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 IJAA  Vol.1 No.2 , June 2011
An Approximation Algorithm for the solution of astrophysics equations using rational scaled generalized Laguerre function collocation method based on transformed Hermite-Gauss nodes
Abstract: In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear or-dinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value problems. We solve this equation by the generalized Laguerre polynomial collocation method based on Her-mite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to a fi-nite domain and transforming domain of the problem to a finite domain. In addition, this method reduces so-lution of the problem to solution of a system of algebraic equations.
Cite this paper: nullA. Pirkhedri, P. Daneshjoo, H. Javadi, H. Navidi, S. Khodamoradi and K. Ghaderi, "An Approximation Algorithm for the solution of astrophysics equations using rational scaled generalized Laguerre function collocation method based on transformed Hermite-Gauss nodes," International Journal of Astronomy and Astrophysics, Vol. 1 No. 2, 2011, pp. 67-72. doi: 10.4236/ijaa.2011.12010.
References

[1]   D. Funaro and O. Kavian, “Approximation of Some Diffusion Evolution Equations in Unbounded Domains by Hermite Functions,” Mathematics of Computation, Vol. 57, No. 196, 1991, pp. 597-619. doi:10.1090/S0025-5718-1991-1094949-X

[2]   O. Coulaud, D. Funaro and O. Kavian, “Laguerre Spectral Approximation of Elliptic Problems in Exterior Domains,” Computer Methods in Applied Mechanics and Engineering, Vol. 80, No. 1-3, 1990, pp. 451-458. doi:10.1016/0045-7825(90)90050-V

[3]   D. Funaro, “Computational Aspects of Pseudospectral La- guerre Approximations,” Applied Numerical Mathematics, Vol. 6, No. 6, 1990, pp. 447-457. doi:10.1016/0168-9274(90)90003-X

[4]   B. Y. Guo, “Error Estimation of Hermite Spectral Method for Nonlinear Partial Differential Equations,” Mathematics of Computation, Vol. 68, No. 227, 1999, pp. 1067- 1078. doi:10.1090/S0025-5718-99-01059-5

[5]   B. Y. Guo and J. Shen, “Laguerre-Galerkin Method for Nonlinear Partial Differential Equations on a Semi-Infinite Interval,” Numerische Mathematik, Vol. 86, No. 4, 2000, pp. 635-654. doi:10.1007/PL00005413

[6]   Y. Maday, B. Pernaud-Thomas and H. Vandeven, “Reappraisal of Laguerre Type Spectral Methods,” La Recherche Aerospatiale, Vol. 6, 1985, pp. 13-35.

[7]   J. Shen, “Stable and Efficient Spectral Methods in Unbounded Domains Using Laguerre Functions,” SIAM Jou- rnal on Numerical Analysis, Vol. 38, No. 4, 2000, pp. 1113-1133. doi:10.1137/S0036142999362936

[8]   H. I. Siyyam, “Laguerre Tau Methods for Solving Higher Order Ordinary Differential Equations,” Journal of Com- putational Analysis and Applications, Vol. 3, No. 2, 2001, pp. 173-182. doi:10.1023/A:1010141309991

[9]   B. Y. Guo, “Gegenbauer Approximation and Its Applications to Differential Equations on the Whole Line,” Jour- nal of Mathematical Analysis and Applications, Vol. 226, No. 1, 1998, pp. 180-206. doi:10.1006/jmaa.1998.6025

[10]   B. Y. Guo, “Jacobi Spectral Approximation and Its Applications to Differential Equations on the Half Line,” Journal of Computational Mathematics, Vol. 18, 2000, pp. 95-112.

[11]   B. Y. Guo, “Jacobi Approximations in Certain Hilbert Spaces and Their Applications to Singular Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 243, No. 2, 2000, pp. 373-408. doi:10.1006/jmaa.1999.6677

[12]   C. I. Christov, “A Complete Orthogonal System of Functions in L2 (?∞, ∞) Space,” SIAM Journal on Numerical Analysis, Vol. 42, No. 6, 1982, pp. 1337-1344.

[13]   J. P. Boyd, “Spectral Methods Using Rational Basis Functions on an Infinite Interval,” Journal of Computational Physics, Vol. 69, No. 1, 1987, pp. 112-142. doi:10.1016/0021-9991(87)90158-6

[14]   J. P. Boyd, “Orthogonal Rational Functions on a Semi- I nfinite Interval,” Journal of Computational Physics, Vol. 70, No. 1, 1987, pp. 63-88. doi:10.1016/0021-9991(87)90002-7

[15]   B. Y. Guo, J. Shen and Z. Q. Wang, “A Rational Approximation and Its Applications to Differential Equations on the Half Line,” Journal of Scientific Computing, Vol. 15, No. 2, 2000, pp. 117-147. doi:10.1023/A:1007698525506

[16]   J. P. Boyd, “Chebyshev and Fourier Spectral Methods,” 2nd Edition, Dover, New York, 2000.

[17]   C. M. Bender, K. A. Milton, S. S. Pinsky, Jr. and L. M. Simmons, “A New Perturbative Approach to Nonlinear Problems,” Journal of Mathematical Physics, Vol. 30, No. 7, 1989, pp. 1447-1455. doi:10.1063/1.528326

[18]   D. C. Biles, M. P. Robinson and J. S. Spraker, “A Generalization of the Lane-Emden Equation,” Journal of Mathe- matical Analysis and Applications, Vol. 273, No. 2, 2002, pp. 654-666. doi:10.1016/S0022-247X(02)00296-2

[19]   G. Bluman, A. F. Cheviakov and M. Senthilvelan, “Solution and Asymptotic/Blow-up Behaviour of a Class of Nonlinear Dissipative Systems,” Journal of Mathematical Analysis and Applications, Vol. 339, No. 2, 2008, pp. 1199-1209. doi:10.1016/j.jmaa.2007.06.076

[20]   G. P. Horedt, “Polytropes Applications in Astrophysics and Related Fields,” Klawer Academic Publishers, Dordrecht, 2004.

[21]   K. Parand and A. Pirkhedri, “Sinc-Collocation Method for Solving Astrophysics Equations,” New Astronomy, Vol. 15, No. 6, 2010, pp. 533-537. doi:10.1016/j.newast.2010.01.001

[22]   S. S. Bayin, “Mathematical Methods in Science and Engineering,” John Wiley & Sons, New York, 2006. doi:10.1002/0470047429

[23]   G. Szegao, “Orthogonal Polynomils,” AMS, New York, 1939.

[24]   D. Funaro, “Polynomial Approximation of Differential Equations,” Springer-Verlag, Berlin, 1992.

[25]   V. Iranzo and A. Falques, “Some Spectral Approximations for Differential Equations in Un-Bounded Domains,” Com- puter Methods in Applied Mechanics and Engineering, Vol. 98, No. 1, 1992, pp. 105-126. doi:10.1016/0045-7825(92)90171-F

[26]   J. P. Boyd, C. Rangan and 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,” Journal of Computational Physics, Vol. 188, No. 1, 2003, pp. 56-74. doi:10.1016/S0021-9991(03)00127-X

[27]   J. Shen and L. Wang, “Some Recent Advances on Spectral Methods for Unbounded Domains,” Communications in Computational Physics, Vol. 5, No. 2, 2009, pp. 195- 241.

[28]   J. Shen, T. Tang, “High Order Numerical Methods and Algorithms,” Chinese Science Press, Beijing, 2005.

[29]   J. Shen, T. Tang and L. Wang, “Spectral Methods Algorithms, Analyses and Applications,” Springer, Berlin, 2010.

 
 
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