Application of the exact operational matrices for solving the Emden-Fowler equations, arising in ‎Astrophysics‎

Subject Areas : International Journal of Industrial Mathematics

S. A. Hossayni‎ ¹ , J. A. Rad ² , K. Parand ³ , S. Abbasbandy ⁴

1 - Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin,Tehran 19839, Iran.
2 - Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin,Tehran 19839, Iran.
3 - Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin,Tehran 19839, Iran.
4 - Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.

Received: 2015-10-15 Accepted : 2015-10-15 Published : 2015-10-01

Keywords: Exact operational matrices, Bernstein polynomials, Emden-Fowler equation, Lane-Emden ‎equation,

Abstract :

The objective of this paper is applying the well-known exact operational matrices (EOMs) idea for solving the Emden-Fowler equations, illustrating the superiority of EOMs over ordinary operational matrices (OOMs). Up to now, a few studies have been conducted on EOMs ; but the solved differential equations did not have high-degree nonlinearity and the reported results could not strongly show the excellence of this new method. So, we chose Emden-Fowler type differential equations and solved them utilizing this method. To confirm the accuracy of the new method and to show the preeminence of EOMs over OOMs, the norm 1 of the residual and error function for both methods are evaluated for multiple $m$ values, where $m$ is the degree of the Bernstein polynomials. We report the results by some plots to illustrate the error convergence of both methods to zero and also to show the primacy of the new method versus OOMs. The obtained results demonstrate the increased accuracy of the new ‎method.‎