The Zhou’s Method for Solving the White-Dwarfs Equation

Affiliation(s)

Department of Mathematics, Universidad Tecnológica de Pereira, Pereira, Colombia.

Department of Physics, Universidad Tecnológica de Pereira, Pereira, Colombia.

Department of Mathematics, Universidad Tecnológica de Pereira, Pereira, Colombia.

Department of Physics, Universidad Tecnológica de Pereira, Pereira, Colombia.

ABSTRACT

In this work
we apply the differential transformation method (Zhou’s method) or DTM for solving white-dwarfs equation which
Chandrasekhar [1] introduced in his study of the gravitational potential of
these degenerate (*white-dwarf*) stars.
DTM may be considered as alternative and efficient for finding the approximate
solutions of the initial values problems. We prove superiority of this method
by applying them on the some Lane-Emden type equation, in this case. The power series solution of the
reduced equation transforms into an approximate implicit solution of the
original equation.

Cite this paper

P. Alzate and W. Urueña, "The Zhou’s Method for Solving the White-Dwarfs Equation,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 28-32. doi: 10.4236/am.2013.410A3005.

P. Alzate and W. Urueña, "The Zhou’s Method for Solving the White-Dwarfs Equation,"

References

[1] S. Chandrasekhar, “Introduction to Study of Stellar Structure,” Dover, New York, 1967.

[2] J. K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits,” Huazhong University of Science & Technology Press, Wuhan, 1986.

[3] S. J. Liao, “A New Analytic Algorithm of Lane-Emden Type Equations,” Advances Applied Mathematics, Vol. 1, No. 142, 2003, pp. 1-16.

[4] A. M. Wazwaz, “A New Algorithm for Solving Differential Equations of Lane-Emden Type,” Applied Mathematics and Computation, Vol. 118, No. 2-3, 2001, pp. 287310.

http://dx.doi.org/10.1016/S0096-3003(99)00223-4

[5] R. A. Gorder, “An Elegant Perturbation Solution for the Lane-Emden Equation of the Second Kind,” New Astronomy, Vol. 2, No. 16, 2011, pp. 65-67. http://dx.doi.org/10.1016/j.newast.2010.08.005

[6] J. I. Ramos, “Series Approach to the Lane-Emden Equation and Comparison with the Homotopy Perturbation Method,” Chaos Solitons Fractals, Vol. 38, No. 2, 2008, pp. 400-408.

http://dx.doi.org/10.1016/j.chaos.2006.11.018

[7] S. Iqbal and A. Javed, “Application of Optimal, Advances,” Applied Mathematics, Vol. 1, No. 42, 2005, pp. 2948.

[8] C. M. Khalique and P. Ntsime, “Exact Solutions of the Lane-Emden Type Equation,” New Astronomy, Vol. 7, No. 13, 2008, pp. 476-480. http://dx.doi.org/10.1016/j.newast.2008.01.002

[9] G. Hojjati and K. Parand, “An Efficient Computational Algorithm for Solving the Nonlinear Lane-Emden Type Equations,” International Journal of Mathematics and Computation, Vol. 4, No. 7, 2011, pp. 182-187.

[10] B. Batiha, “Numerical Solution of a Class of Singular Second-Order IVPs by Variational Iteration Method,” International Journal of Mathematical Analysis, Vol. 3, No. 40, 2009, pp. 1953-1968.

[11] H. Davis, “Introduction to Nonlinear Differential and Integral Equations,” Dover, New York, 1962.

[12] N. Kumar and R. Pandey, “Solution of the Lane-Emden Equation Using the Bernstein Operational Matrix of Integration,” ISRN Astronomy and Astrophysics, Vol. 2011, 2011, Article ID: 351747. http://dx.doi.org/10.5402/2011/351747

[13] S. Motsa and S. Shate, “New Analytic Solution to the Lane-Emden Equation of Index 2,” Mathematics Problems Engineering, Vol. 2012, 2012, Article ID: 614796.

[1] S. Chandrasekhar, “Introduction to Study of Stellar Structure,” Dover, New York, 1967.

[2] J. K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits,” Huazhong University of Science & Technology Press, Wuhan, 1986.

[3] S. J. Liao, “A New Analytic Algorithm of Lane-Emden Type Equations,” Advances Applied Mathematics, Vol. 1, No. 142, 2003, pp. 1-16.

[4] A. M. Wazwaz, “A New Algorithm for Solving Differential Equations of Lane-Emden Type,” Applied Mathematics and Computation, Vol. 118, No. 2-3, 2001, pp. 287310.

http://dx.doi.org/10.1016/S0096-3003(99)00223-4

[5] R. A. Gorder, “An Elegant Perturbation Solution for the Lane-Emden Equation of the Second Kind,” New Astronomy, Vol. 2, No. 16, 2011, pp. 65-67. http://dx.doi.org/10.1016/j.newast.2010.08.005

[6] J. I. Ramos, “Series Approach to the Lane-Emden Equation and Comparison with the Homotopy Perturbation Method,” Chaos Solitons Fractals, Vol. 38, No. 2, 2008, pp. 400-408.

http://dx.doi.org/10.1016/j.chaos.2006.11.018

[7] S. Iqbal and A. Javed, “Application of Optimal, Advances,” Applied Mathematics, Vol. 1, No. 42, 2005, pp. 2948.

[8] C. M. Khalique and P. Ntsime, “Exact Solutions of the Lane-Emden Type Equation,” New Astronomy, Vol. 7, No. 13, 2008, pp. 476-480. http://dx.doi.org/10.1016/j.newast.2008.01.002

[9] G. Hojjati and K. Parand, “An Efficient Computational Algorithm for Solving the Nonlinear Lane-Emden Type Equations,” International Journal of Mathematics and Computation, Vol. 4, No. 7, 2011, pp. 182-187.

[10] B. Batiha, “Numerical Solution of a Class of Singular Second-Order IVPs by Variational Iteration Method,” International Journal of Mathematical Analysis, Vol. 3, No. 40, 2009, pp. 1953-1968.

[11] H. Davis, “Introduction to Nonlinear Differential and Integral Equations,” Dover, New York, 1962.

[12] N. Kumar and R. Pandey, “Solution of the Lane-Emden Equation Using the Bernstein Operational Matrix of Integration,” ISRN Astronomy and Astrophysics, Vol. 2011, 2011, Article ID: 351747. http://dx.doi.org/10.5402/2011/351747

[13] S. Motsa and S. Shate, “New Analytic Solution to the Lane-Emden Equation of Index 2,” Mathematics Problems Engineering, Vol. 2012, 2012, Article ID: 614796.