) 53-59.

[20] M. Tatari, M. Dehghan, On the convergence of He-s variational iteration method, J. Comput. Appl. Math., 207(2007) 121-128.

[21] M. Dehghan, A. Saadatmandi, Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos, Solitons and Fractals, 41(2009) 1448-1453.

[22] M. Dehghan, M. Tatari, The use of Adomian decomposition method for solving problems in calculus of variations, Mathematical Problems in Engineering, 2006(2006) 1-12

[23] W.H. Enright, Second Derivative Multistep methods for Stiff Ordinary Differential Equations, SIAM J. Numerical Anal., 11(1974) 321-331.

[24] D. Funaro, O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57(1991) 597-619.

[25] B.Y. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comput. 68 227 (1999) 1067-1078.

[26] B.Y. Guo, Gegenbauer approximation and its applications to differential equations on the whole line, J. Math. Anal. Appl., 226(1998) 180-206.

[27] B.Y. Guo, J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86(2000) 635-654.

[28] B.Y. Guo, Jacobi spectral approximation and its applications to differential equations on the half line, J. Comput. Math., 18(2000) 95-112.

[29] B.Y. Guo, J. Shen, Z.Q. Wang, A rational approximation and its applications to differential equations on the half line, J. Sci. Comput., 15(2000) 117-147.

[30] B.Y. Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl., 243(2000) 373-408.

[31] J.H. He, Variational approach to the Lane-Emden equation, Appl. Math. Comput., 143(2003) 539-541.

[32] G. Hojjati, M.Y. Rahimi Ardabili, S.M. Hosseini, New Second Derivative Multistep Methods for Stiff Systems, Applied Math. Modeling, 30(2006) 466-476.

[33] G.P. Horedt, Polytropes Applications in Astrophysics and Related Fieldsa, Klawer Academic Publishers., Dordrecht, 2004.

[34] G. Ismail, I. Ibrahim, New Efficient Second Derivative Multistep Methods for Stiff Systems, Applied Math. Modeling, 23(1999) 279-288.

[35] A.H. Kara, F.M. Mahomed, Equivalent Lagrangians and solutions of some classes of nonlinear equations, Int. J. Nonlinear Mech., 27(1992) 919-927.

[36] A.H. Kara, F.M. Mahomed, A note on the solutions of the Emden- Fowler equation, Int. J. Nonlinear Mech., 28(1993) 379-384.

[37] S. Liao, A new analytic algorithm of Lane-Emden type equations, Appl. Math. Comput., 142(2003) 1-16.

[38] Y. Maday, B. Pernaud-Thomas, H. Vandeven, Reappraisal of Laguerre type spectral methods, La. Rech. Aerospatiale, 6(1985) 13-35.

[39] V.B. Mandelzweig, F. Tabakin, Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Comput. Phys. Commun., 141(2001) 268-281.

[40] H.R. Marzban, H.R. Tabrizidooz, M. Razzaghi, Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations, Phys. Lett. A., 372 37 (2008) 5883-5886.

[41] K. Parand, M. Razzaghi, Rational Chebyshev tau method for solving Volterra-s population model, Appl. Math. Comput., 149(2004) 893-900.

[42] K. Parand, M. Razzaghi, Rational Chebyshev tau method for solving higher-order ordinary differential equations, Int. J. Comput. Math., 81(2004) 73-80.

[43] K. Parand, M. Razzaghi, Rational Legendre approximation for solving some physical problems on semi-infinite intervals, Phys. Scripta., 69(2004) 353-357.

[44] K. Parand, M. Shahini, Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation, Phys. Lett. A., 373(2009) 210-213.

[45] K. Parand, A. Taghavi, Rational scaled generalized Laguerre function collocation method for solving the Blasius equation, J. Comput. Appl. Math., 233(2009) 980-989.

[46] K. Parand, M. Shahini, M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, J. Comput. Phys., 228(2009) 8830-8840.

[47] J.I. Ramos, Linearization methods in classical and quantum mechanics, Comput. Phys. Commun., 153(2003) 199-208.

[48] J.I. Ramos, Linearization techniques for singular initial-value problems of ordinary differential equations, Appl. Math. Comput., 161(2005) 525- 542.

[49] J.I. Ramos, Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method, Chaos. Solit. Fract., 38(2008) 400-408.

[50] J.I. Ramos, Piecewise-adaptive decomposition methods, Chaos, Solit. Fract., 40(2009) 1623-1636.

[51] O.U. Richardson, The Emission of Electricity from Hot Bodies, London, 1921.

[52] A. Saadatmandi, M. Dehghan, A. Eftekhari, Application of He-s homotopy perturbation method for non-linear system of second-order boundary value problems, Nonlinear Analysis: Real World Applications, 10(2009) 1912-1922.

[53] F. Shakeri, M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Modelling, 48(2008) 486-498.

[54] N.T. Shawagfeh, Nonperturbative approximate solution for Lane-Emden equation, J. Math. Phys., 34(1993) 4364-4369.

[55] J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38(2000) 1113-1133.

[56] O.P. Singh, R.K. Pandey, V.K. Singh, An analytic algorithm of Lane- Emden type equations arising in astrophysics using modified Homotopy analysis method, Comput. Phys. Commun., 180(2009) 1116-1124.

[57] H.I. Siyyam, Laguerre tau methods for solving higher order ordinary differential equations, J. Comput. Anal. Appl., 3(2001) 173-182.

[58] A. Wazwaz, A new algorithm for solving differential equations of Lane- Emden type, Appl. Math. Comput., 118(2001) 287-310.

[59] A. Wazwaz, The modified decomposition method for analytic treatment of differential equations, Appl. Math. Comput., 173(2006) 165-176.

[60] A. Yildirim, T. O┬¿ zis┬©, Solutions of Singular IVPs of Lane-Emden type by the variational iteration method, Nonlinear. Anal. Ser. A. Theor. Method. Appl., 70(2009) 2480-2484.

[61] S.A. Yousefi, Legendre wavelets method for solving differential equations of Lane-Emden type, Appl. Math. Comput., 181(2006) 1417-1422.