# 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.## Abstract

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.

## References

Bengochea, G.(2014). Algebraic approach to the Lane–

Emden equation, Applied Mathematics and Computation,

:424–430.

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

Sciences, 34:2218–2230.

Zheng, J., Sederberg, T. W. & Johnson, R.W. (2004).

Least squares methods for solving differential equations

using Bézier control points, Applied Numerical Mathematics,

:237–252.