# NUMERICAL SOLUTION OF GENERALIZED EMDEN- FOWLER EQUATIONS BY SOME APPROXIMATION TECHNIQUES

## Abstract

In this paper, we provide reliable approximations to the generalized Emden - Fowler equation using two semi - analytic methods; Adomian decomposition method and variational iteration method, and the recursive Tau method that employed Newton-Kantorovich approach. The three methods give very close results, with the semi - analytic methods givingresults that agree completely with some existing results in the literature when certain parameters are xed. The results are presented in both tabular and graphical forms.

## References

K. Issa and R. B. Adeniyi A generalized scheme for the numerical solution of initial value problems in ordinary differential equations by the recursive formulation of Tau Method, Intl. Jour. of Pure and Appl. Math. 88(1), 1-13, 2013.

K. Issa and R. B. Adeniyi, Extension of Generalized Recursive Tau Method to Non-linear Ordinary Differential Equations, Journal of Nigeria Mathematics Society, 35, 18-24, 2016.

J. Biazar and K. Hosseini, An effective Modification of Adomian Decomposition Method for Solving EmdenFowler Type Systems, Natl. Acad. Sci. Lett., DOI: 10.1007/s40009-017-0571-4

J. I. Ramos, Linearization techniques for singular initial value problems of ordinary differential equations, Appl Math. Comput., 161, 525542, 2005.

J. I. Ramos, Series approach to the LaneEmden equation and comparison with the homotopy perturbation method, Chaos Solitons Fractals, 38, 400408, 2008.

K. Parand, M. Dehghan, A. R. Rezaei and S. M. Ghaderi, An approximation algorithm for the solution of the nonlinear LaneEmden type equations arising in astrophysics using Hermite functions collocation method, Comput Phys Commun, 181, 1096-11-8, 2010.

O. P. Singh, R. K. Pandey and V. K. Singh, An analytic algorithm of LaneEmden type equations arising in astrophysics using modified homotopy analysis method, Comput. Phys. Commun., 180, 1116 - 1124, 2009.

J. H. He, Variational approach to the LaneEmden equation, Appl Math Comput., 143, 539541, 2003.

L. M. Berkorich, The Generalized Emden-Fowler Equation, Sym. Nonlinear Math. Phy., 1, 155-163, 1997.

E. L. Ortiz and H. Samara, An operational approach to the Tau Method for the numerical solution of non-linear differential equations, Computing, 27, 15-25, 1981.

B. M. Yisa and R. B. Adeniyi, Generalization of canonical polynomials for overde-

termined mth order ordinary differential equations(ODEs), IJERT, 1(6), 1-15, 2012.

M. Hermann and M. Seravi, Nonlinear Ordinary Dierential Equations, Springer(Indian), 2016.

A. M. Wazwaz, R. Rach, L. Bougoa and J. S. Duan, Solving the Two - Dimensional Lane- Emdem Type Equations of Higher Orders by the Adomian Decomposition Method, CMES, 100(6),507-529, 2017.

A. M. Wazwaz, Solving the Two - Dimensional Emdem-Fowler Type Equations of Third Order by the Variational Iteration Method, Appl. Math. Info.Sci, 9(5), 2429-2436, 2015.

A. Nazari-Golshan, S. S. Nourazar, H. Ghafoori-Fard, A. Yildirim, A. Campo, A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular LaneEmden equations, Appl. Math. Lett., 26, 1018-1025, 2013.

A. M. Wazwaz, The variational iteration method for solving systems of equations of EmdenFowler type, Int J Comput Math, 88, 34063415, 2011.

J. Biazar and K. Hosseini, A modified Adomian decomposition method for singular initial value EmdenFowler type equations, Int. J. Appl. Math. Res., 5, 6972, 2016.

H. Goenner and P. Havas, Exact solutions of the generalized Lane-Emden equation, Jour. of Mathematical Physics, 41(10), 70297042, 2000.

N. T. Shawagfeh, Non-perturbative approximate solution for Lane-Emden equation, Journal of Mathematical Physics,34(9), 43644369, 1993.

A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Applied Mathematics and Computation, 118,(2-3), 287310, 2001.

K. Issa, R. B. Adeniyi and B. M. Yisa, Generalized Error Estimation of the Tau method in Ordinary Differential Equations, J. Nig. Math. Soc., 36, 113-137, 2017.

R. Mohammadzadeh, M. Lakestani and M. Dehghan, Collocation method for the numerical solutions of Lane-Emden type equations using cubic Hermite spline functions, Math. Method. Appl. Sci., 37 (9), 1303-1717, 2014.

B. T. Polyak, Newton-Kantorovich method and its global convergence, J. Math. Sci., 133(4), 1513-1523, 2006.

M. A. Waleed, H. Y. Youssri and H. D. Eid, New Solutions for Singular Lane-Emden Equations Arising in Astrophysics Based on Shifted Ultra spherical Operational Matrices of Derivatives, Computational Methods for Dierential Equations, 2(3), 171-185, 2014

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*Journal of the Nigerian Mathematical Society*,

*37*(1), 23–40. Retrieved from https://ojs.ictp.it/jnms/index.php/jnms/article/view/287