Author:
Wazwaz Abdul‐Majid,Rach Randolph
Abstract
PurposeThe purpose of this paper is to provide a comparison of the Adomian decomposition method (ADM) with the variational iteration method (VIM) for solving the Lane‐Emden equations of the first and second kinds.Design/methodology/approachThe paper examines the theoretical framework of the Adomian decomposition method and compares it with the variational iteration method. The paper seeks to determine the relative merits and computational benefits of both the Adomian decomposition method and the variational iteration method in the context of the important physical models of the Lane‐Emden equations of the first and second kinds.FindingsThe Adomian decomposition method is shown to readily solve the Lane‐Emden equations of both the first and second kinds for all positive real values of the system coefficient α and for all positive real values of the nonlinear exponent m. The decomposition series solution of these nonlinear differential equations requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The paper shows that the variational iteration method works effectively to solve the Lane‐Emden equation of the first kind for system coefficient values α=1, 2, 3, 4 but only for positive integer values of the nonlinear exponent m. The successive approximations of the solution of these nonlinear differential equations require the determination of the appropriate Lagrange multipliers, which are established in this paper. These two methodologies overcome the singular behavior at the origin x=0. The paper shows that the variational iteration method is impractical for solving either the Lane‐Emden equation of the first kind for non‐integer values of the system exponent m or the Lane‐Emden equations of the second kind. Indeed the Adomian decomposition method is shown to solve even the generalized Lane‐Emden equation for any analytic nonlinearity and all positive values of the system coefficient α in a practical and straightforward manner. The conclusions are supported by several numerical examples.Originality/valueThis paper presents an accurate comparison of the Adomian decomposition method with the variational iteration method for solving the Lane‐Emden equations of the first and second kinds. The paper presents a new solution algorithm for the generalized Lane‐Emden equation for any analytic system nonlinearity and for any model geometry as characterized by all possible positive real values of the system coefficient α.
Subject
Computer Science (miscellaneous),Social Sciences (miscellaneous),Theoretical Computer Science,Control and Systems Engineering,Engineering (miscellaneous)
Reference17 articles.
1. Adomian, G. (1994), Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht.
2. Adomian, G. and Rach, R. (1983), “Inversion of nonlinear stochastic operators”, Journal of Mathematical Analysis and Applications, Vol. 91 No. 1, pp. 39‐46.
3. Adomian, G., Rach, R. and Shawagfeh, N.T. (1995), “On the analytic solution of the Lane‐Emden equation”, Foundations of Physics Letters, Vol. 8 No. 2, pp. 161‐81.
4. Chandrasekhar, S. (1967), Introduction to the Study of Stellar Structure, Dover, New York, NY.
5. Cherruault, Y. (1998), Modéles et méthodes mathématiques pour les sciences du vivant, Presses universitaires de France, Paris.
Cited by
56 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献