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A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations. (English) Zbl 1244.65099
Summary: A shifted Jacobi-Gauss collocation spectral method is proposed for solving the nonlinear Lane-Emden type equation. The spatial approximation is based on shifted Jacobi polynomials $$P_{T,n}^{(\alpha ,\beta)}(x)$$ with $$\alpha , \beta \in ( - 1, \infty ), ~T > 0$$, and $$n$$ is the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results. The method is easy to implement and yields very accurate results.

##### MSC:
 65L05 Numerical methods for initial value problems 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems, general theory
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##### References:
 [1] Adibi, H.; Rismani, A.M., On using a modified Legendre-spectral method for solving singular IVPs of lane – emden type, Comput math appl, 60, 2126-2130, (2010) · Zbl 1205.65201 [2] Agarwal, R.P.; O’Regan, D., Second order initial value problems of lane – emden type, Appl math lett, 20, 1198-1205, (2007) · Zbl 1147.34300 [3] Aslanov, A., A generalization of the lane – emden equation, Int J comput math, 85, 1709-1725, (2008) · Zbl 1154.65059 [4] Bataineh, A.S.; Noorani, M.S.M.; Hashim, I., Homotopy analysis method for singular IVPs of emden – fowler type, Commun nonlinear sci numer simulat, 14, 1121-1131, (2009) · Zbl 1221.65197 [5] Bhrawy, A.H., Legendre – galerkin method for sixth-order boundary value problems, J Egypt math soc, 17, 173-188, (2009) · Zbl 1190.65177 [6] Bhrawy, A.H.; El-Soubhy, S.I., Jacobi spectral Galerkin method for the integrated forms of second-order differential equations, Appl math comput, 217, (2010) · Zbl 1204.65132 [7] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer New York · Zbl 0658.76001 [8] Caglar, H.; Caglar, N.; Ozer, M., B-spline solution of non-linear singular boundary value problems arising in physiology, Chaos soliton fract, 39, 1232-1237, (2009) · Zbl 1197.65107 [9] Chandrasekhar, S., Introduction to the study of stellar structure, (1967), Dover New York · Zbl 0022.19207 [10] Chowdhury, M.S.H.; Hashim, I., Solutions of a class of singular second-order IVPs by homotopy-perturbation method, Phys lett A, 365, 439-447, (2007) · Zbl 1203.65124 [11] Chowdhury, M.S.H.; Hashim, I., Solutions of emden – fowler equations by homotopy-perturbation method, Nonlinear anal-real, 10, 104-115, (2009) · Zbl 1154.34306 [12] Dehghan, M.; Shakeri, F., Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New astron, 13, 53-59, (2008) [13] Doha, E.H.; Abd-Elhameed, W.M.; Bhrawy, A.H., Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nd-order linear differential equations, Appl math model, 33, 1982-1996, (2009) · Zbl 1205.65224 [14] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer algorithms, 42, 137-164, (2006) · Zbl 1103.65119 [15] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl numer math, 58, 1224-1244, (2008) · Zbl 1152.65112 [16] Doha, E.H.; Bhrawy, A.H., A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations, Numer methods partial differential eq, 25, 712-739, (2009) · Zbl 1170.65099 [17] Fornberg, B., A practical guide to pseudospectral methods, (1998), Cambridge University Press Cambridge, MA · Zbl 0912.65091 [18] Genga, F.; Cui, M.; Zhanga, B., Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods, Nonlinear anal-real, 11, 637-644, (2010) · Zbl 1187.34012 [19] Guo, B.-y.; Yan, J.-p., Legendre – gauss collocation method for initial value problems of second order ordinary differential equations, Appl numer math, 59, 1386-1408, (2009) · Zbl 1162.65374 [20] Karimi Vanani, S.; Aminataei, A., On the numerical solution of differential equations of lane – emden type, Comput math appl, 59, 2815-2820, (2010) · Zbl 1193.65151 [21] Khuri, S.A.; Sayf, A., A novel approach for the solution of a class of singular boundary value problems arising in physiology, Math comput model, 52, 626-636, (2010) · Zbl 1201.65135 [22] Parand, K.; Pirkhedri, A., Sinc-collocation method for solving astrophysics equations, New astron, 15, 533-537, (2010) [23] Parand, K.; Dehghan, M.; Rezaeia, A.; Ghaderi, S., 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 [24] Peyret, R., Spectral methods for incompressible viscous flow, (2002), Springer New York · Zbl 1005.76001 [25] Ramos, J.I., Linearization techniques for singular initial-value problems of ordinary differential equations, Appl math comput, 161, 525-542, (2005) · Zbl 1061.65061 [26] Momoniat, E.; Harley, C., An implicit series solution for a boundary value problem modelling a thermal explosion, Math comput model, 53, 249-260, (2011) · Zbl 1211.65101 [27] Trefethen, L.N., Spectral methods in MATLAB, (2000), SIAM Philadelphia, PA · Zbl 0953.68643 [28] Yildirim, A.; Özi, T., Solutions of singular IVPs of lane – emden type by homotopy perturbation method, Phys lett A, 369, 70-76, (2007) · Zbl 1209.65120 [29] Yildirim, A.; Özi, T., Solutions of singular IVPs of lane – emden type by the variational iteration method, Nonlinear anal-theor, 70, 2480-2484, (2009) · Zbl 1162.34005 [30] Om, P.S.; Rajesh, K.P.; Vineet, K.S., An analytic algorithm of lane – emden type equations arising in astrophysics using modified homotopy analysis method, Comput phys commun, 180, 1116-1124, (2009) · Zbl 1198.65250 [31] Shawagfeh, N.T., Nonperturbative approximate solution for lane – emden equation, J math phys, 34, 4364-4369, (1993) · Zbl 0780.34007 [32] Shen, T.-T.; Xing, K.-Z.; Ma, H.-P., A Legendre petrov – galerkin method for fourth-order differential equations, Comput math appl, 61, 8-16, (2011) · Zbl 1207.65142 [33] Van Gorder, R.A., An elegant perturbation solution for the lane – emden equation of the second kind, New astron, 16, 65-67, (2011) [34] Van Gorder, R.A.; Vajravelu, K., Analytic and numerical solutions to the lane – emden equation, Phys lett A, 372, 6060-6065, (2008) · Zbl 1223.85004 [35] Zhang, B.-Q.; Wu, Q.-B.; Luo, X.-G., Experimentation with two-step Adomian decomposition method to solve evolution models, Appl math comput, 175, 1495-1502, (2006) · Zbl 1093.65100
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