Solving non-linear Lane–Emden type equations using Bessel orthogonal functions collocation method
Tóm tắt
The Lane–Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. These equations are categorized as non-linear singular ordinary differential equations on the semi-infinite domain
$$[0,\infty )$$
. In this research we introduce the Bessel orthogonal functions as new basis for spectral methods and also, present an efficient numerical algorithm based on them and collocation method for solving these well-known equations. We compare the obtained results with other results to verify the accuracy and efficiency of the presented scheme. To obtain the orthogonal Bessel functions we need their roots. We use the algorithm presented by Glaser et al. (SIAM J Sci Comput 29:1420–1438, 2007) to obtain the
$$N$$
roots of Bessel functions.
Tài liệu tham khảo
Agrawal, R.P., O’Regan, D.: Second order initial value problems of Lane–Emden type. Appl. Math. Lett. 20, 1198–1205 (2007)
Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Homotopy analisis method for singular IVPs of Emden-Fowler type. Commun. Nonlinear. Sci. Numer. Simul. 14, 1121–1131 (2009)
Bell, W.W.: Special Functions For Scientists And Engineers. D. Van Nostrand Company, Canada (1967)
Boubaker, K., Van Gorder, R.A.: Application of the BPES to Lane–Emden equations governing polytropic and isothermal gas spheres. New Astron. 17, 565–569 (2012)
Boyd, J.P.: Chebyshev spectral methods and the Lane–Emden problem. Numer. Math. Theor. Meth. Appl. 4, 142–157 (2011)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2000)
Chandrasekhar, S.: Introduction to the Study of Stellar Structure. Dover, New York (1967)
Chowdhury, M.S.H., Hashim, I.: Solution of a class of singular second-order IVPs by Homotopy-Perturbation method. Phys. Lett. A 365, 439–447 (2007)
Davis, H.T.: Introduction to Nonlinear Differential and Integral Equations. Dover, New York (1962)
Dehghan, M., Shakeri, F.: Approximat solution of a differential equation arising in astrophysics using the VIM. New Astron. 13, 53–59 (2008)
Glaser, A., Liu, X., Rokhlin, V.: A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput. 29, 1420–1438 (2007)
Guo, B.Y., Shen, J., Wang, Z.: A rational approximation and its applications to differential equations on the half line. SIAM. J. Sci. Comput. 30, 697–701 (2006)
Horedt, G.P.: Polytropes: Applications in Astrophysics and Related Fileds. Kluwer, Dordecht (2004)
Marzban, H.R., Tabrizidooz, H.R., Razzaghi, M.: Hybrid functions for nonlinear initial-value problems with applications to Lane–Emden equations. Phys. Lett. A. 37, 5883–5886 (2008)
Parand, K., Nikarya, M., Rad, J.A., Baharifard, F.: A new reliable numerical algorithm based on the first kind of Bessel functions to solve prandtlBlasius laminar viscous flow over a semi-infinite flat plate. Z. Naturforch. A 67, 665–673 (2012)
Parand, K., Dehghan, M., Rezaei, A.R., Ghaderi, S.M.: An approximation algorthim for the solution of the nonlinear Lane–Emden type equation arising in astorphisics using Hermite functions collocation method. Comput. Phys. Commun. 181, 1096–1108 (2010)
Prüfer, H.: Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen. J. Math. Ann. 95, 499–518 (1960)
Shawagfeh, N.T.: Nonperturbative approximate solution of Lane–Emden equation. J. Math. Phys. 34, 4364–4369 (1993)
Shen, J., Tang, T., Wang, L.L.: Spectral Methods Algorithms, Analysics and Applicattions, 1st edn. Springer, New York (2001)
Shen, J., Tang, T.: High Order Numerical Methods and Algorithms. Chinese Science Press, Chinese (2005)
Singh, O.P., Pandey, R.K., Singh, V.K.: An analytic algorithm of Lane–Emden type equations arising in astropysics using modified Homotopy analisis method. Comput. Phys. Commun. 180, 1116–1124 (2009)
Van Gorder, R.A.: Analytical solutions to a quasilinear differential equation related to the Lane–Emden equation of the second kind. Celes. Mech. Dyn. Astron. 109, 137–145 (2011)
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1967)
Wazwaz, A.M.: A new algorithm for solving differential equations of Lane–Emden type. Appl. Math. Comput. 118, 287–310 (2001)
Yildirim, A., Ozis, T.: Solutions of singular IVPs of Lane–Emden type equations by the VIM method. J. Nonlinear Anal. Ser. A Theor. Method 70, 2480–2484 (2009)