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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access February 5, 2015

The variational iteration method for solving the Volterra integro-differential forms of the Lane-Emden and the Emden-Fowler problems with initial and boundary value conditions

  • Abdul-Majid Wazwaz and Suheil A. Khuri
From the journal Open Engineering

Abstract

In this paper, the variational iteration method (VIM) is used to examine the Volterra integro-differential forms of the singular Lane–Emden and the Emden–Fowler initial value problems and boundary value problems arising in physics, astrophysics and stellar structures. The Volterra integro-differential forms of the Lane–Emden and the Emden–Fowler equations overcome the singularity behavior at the origin x = 0. The Lagrange multiplier, needed for the VIM, is λ = −1 for the various cases of the specified equations having distinct shape factors. We illustrate our work by analyzing few initial value problems and boundary value problems to emphasize the convergence of the acquired results.

References

[1] A.M. Wazwaz, R. Rach, J.-S. Duan, Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, App. Math. Comput. 2013, 219, 5004–5019. Search in Google Scholar

[2] A.M. Wazwaz, Analytical solution for the time–dependent Emden–Fowler type of equations by Adomian decomposition method, Appl. Math. Comput. 2005, 166, 638–651. Search in Google Scholar

[3] A.M. Wazwaz, A new method for solving differential equations of the Lane–Emden type, Appl. Math. Comput. 2001, 118(2/3), 287–310. Search in Google Scholar

[4] A.M. Wazwaz, A new method for solving singular initial value problems in the second order ordinary diff erential equations, Appl. Math. Comput. 2002, 128, 47–57. Search in Google Scholar

[5] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Emden–Fowler equation, Appl. Math. Comput. 2005, 161, 543–560. Search in Google Scholar

[6] A.M. Wazwaz, Partial differential equations and solitary waves theory, HEP and Springer, Beijing and Berlin, 2009. 10.1007/978-3-642-00251-9Search in Google Scholar

[7] A. Aslanov, Approximate solutions of Emden–Fowler type equations, Int. J. Comput. Math. 2009, 86(5), 807–826. Search in Google Scholar

[8] H. Kaur, R.C. Mittal, V. Mishra, Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics, Comp. Phys. Commun. 2013, 184, 2169–2177. Search in Google Scholar

[9] A. Yildirim, T. Ozis, Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method, Phys. Lett. A 2007, 369, 70–76. 10.1016/j.physleta.2007.04.072Search in Google Scholar

[10] A. Yildirim, T. Ozis, Solutions of singular IVPs of Lane–Emden type by the variational iteration method, Nonlinear Anal. 2009, 70, 2480–2484. Search in Google Scholar

[11] X. Shang, P. Wu, X. Shao, An efficient method for solving Emden–Fowler equations, J. Franklin Institute 2009, 346, 889– 897. 10.1016/j.jfranklin.2009.07.005Search in Google Scholar

[12] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron. 2008, 13, 53–59. Search in Google Scholar

[13] K. Parand, M. Dehghan, A.R. Rezaeia, S.M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun. 2010, 181, 1096– 1108. Search in Google Scholar

[14] K. Parand, M. Shahini, M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type, J. Comput. Phys. 2009, 228, 8830– 8840. Search in Google Scholar

[15] K. Parand, M. Dehghan, A.R. Rezaeia, S.M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun. 2010, 181, 1096– 1108. Search in Google Scholar

[16] H. Tabriziodooz, H. Marzban, M. Razzaghi, Solution of the generalized Emden–Fowler equations by the hybrid functions method, Physica Scripta 2009, 80, 025001. 10.1088/0031-8949/80/02/025001Search in Google Scholar

[17] A.S.V. Ravi Kanth, K. Aruna, He’s variational iteration method for treating nonlinear singular boundary value problems, Comput. Math. Applic. 2010, 60, 821–829. Search in Google Scholar

[18] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962. Search in Google Scholar

[19] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover Publications, New York, 1967. Search in Google Scholar

[20] O.U. Richardson, The Emission of Electricity from Hot Bodies, London 1921. Search in Google Scholar

[21] G. Adomian, R. Rach, N.T. Shawagfeh,Onthe analytic solution of Lane–Emden equation, Found. Phys. Lett. 1995, 8(2), 161–181. Search in Google Scholar

[22] J.H. He, A variational iteration approach to nonlinear problems and its applications, Mech. Appl. 1998, 20(1), 30–31. Search in Google Scholar

[23] J. Biazar, M. Eslami, H. Ghazvini, He’s homotopy perturbation method for systems of integro-differential equations, Chaos Soliton. Fract. 2009, 39, 1253–1258. Search in Google Scholar

[24] M. Eslami, M. Mirzazadeh, Study of convergence of Homotopy perturbation method for two dimensional linear Volterra integral equations of the first kind, IJCSM 2014, 5(1), 72–80. 10.1504/IJCSM.2014.059379Search in Google Scholar

[25] J. Biazar, M. Eslami, Application of homotopy perturbation method for systems of Volterra integral equations of the first kind, Chaos Soliton. Fract. 2009, 42, 2597–3246. Search in Google Scholar

[26] M. Eslami, New Homotopy Perturbation Method for Special Kind of Systems of Volterra Integral Equations in Two-dimensional Spaces, Comput. Math. Model. 2014, 25(1), 135–148. Search in Google Scholar

[27] J. Biazar, M. Eslami, Differential transform method for systems of Volterra integral equations of the second kind and comparison with homotopy perturbation method, Int. J. Phys. Sci. 2011, 6(5), 1207–1212. Search in Google Scholar

[28] J. Biazar, M. Eslami, Acceleration of the convergence of He’s homotopy perturbation method for solving Fredholm integral equations of the second kind, JARAM 2010, 2(2), 58–67. 10.5373/jaram.286.110909Search in Google Scholar

Received: 2014-7-7
Accepted: 2014-10-14
Published Online: 2015-2-5

© 2015 A.M. Wazwaz and S.A. Khuri

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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