Higher order numeric solutions of the Lane–Emden-type equations derived from the multi-stage modified Adomian decomposition method

References

• K. Abbaoui and Y. Cherruault, Convergence of Adomian's method applied to nonlinear equations, Math. Comput. Model. 20 (1994), pp. 60–73. doi: 10.1016/0895-7177(94)00163-4
   Web of Science ®Google Scholar
• K. Abbaoui, Y. Cherruault, and V. Seng, Practical formulae for the calculus of multivariable Adomian polynomials, Math. Comput. Model. 22 (1995), pp. 89–93. doi: 10.1016/0895-7177(95)00103-9
   Web of Science ®Google Scholar
• A. Abdelrazec and D. Pelinovsky, Convergence of the Adomian decomposition method for initial-value problems, Numer. Methods Partial Diff. Equ. 27 (2011), pp. 749–766. doi: 10.1002/num.20549
   Web of Science ®Google Scholar
• F. Abdelwahid, A mathematical model of Adomian polynomials, Appl. Math. Comput. 141 (2003), pp. 447–453. doi: 10.1016/S0096-3003(02)00266-7
   Web of Science ®Google Scholar
• G. Adomian, Stochastic Systems, Academic, New York, 1983.
   Google Scholar
• G. Adomian, Nonlinear Stochastic Operator Equations, Academic, Orlando, OR, 1986.
   Google Scholar
• G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, 1994.
   Google Scholar
• G. Adomian and R. Rach, Inversion of nonlinear stochastic operators, J. Math. Anal. Appl. 91 (1983), pp. 39–46. doi: 10.1016/0022-247X(83)90090-2
   Web of Science ®Google Scholar
• G. Adomian and R. Rach, Transformation of series, Appl. Math. Lett. 4 (1991), pp. 69–71. doi: 10.1016/0893-9659(91)90058-4
   Web of Science ®Google Scholar
• G. Adomian and R. Rach, Nonlinear transformation of series – part II, Comput. Math. Appl. 23 (1992), pp. 79–83. doi: 10.1016/0898-1221(92)90058-P
   Web of Science ®Google Scholar
• G. Adomian and R. Rach, Modified decomposition solution of nonlinear partial differential equations, Appl. Math. Lett. 5 (1992), pp. 29–30. doi: 10.1016/0893-9659(92)90008-W
   Web of Science ®Google Scholar
• G. Adomian and R. Rach, Generalization of Adomian polynomials to functions of several variables, Comput. Math. Appl. 24 (1992), pp. 11–24. doi: 10.1016/0898-1221(92)90037-I
   Web of Science ®Google Scholar
• G. Adomian and R. Rach, Solution of nonlinear partial differential equations in one, two, three, and four dimensions, World Sci. Ser. Appl. Anal. 2 (1993), pp. 1–13.
   Google Scholar
• G. Adomian and R. Rach, Modified decomposition solution of linear and nonlinear boundary-value problems, Nonlinear Anal. 23 (1994), pp. 615–619. doi: 10.1016/0362-546X(94)90240-2
   Web of Science ®Google Scholar
• G. Adomian, R. Rach, and N.T. Shawagfeh, On the analytic solution of the Lane–Emden equation, Found. Phys. Lett. 8 (1995), pp. 161–181. doi: 10.1007/BF02187585
   Web of Science ®Google Scholar
• G. Adomian, R.C. Rach, and R.E. Meyers, Numerical integration, analytic continuation, and decomposition, Appl. Math. Comput. 88 (1997), pp. 95–116. doi: 10.1016/S0096-3003(96)00052-5
   Web of Science ®Google Scholar
• M. Azreg-Aïnou, A developed new algorithm for evaluating Adomian polynomials, Comput. Model. Eng. Sci. 42 (2009), pp. 1–18.
   Web of Science ®Google Scholar
• V.K. Baranwal, R.K. Pandey, M.P. Tripathi, and O.P. Singh, An analytic algorithm of Lane–Emden-type equations arising in astrophysics - a hybrid approach, J. Theor. App. Phys. 6(22) (2012), pp. 1–16.
   Google Scholar
• D.C. Biles, M.P. Robinson, and J.S. Spraker, A generalization of the Lane–Emden equation, J. Math. Anal. Appl. 273 (2002), pp. 654–666. doi: 10.1016/S0022-247X(02)00296-2
   Web of Science ®Google Scholar
• S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, New York, 1967.
   Google Scholar
• Y. Cherruault and G. Adomian, Decomposition methods: A new proof of convergence, Math. Comput. Model. 18 (1993), pp. 103–106. doi: 10.1016/0895-7177(93)90233-O
   Web of Science ®Google Scholar
• H. Chu and Y. Liu, The new ADM-Padé technique for the generalized Emden–Fowler equations, Modern Phys. Lett. B 24 (2010), pp. 1237–1254. doi: 10.1142/S0217984910023268
   Web of Science ®Google Scholar
• H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962.
   Google Scholar
• M. Dehghan and F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron. 13 (2008), pp. 53–59. doi: 10.1016/j.newast.2007.06.012
   Web of Science ®Google Scholar
• M. Dehghan and F. Shakeri, The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Phys. Scr. 78 (2008), pp. 1–11. 065004. doi: 10.1088/0031-8949/78/06/065004
   Web of Science ®Google Scholar
• M. Dehghan and M. Tatari, Finding approximate solutions for a class of third-order non-linear boundary value problems via the decomposition method of Adomian, Int. J. Comput. Math. 87 (2010), pp. 1256–1263. doi: 10.1080/00207160802270853
   Web of Science ®Google Scholar
• M. Dehghan, M. Shakourifar, and A. Hamidi, The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique, Chaos Solitons Fractals 39 (2009), pp. 2509–2521. doi: 10.1016/j.chaos.2007.07.028
   Web of Science ®Google Scholar
• J.-S. Duan, An efficient algorithm for the multivariable Adomian polynomials, Appl. Math. Comput. 217 (2010), pp. 2456–2467. doi: 10.1016/j.amc.2010.07.046
   Web of Science ®Google Scholar
• J.-S. Duan, Recurrence triangle for Adomian polynomials, Appl. Math. Comput. 216 (2010), pp. 1235–1241. doi: 10.1016/j.amc.2010.02.015
   Web of Science ®Google Scholar
• J.-S. Duan, Convenient analytic recurrence algorithms for the Adomian polynomials, Appl. Math. Comput. 217 (2011), pp. 6337–6348. doi: 10.1016/j.amc.2011.01.007
   Web of Science ®Google Scholar
• J.-S. Duan and R. Rach, New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods, Appl. Math. Comput. 218 (2011), pp. 2810–2828. doi: 10.1016/j.amc.2011.08.024
   Web of Science ®Google Scholar
• J.-S. Duan and R. Rach, Higher-order numeric Wazwaz–El-Sayed modified Adomian decomposition algorithms, Comput. Math. Appl. 63 (2012), pp. 1557–1568. doi: 10.1016/j.camwa.2012.03.050
   Web of Science ®Google Scholar
• J.S. Duan, R. Rach, D. Baleanu, and A.M. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Fract. Calc. 3 (2012), pp. 73–99.
   Google Scholar
• G.P. Horedt, Polytropes: Applications in Astrophysics and Related Fields, Kluwer Academic, Dordrecht, 2004.
   Google Scholar
• J.-C. Hsu, H.-Y. Lai, and C.-K. Chen, An innovative eigenvalue problem solver for free vibration of uniform Timoshenko beams by using the Adomian modified decomposition method, J. Sound Vib. 325 (2009), pp. 451–470. doi: 10.1016/j.jsv.2009.03.015
   Web of Science ®Google Scholar
• C.M. Khalique, F.M. Mahomed, and B. Muatjetjeja, Lagrangian formulation of a generalized Lane–Emden equation and double reduction, J. Nonlinear Math. Phys. 15 (2008), pp. 152–161. doi: 10.2991/jnmp.2008.15.2.3
   Web of Science ®Google Scholar
• E. Momoniat and C. Harley, Approximate implicit solution of a Lane–Emden equation, New Astron. 11 (2006), pp. 520–526. doi: 10.1016/j.newast.2006.02.004
   Web of Science ®Google Scholar
• E. Momoniat and C. Harley, An implicit series solution for a boundary value problem modelling a thermal explosion, Math. Comput. Model. 53 (2011), pp. 249–260. doi: 10.1016/j.mcm.2010.08.013
   Web of Science ®Google Scholar
• B. Muatjetjeja and C.M. Khalique, Exact solutions of the generalized Lane–Emden equations of the first and second kind, Pramana 77 (2011), pp. 545–554. doi: 10.1007/s12043-011-0174-4
   Web of Science ®Google Scholar
• K. Parand, M. Shahini, and M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type, J. Comput. Phys. 228 (2009), pp. 8830–8840. doi: 10.1016/j.jcp.2009.08.029
   Web of Science ®Google Scholar
• K. Parand, M. Dehghan, A.R. Rezaeia, and 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. 181 (2010), pp. 1096–1108. doi: 10.1016/j.cpc.2010.02.018
   Web of Science ®Google Scholar
• R. Rach, A convenient computational form for the Adomian polynomials, J. Mathl. Anal. Appl. 102 (1984), pp. 415–419. doi: 10.1016/0022-247X(84)90181-1
   Web of Science ®Google Scholar
• R. Rach, A new definition of the Adomian polynomials, Kybernetes 37 (2008), pp. 910–955. doi: 10.1108/03684920810884342
   Web of Science ®Google Scholar
• R. Rach, A bibliography of the theory and applications of the Adomian decomposition method, 1961–2011, Kybernetes 41 (2012), pp. 1087–1148.
   Web of Science ®Google Scholar
• R. Rach and J.-S. Duan, Near-field and far-field approximations by the Adomian and asymptotic decomposition methods, Appl. Math. Comput. 217 (2011), pp. 5910–5922. doi: 10.1016/j.amc.2010.12.093
   Web of Science ®Google Scholar
• R. Rach, G. Adomian, and R.E. Meyers, A modified decomposition, Comput. Math. Appl. 23 (1992), pp. 17–23. doi: 10.1016/0898-1221(92)90076-T
   Web of Science ®Google Scholar
• A.S.V. Ravi Kanth and K. Aruna, He's variational iteration method for treating nonlinear singular boundary value problems, Comput. Math. Appl. 60 (2010), pp. 821–829. doi: 10.1016/j.camwa.2010.05.029
   Web of Science ®Google Scholar
• O.W. Richardson, The Emission of Electricity from Hot Bodies, Longmans, Green and Company, London, 1921.
   Google Scholar
• F. Shakeri and M. Dehghan, Application of the decomposition method of Adomian for solving the pantograph equation of order m, Z. Naturforsch. 65a (2010), pp. 453–460.
   Google Scholar
• X. Shang, P. Wu, and X. Shao, An efficient method for solving Emden–Fowler equations, J. Franklin Instit. 346 (2009), pp. 889–897. doi: 10.1016/j.jfranklin.2009.07.005
   Web of Science ®Google Scholar
• H.R. Tabrizidooz, H.R. Marzban, and M. Razzaghi, Solution of the generalized Emden–Fowler equations by the hybrid functions method, Phys. Scr. 80 (2009), 1–5. 025001. doi: 10.1088/0031-8949/80/02/025001
   Web of Science ®Google Scholar
• A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput. 111 (2000), pp. 53–69. doi: 10.1016/S0096-3003(99)00063-6
   Web of Science ®Google Scholar
• A.-M. Wazwaz, A new algorithm for solving differential equations of Lane–Emden type, Appl. Math. Comput. 118 (2001), pp. 287–310. doi: 10.1016/S0096-3003(99)00223-4
   Web of Science ®Google Scholar
• A.-M. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations, Appl. Math. Comput. 128 (2002), pp. 45–57. doi: 10.1016/S0096-3003(01)00021-2
   Web of Science ®Google Scholar
• A.-M. Wazwaz, Adomian decomposition method for a reliable treatment of the Emden–Fowler equation, Appl. Math. Comput. 161 (2005), pp. 543–560. doi: 10.1016/j.amc.2003.12.048
   Web of Science ®Google Scholar
• A.-M. Wazwaz, Analytical solution for the time–dependent Emden–Fowler type of equations by Adomian decomposition method, Appl. Math. Comput. 166 (2005), pp. 638–651. doi: 10.1016/j.amc.2004.06.058
   Web of Science ®Google Scholar
• A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, HEP and Springer, Beijing, 2009.
   Google Scholar
• A.-M. Wazwaz, The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), pp. 3881–3886. doi: 10.1016/j.cnsns.2011.02.026
   Web of Science ®Google Scholar
• A.-M. Wazwaz and R. Rach, Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane–Emden equations of the first and second kinds, Kybernetes 40 (2011), pp. 1305–1318. doi: 10.1108/03684921111169404
   Web of Science ®Google Scholar
• A.-M. Wazwaz, R. Rach, and J.-S. Duan, Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, Appl. Math. Comput. 219 (2013), pp. 5004–5019. doi: 10.1016/j.amc.2012.11.012
   Web of Science ®Google Scholar
• A.-M. Wazwaz, R. Rach, and J.-S. Duan, A study on the systems of the Volterra integral forms of the Lane–Emden equations by the Adomian decomposition method, Math. Method. Appl. Sci. 37 (2014), pp. 10–19. doi: 10.1002/mma.2776
   Web of Science ®Google Scholar
• A. Yildirim and T. Özis, Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method, Phys. Lett. A 369 (2007), pp. 70–76. doi: 10.1016/j.physleta.2007.04.072
   Web of Science ®Google Scholar
• A. Yildirim and T. Özis, Solutions of singular IVPs of Lane–Emden type by the variational iteration method, Nonlinear Anal. 70 (2009), pp. 2480–2484. doi: 10.1016/j.na.2008.03.012
   Web of Science ®Google Scholar
• Y. Zhu, Q. Chang, and S. Wu, A new algorithm for calculating Adomian polynomials, Appl. Math. Comput. 169 (2005), pp. 402–416. doi: 10.1016/j.amc.2004.09.082
   Web of Science ®Google Scholar