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Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method

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  • Ramos, J.I.

Abstract

Series solutions of the Lane–Emden equation based on either a Volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of the dependent variables. It is shown that these four series solutions are the same as those obtained by a direct application of Adomian’s decomposition method to the original differential equation, He’s homotopy perturbation technique, and Wazwaz’s two implementations of the Adomian method based on either the introduction of a new differential operator that overcomes the singularity of the Lane–Emden equation at the origin or the elimination of the first-order derivative term of the original equation. It is also shown that Adomian’s decomposition technique can be interpreted as a perturbative approach which coincides with He’s homotopy perturbation method. An iterative technique based on Picard’s fixed-point theory is also presented and its convergence is analyzed. The convergence of this iterative approach depends on the independent variable and, therefore, this technique is not as convenient as the series solutions derived by the four methods presented in this paper, He’s homotopy perturbation technique, and Adomian’s decomposition method.

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  • Ramos, J.I., 2008. "Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 400-408.
    • Handle: RePEc:eee:chsofr:v:38:y:2008:i:2:p:400-408
    DOI: 10.1016/j.chaos.2006.11.018
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    References listed on IDEAS

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    1. Al-Khaled, Kamel, 2007. "Theory and computation in singular boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 678-684.
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    Cited by:

    1. Ahmad Sami Bataineh & Osman Rasit Isik & Abedel-Karrem Alomari & Mohammad Shatnawi & Ishak Hashim, 2020. "An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials," Mathematics, MDPI, vol. 8(9), pages 1-17, September.
    2. Bengochea, Gabriel, 2014. "Algebraic approach to the Lane–Emden equation," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 424-430.
    3. Aydinlik, Soner & Kiris, Ahmet & Roul, Pradip, 2022. "An effective approach based on Smooth Composite Chebyshev Finite Difference Method and its applications to Bratu-type and higher order Lane–Emden problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 193-205.
    4. Ramos, J.I., 2009. "Generalized decomposition methods for singular oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1149-1155.
    5. Ramos, J.I., 2009. "Generalized decomposition methods for nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1078-1084.
    6. Amit K. Verma & Biswajit Pandit & Lajja Verma & Ravi P. Agarwal, 2020. "A Review on a Class of Second Order Nonlinear Singular BVPs," Mathematics, MDPI, vol. 8(7), pages 1-50, June.
    7. Izadi, Mohammad, 2021. "A discontinuous finite element approximation to singular Lane-Emden type equations," Applied Mathematics and Computation, Elsevier, vol. 401(C).

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