A Residual Method Using Bézier Curves for Singular Nonlinear Equations of Lane-Emden Type
Keywords:Bézier curves, Bernstein polynomials, continuous linear approximation, Lane- Emden equations, singular nonlinear differential equations.
In this article, we introduce a new method to solve a singular non-linear equation of the Lane-Emden type by approximating the solution with Bernstein polynomials. This method is based on the minimization of a residual function using Taylor’s series expansion. We also apply this method to problems that are solved by other methods and the obtained results show that
our method is efficient, applicable and has great potential than others.
Bengochea, G.(2014). Algebraic approach to the Lane–
Emden equation, Applied Mathematics and Computation,
Evrenosoglu, M. & Somali, S. (2008). Least squares
methods for solving singularly perturbed two-point
boundary value problems using Bézier control points,
Applied Mathematics Letters, 21:1029–1032.
Ghomanjani, F. , Farahi, M. & Gachpazan M. (2012).
Bézier control points method to solve constrained
quadratic optimal control of time varying linear systems,
Computational & Applied Mathematics, 31:433–456.
Kajani, M. T., Tabatabaei, F. G. & Maleki, M. (2012)
Rational second kind Chebyshev approximation for
solving some physical problems on semi-infinite intervals,
Kuwait Journal of Science & Engineering, 39:15–29.
Marzban, H., Tabrizidooz, H. & Razzaghi, M. (2008).
Hybrid functions for non-linear initial-value problems
with applications to Lane–Emden type equations, Physics
Letters A, 372:5883–5886.
Parand, K., Dehghan, M., Rezaei, A. & Ghaderi S. (2010).
An approximation algorithm for the solution of the nonlinear
Lane–Emden type equations arising in astrophysics
using Hermite functions collocation method, Computer
Physics Communications, 181:1096–1108.
Turkyilmazoglu, M. (2013). Effective computation of
exact and analytic approximate solutions to singular nonlinear
equations of Lane–Emden– Fowler type, Applied
Mathematical Modelling, 37:7539–7548.
Wang, Y., Yu, H., Tan, F. & Li, S. (2014). Using an
effective numerical method for solving a class of Lane-
Emden equations, Abstract and Applied Analysis.
Wazwaz, A. M. (2001). A new algorithm for solving
differential equations of Lane–Emden type, Applied
Mathematics and Computation, 118:287–310.
Wazwaz, A. M. (2014). The variational iteration method for
solving the Volterra integro-differential forms of the Lane-
Emden equations of the first and the second kind, Journal of
Mathematical Chemistry, 52:613–626.
Wu J. (2012). Least squares methods for solving partial
differential equations by using Bézier control points,
Applied Mathematics and Computation, 219:3655–3663.
Yüzbaşı, Ş. (2011). A numerical approach for solving a
class of the nonlinear Lane–Emden type equations arising
in astrophysics, Mathematical Methods in the Applied
Zheng, J., Sederberg, T. W. & Johnson, R.W. (2004).
Least squares methods for solving differential equations
using Bézier control points, Applied Numerical Mathematics,