Higher order numeric solutions of the Lane–Emden-type equations derived from the multi-stage modified Adomian decomposition method. Issue 1 (2nd January 2017)

• Record Type: Journal Article
• Authors: Duan, Jun-Sheng; Rach, Randolph; Wazwaz, Abdul-Majid
• Is Part Of: International journal of computer mathematics. Volume 94:Issue 1(2017)
• Page Start: 197
• Page End: 215
• Publication Date: 2017-01-02
• Subjects: Lane–Emden equation -- Emden–Fowler equation -- Adomian decomposition method -- nonlinear differential equation -- multi-stage decomposition -- numeric solutions; 34A45 -- 34K28; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2015.1100299
• Languages: English
• ISSNs: 0020-7160

Abstract

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 originx = 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.