Higher order numeric solutions of the Lane–Emden-type equations derived from the multi-stage modified Adomian decomposition method

ABSTRACT

In this paper, we establish higher order numeric solutions for the IVP of the singular Lane–Emden-type equation, including the Emden–Fowler equation. We use the multi-stage modified decomposition method to effectively treat these types of equations and develop numeric solutions that are effective in the large. The step-size and the order in our numeric solutions are two parameters that may be arbitrarily specified. Fast algorithms of the Adomian polynomials guarantee the efficiency of our approach, and a higher order numeric solution can be readily generated at will. The proposed method overcomes the singular behaviour at the origin $x=0$ and exhibits approximations of high accuracy with a large effective region of convergence. Several numerical examples are examined to demonstrate the reliability of our new approach. In these examples, we have demonstrated that our numeric solutions are consistent by halving the step-size, i.e. the numeric solutions of different step-sizes nearly coincide.

KEYWORDS:

• Lane–Emden equation
• Emden–Fowler equation
• Adomian decomposition method
• nonlinear differential equation
• multi-stage decomposition
• numeric solutions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34A45
• 34K28

Acknowledgments

This work was supported by the Natural Science Foundation of Shanghai [No.14ZR1440800] and the Innovation Program of Shanghai Municipal Education Commission [No.14ZZ161].

Disclosure statement

No potential conflict of interest was reported by the authors.