A modified Adomian decomposition method for singular initial value Emden-Fowler type equations
About this article
Abstract
Traditional Adomian decomposition method (ADM) usually fails to solve singular initial value problems of Emden-Fowler type. To overcome this shortcoming, a new and effective modification of ADM that only requires calculation of the first Adomian polynomial is formally proposed in the present paper. Three singular initial value problems of Emden-Fowler type with alpha=1, 2, and >2, have been selected to demonstrate the efficiency of the method.
References
[1] A.M. Wazwaz, A reliable treatment of singular Emden-Fowler initial value problems and boundary value problems, Applied Mathematics and Computation, 217 (2011), 10387–10395. http://dx.doi.org/10.1016/j.amc.2011.04.084.
[2] K. Parand, M. Dehghan, A.R. Rezaei, S.M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method, Computer Physics Communications, 181 (2010), 1096–1108. http://dx.doi.org/10.1016/j.cpc.2010.02.018.
[3] K. Parand, M. Shahini, M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, Journal of Computational Physics, 228 (2009), 8830–8840. http://dx.doi.org/10.1016/j.jcp.2009.08.029.
[4] A.M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Applied Mathematics and Computation, 118 (2001), 287–310. http://dx.doi.org/10.1016/S0096-3003(99)00223-4.
[5] S.J. Liao, A new analytic algorithm of Lane-Emden type equations, Applied Mathematics and Computation, 142 (2003), 1–16. http://dx.doi.org/10.1016/S0096-3003(02)00943-8.
View more references (18)
[6] O.P. Singh, R.K. Pandey, V.K. Singh, An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified homotopy analysis method, Computer Physics Communications, 180 (2009), 1116–1124. http://dx.doi.org/10.1016/j.cpc.2009.01.012.
[7] J.I. Ramos, Linearization techniques for singular initial-value problems of ordinary differential equations, Applied Mathematics and Computation, 161 (2005), 525–542. http://dx.doi.org/10.1016/j.amc.2003.12.047.
[8] A. Yildirim, T. Ozis, Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method, Physics Letters A, 369 (2007), 70–76. http://dx.doi.org/10.1016/j.physleta.2007.04.072.
[9] X. Shang, P. Wu, X. Shao, An efficient method for solving Emden-Fowler equations, Journal of the Franklin Institute, 346 (2009), 889–897. http://dx.doi.org/10.1016/j.jfranklin.2009.07.005.
[10] J.H. He, Variational approach to the Lane-Emden equation, Applied Mathematics and Computation, 143 (2003), 539–541. http://dx.doi.org/10.1016/S0096-3003(02)00382-X.
[11] J.I. Ramos, Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method, Chaos, Solitons & Fractals, 38 (2008), 400–408. http://dx.doi.org/10.1016/j.chaos.2006.11.018.
[12] S.A. Yousefi, Legendre wavelets method for solving differential equations of Lane-Emden type, Applied Mathematics and Computation, 181 (2006), 1417–1422. http://dx.doi.org/10.1016/j.amc.2006.02.031.
[13] N.T. Shawagfeh, Nonperturbative approximate solution for Lane-Emden equation, Journal of Mathematical Physics, 34 (1993), 4364–4369. http://dx.doi.org/10.1063/1.530005.
[14] A.S. Bataineh, M.S.M. Noorani, I. Hashim, Homotopy analysis method for singular IVPs of Emden-Fowler type, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 1121–1131. http://dx.doi.org/10.1016/j.cnsns.2008.02.004.
[15] M.S.H. Chowdhury, I. Hashim, Solutions of Emden-Fowler equations by homotopy perturbation method, Nonlinear Analysis: Real World Applications, 10 (2009), 104–115. http://dx.doi.org/10.1016/j.nonrwa.2007.08.017.
[16] H.R. Marzban, H.R. Tabrizidooz, M. Razzaghi, Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations, Physics Letters A, 372 (2008), 5883–5886. http://dx.doi.org/10.1016/j.physleta.2008.07.055.
[17] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy, 13 (2008), 53–59. http://dx.doi.org/10.1016/j.newast.2007.06.012.
[18] R.K. Pandey, N. Kumar, Solution of Lane-Emden type equations using Bernstein matrix of differentiation, New Astronomy, 17 (2012), 303–308. http://dx.doi.org/10.1016/j.newast.2011.09.005.
[19] B. Muatjetjeja, C.M. Khalique, Exact solutions of the generalized Lane-Emden equations of the first and second kind, Pramana Journal of Physics, 77 (2011), 545–554. http://dx.doi.org/10.1007/s12043-011-0174-4.
[20] J.S. Duan, Convenient analytic recurrence algorithms for the Adomian polynomials, Applied Mathematics and Computation, 217 (2011), 6337–6348. http://dx.doi.org/10.1016/j.amc.2011.01.007.
[21] H. Aminikhah, J. Biazar, A new HPM for ordinary differential equations, Numerical Methods for Partial Differential Equations, 26 (2009), 480–489. http://dx.doi.org/10.1002/num.20413.
[22] K. Hosseini, J. Biazar, R. Ansari, P. Gholamin, A new algorithm for solving differential equations, Mathematical Methods in the Applied Sciences, 35 (2012), 993–999. http://dx.doi.org/10.1002/mma.1601.
[23] B. Muatjetjeja, C.M. Khalique, Lagrangian approach to a generalized coupled Lane-Emden system: Symmetries and first integrals, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 1166–1171. http://dx.doi.org/10.1016/j.cnsns.2009.06.002.