Comparative Study of Padé Approximation Methods in Solving Lane-Emden Type Differential Equations
Abstract
This study examines how effective Padé approximation methods are for solving Lane-Emden type differential equations. These are special types of equations that are often used in astronomy, like when understanding the structure of some stars and how gases act in round shapes. These equations are hard because there’s an issue at the beginning point and they get more complicated due to the polytropic index. Traditional power series methods usually don't work well beyond a small range because they are only effective within specific limits. Padé approximants are a kind of math tool that can give more accurate estimates. They are effective because they can be used in many different situations and show what happens accurately at places where functions act strangely. In this study, we apply different Padé approximations of different levels to the Lane-Emden equation. They are evaluated based on how correct they are, how fast they find a solution, and how well they use computer resources. Computer simulations are performed, and the results are compared with older techniques, such as the Runge-Kutta method and standard series solutions. The comparison shows that Padé approximants are better because they give accurate results faster over larger areas, especially when the polytropic index goes up. But there are still issues with picking the best orders for the estimates and handling more complicated nonlinear problems. The results show that Padé approximation is a helpful way to understand things while being efficient with computers. This makes it a helpful tool in both ideas and real life, especially when regular methods don’t work well due to unusual situations or high computer expenses.
References
M. Sajid, T. Hayat, and S. Asghar, ‘Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt’, Nonlinear Dyn., vol. 50, pp. 27–35, 2007.
A. Aslanov, ‘Approximate solutions of Emden-Fowler type equations’, Int. J. Comput. Math., vol. 86, no. 5, pp. 807–826, 2009.
A. Aslanov, ‘Determination of convergence intervals of the series solutions of Emden--Fowler equations using polytropes and isothermal spheres’, Phys. Lett. A, vol. 372, no. 20, pp. 3555–3561, 2008.
E. Momoniat and C. Harley, ‘Approximate implicit solution of a Lane-Emden equation’, New Astron., vol. 11, no. 7, pp. 520–526, 2006.
S. A. Yousefi, ‘Legendre wavelets method for solving differential equations of Lane--Emden type’, Appl. Math. Comput., vol. 181, no. 2, pp. 1417–1422, 2006.
A.-M. Wazwaz, ‘A new algorithm for solving differential equations of Lane--Emden type’, Appl. Math. Comput., vol. 118, no. 2–3, pp. 287–310, 2001.
S. Abbasbandy, T. Hayat, A. Alsaedi, and M. M. Rashidi, ‘Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid’, Int. J. Numer. Methods Heat & Fluid Flow, vol. 24, no. 2, pp. 390–401, 2014.
M. Dehghan and F. Shakeri, ‘Approximate solution of a differential equation arising in astrophysics using the variational iteration method’, New Astron., vol. 13, no. 1, pp. 53–59, 2008.
A. Yildirim and T. Özics, ‘Solutions of singular IVPs of Lane--Emden type by the variational iteration method’, Nonlinear Anal. Theory, Methods & Appl., vol. 70, no. 6, pp. 2480–2484, 2009.
J. I. Ramos, ‘Series approach to the Lane--Emden equation and comparison with the homotopy perturbation method’, Chaos, Solitons & Fractals, vol. 38, no. 2, pp. 400–408, 2008.
S. K. Vanani and A. Aminataei, ‘On the numerical solution of differential equations of Lane--Emden type’, Comput. & Math. with Appl., vol. 59, no. 8, pp. 2815–2820, 2010.
A. Yildirim and T. Özics, ‘Solutions of singular IVPs of Lane--Emden type by homotopy perturbation method’, Phys. Lett. A, vol. 369, no. 1–2, pp. 70–76, 2007.
J. K. Zhou, ‘Differential Transformation and Its Applications for Electrical Circuits’, Huazhong Univ. Press. Wuhan China (In Chinese) google Sch., vol. 2, pp. 413–420, 1986.
G. Adomian, R. Rach, and N. T. Shawagfeh, ‘On the analytic solution of the Lane-Emden equation’, Found. Phys. Lett., vol. 8, pp. 161–181, 1995.
R. A. Van Gorder and K. Vajravelu, ‘Analytic and numerical solutions to the Lane--Emden equation’, Phys. Lett. A, vol. 372, no. 39, pp. 6060–6065, 2008.
B. W. Carroll and D. A. Ostlie, An introduction to modern astrophysics. Cambridge University Press, 2017.
R. Emden, Gaskugeln: Anwendungen der mechanischen Warmetheorie auf kosmologische und meteorologische Probleme... BG Teubner, 1907.
S. Chandrasekhar and S. Chandrasekhar, An introduction to the study of stellar structure, vol. 2. Courier Corporation, 1957.
E. Celik and M. Bayram, ‘Arbitrary order numerical method for solving differential-algebraic equation by Padé series’, Appl. Math. Comput., vol. 137, no. 1, pp. 57–65, 2003.
H. J. Lane, ‘On the theoretical temperature of the sun, under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment’, Am. J. Sci., vol. 2, no. 148, pp. 57–74, 1870.
S. Liao, ‘Notes on the homotopy analysis method: some definitions and theorems’, Commun. Nonlinear Sci. Numer. Simul., vol. 14, no. 4, pp. 983–997, 2009.
S. Liao, ‘A new analytic algorithm of Lane--Emden type equations’, Appl. Math. Comput., vol. 142, no. 1, pp. 1–16, 2003.
S.-J. Liao, ‘The proposed homotopy analysis technique for the solution of nonlinear problems’, Ph. D. Thesis, Shanghai Jiao Tong University Shanghai, 1992.
M. M. Rashidi, S. Abbasbandy, and others, ‘Analytic approximate solutions for heat transfer of a micropolar fluid through a porous medium with radiation’, Commun. Nonlinear Sci. Numer. Simul., vol. 16, no. 4, pp. 1874–1889, 2011.
