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A Domain Decomposition Based Spectral Collocation Method for Lane-Emden Equations

Part of: Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  21 June 2017

Yuling Guo  and
Jianguo Huang
Yuling Guo*
Affiliation:
School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, P.R. China
Jianguo Huang*
Affiliation:
School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, P.R. China
*
*Corresponding author. Email addresses:jghuang@sjtu.edu.cn (J. Huang), guoyuling56@163.com (Y. Guo)
*Corresponding author. Email addresses:jghuang@sjtu.edu.cn (J. Huang), guoyuling56@163.com (Y. Guo)
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Abstract

A domain decomposition based spectral collocation method is proposed for numerically solving Lane-Emden equations, which are frequently encountered in mathematical physics and astrophysics. Compared with the existing methods, this method requires less computational cost and is particularly suitable for long-term computation. The related error estimates are also established, indicating the spectral accuracy of the method. The numerical performance and efficiency of the method are illustrated by several numerical experiments.

Keywords

Multi-step Legendre-Gauss-Radau spectral collocation methodVolterra integro-differential equationdecomposition methodlong-term computationerror estimates

MSC classification

Secondary: 65L05: Initial value problems 65L06: Multistep, Runge-Kutta and extrapolation methods 65L20: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 2 , August 2017 , pp. 542 - 571
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bhrawy, A.H. and Alofi, A. S., A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations, Commun. Nonlinear. Sci. Numer. Simulat., 17 (2012), 62–70.CrossRefGoogle Scholar
[2] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004.Google Scholar
[3] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.Google Scholar
[4] Chandrasekhar, S., Introduction to the Study of Stellar Structure, Dover, New York, 1967.Google Scholar
[5] Doha, E. H., Bhrawy, A. H., Hafez, R. M. and Gorder, R. A. Van, A Jacobi rational pseudospectral method for Lane-Emden initial value problems arising in astrophysics on a semi-infinite interval, Comp. Appl. Math., 33 (2014), 607–619.Google Scholar
[6] Doha, E. H., Bhrawy, A. H., Hafez, R. M. and Van Gorder, R. A., Jacobi rational-Gauss collocation method for Lane-Emden equations of astrophysical significance, Nonlinear. Anal-Model., 19 (2014), 1–14.Google Scholar
[7] Gürbü, B. and Sezer, M., Laguerre polynomial approach for solving Lane-Emden type functional differential equations, Appl. Math. Comput., 242 (2014), 255–264.Google Scholar
[8] Guo, B. Y., Spectral and pseudospectral methods for unbounded domains (in Chinese), Sci. Sin. Math., 45 (2015), 975–1024.CrossRefGoogle Scholar
[9] Guo, B. Y. and Wang, Z. Q., Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math., 30 (2009), 249–280.CrossRefGoogle Scholar
[10] Guo, B. Y. and Yan, J. P., Legendre-Guass collocation for initial value problems of second order ordinary differential equations, Appl. Numer. Math., 59 (2009), 1386–1408.CrossRefGoogle Scholar
[11] Horedt, G.P., Polytropes: Applications in Astrophysics and Related Fields, Kluwer Academic Publishers, Dordrecht, 2004.Google Scholar
[12] Hosseini, S. G. and Abbasbandy, S., Solution of Lane Emden Type Equations by Combination of the Spectral Method and Adomian Decomposition Method, Math. Prob. Eng., Volume 2015, Article ID 534754, 10 pages.CrossRefGoogle Scholar
[13] Nazari-Golshan, A., Nourazar, S.S., Ghafoori-Fard, H., Yildirim, A. and Campo, A., A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane-Emden equations, Appl. Math. Lett., 26 (2013), 1018–1025.Google Scholar
[14] Parand, K., Dehghan, M., Rezaeia, A. and 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 (2010), 1096–1108.CrossRefGoogle Scholar
[15] Parand, K., Shahini, M. and Dehghan, M., Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, J. Comput. Phys., 228 (2009), 8830–8840.CrossRefGoogle Scholar
[16] Rach, R., Wazwaz, A. M. and Duan, J. S., The Volterra integral form of the Lane-Emden equation: new derivations and solution by the Adomian decomposition method, J. Appl. Math. Comput., 47 (2015), 365–379.CrossRefGoogle Scholar
[17] Richardson, O. U., The Emission of Electricity from Hot Bodies, Longman's Green and Company, London, 1921.Google Scholar
[18] Sahu, P. K. and Saha Ray, S., Numerical solutions for integro-differential forms of Lane-Emden equations of first and second kind using Legendre Multi-Wavelets, Electron. J. Differ. Eq., 28 (2015), 1–11.Google Scholar
[19] Shen, J., Tang, T. and Wang, L. L., Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin, 2011.Google Scholar
[20] Sheng, C. T., Wang, Z. Q. and Guo, B. Y., A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations, SIAM J. Numer. Anal., 52 (2014), 1953–1980.Google Scholar
[21] Wang, Z. Q., Guo, Y. L. and Yi, L. J., An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Comp., in press.Google Scholar
[22] Wazwaz, A. M., Linear and Nonlinear Intergral Equations Methods and Applications, Higher Education Press, Beijing, 2011.Google Scholar
[23] Wazwaz, A.M., The variational iteration method for solving the Volterra integro-differential forms of the Lane-Emden equations of the first and the second kind, J. Math. Chem., 52 (2014), 613–626.Google Scholar
[24] Yan, J. P. and Guo, B. Y., Laguerre-Gauss collocation method for initial value problems of second order ODEs, Appl. Math. Mech. Engl. Ed., 32 (2011), 1541–1564.Google Scholar
[25] Yi, L. J. and Wang, Z. Q., A Legendre-Gauss-Radau spectral collocation method for first order nonlinear delay differential equations, Calcolo, 53(2016), 691–721.Google Scholar
[26] Youssri, Y. H., Abd-Elhameed, W. M. and Doha, E. H., Ultraspherical Wavelets method for solving Lane-Emden type equations, Rom. Journ. Phys., 60 (2015), 1298–1314.Google Scholar