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A novel wavelet approximation method for the solution of nonlinear differential equations with variable coefficients arising in astrophysics

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Abstract

Lane-Emden type equations are well known model equations for the singular nonlinear boundary value problem that arise in the field of astrophysics. In this paper, a new method incorporated by the Legendre wavelet operational matrix of derivatives and block-pulse function approximation is adopted to investigate the solution of Lane-Emden type equations. Illustrative examples are given to demonstrate the simplicity and effectiveness of the proposed method. The obtained results are compared with other available results.

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References

  • Abualrub, T., Sadek, I.: Legendre wavelet operational matrix of derivative for optimal control in a connective—diffusive fluid problem. J. Franklin Inst. 351, 682–693 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Balaji, S.: Solution of nonlinear Riccati differential equation using Chebyshev wavelets. WSEAS Trans. Math. 13, 441–451 (2014a)

    Google Scholar 

  • Balaji, S.: A new approach for the solution of Duffing equation involving both integral and non-integral forcing terms. Ain Shams Eng. J. 5, 985–990 (2014b)

    Article  Google Scholar 

  • Balaji, S.: Legendre wavelet operational matrix method for solution of Riccati differential equation. Int. J. Math. Math. Sci. 2014, 304745 (2014c)

    Article  MathSciNet  Google Scholar 

  • Balaji, S.: A new Bernoulli wavelet operational matrix of derivative method for the solution of nonlinear singular Lane–Emden type equations arising in astrophysics. ASME J. Comput. Nonlinear Dyn. 11, 051013 (2016)

    Article  Google Scholar 

  • Bender, C.M., Milton, K.A., Pinsky, S.S., Simmons, L.M. Jr.: A new perturbative approach to nonlinear problems. J. Math. Phys. 30, 1447–1455 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Căruntu, B., Bota, C.: Approximate polynomial solutions of the nonlinear Lane–Emden type equations arising in astrophysics using the squared remainder minimization method. Comput. Phys. Commun. 184, 1643–1648 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  • Chandrasekhar, S.: Introduction to Study of Stellar Structure. Dover, New York (1967)

    Google Scholar 

  • Chowdhury, M.S.H., Hashim, I.: Solutions of Emden–Fowler equations by homotopy perturbation method. Nonlinear Anal., Real World Appl. 10, 104–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Davis, H.T.: Introduction to Nonlinear Differential and Integral Equations. Dover, New York (1962)

    Google Scholar 

  • Dehghan, M., Shakeri, F.: Approximate solution of differential equation arising in astrophysics using the variation method. New Astron. 13, 53–59 (2008)

    Article  ADS  Google Scholar 

  • Horedt, G.P.: Seven-digit tables of Lane-Emden functions. Astrophys. Space Sci. 126, 357–408 (1986)

    Article  ADS  Google Scholar 

  • Horedt, G.P.: Polytropes: Applications in Astrophysics and Related Fields. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  • Iqbal, S., Javed, A.: Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation. Appl. Math. Comput. 217, 7753–7761 (2011)

    MathSciNet  MATH  Google Scholar 

  • Kaur, H., Mittal, R.C., Mishra, V.: Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics. Comput. Phys. Commun. 184, 2169–2177 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kumar, N., Pandey, R.K., Cattani, C.: Solution of the Lane-Emden equation using the Bernstein operational matrix of integration. ISRN Astron. Astrophys. 2011, 351747 (2011), 7 p.

    Google Scholar 

  • Lakestani, M., Dehghan, M.: Four techniques based on the B-spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation. Math. Methods Appl. Sci. 16, 2243–2253 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Lane, J.H.: On theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its internal heat and depending on the laws of gases known to terrestrial experiment. Am. J. Sci. Arts Ser. 2(50), 57–74 (1870)

    Article  ADS  Google Scholar 

  • Liao, S.: A new analytic algorithm of Lane–Emden type equations. Appl. Math. Comput. 142, 1–16 (2003)

    MathSciNet  MATH  Google Scholar 

  • Mohammadi, F., Hosseini, M.M.: A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations. J. Franklin Inst. 348, 1787–1796 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Mukherjee, S., Roy, B., Chaterjee, P.K.: Solution of Lane-Emden equation by differential transform method. Int. J. Nonlinear Sci. 12, 478–484 (2011)

    MathSciNet  Google Scholar 

  • Nasab, A.K., Kılıçman, A., Atabakan, Z.P., Leong, W.J.: A numerical approach for solving singular nonlinear Lane–Emden type equations arising in astrophysics. New Astron. 34, 178–186 (2015)

    Article  ADS  Google Scholar 

  • Öztürk, Y., Gülsu, M.: An approximation algorithm for the solution of the Lane–Emden type equations arising in astrophysics and engineering using Hermite polynomials. Comput. Appl. Math. 33, 131–145 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Pandey, R.K., Kumar, N.: Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation. New Astron. 17, 303–308 (2012)

    Article  ADS  Google Scholar 

  • Pandey, R.K., Kumar, N., Bhardwaj, A., Dutta, G.: Solution of Lane–Emden type equations using Legendre operational matrix of differentiation. Appl. Math. Comput. 218, 7629–7637 (2012)

    MathSciNet  MATH  Google Scholar 

  • Parand, K., Shahini, M., Dehghan, M.: Rational Legendre pseudo-spectral approach for solving nonlinear differential equations of Lane–Emden type. J. Comput. Phys. 228, 8830–8840 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Parand, K., Dehghan, M., Rezaei, A.R., Ghaderi, S.M.: 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, 1096–1108 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Ramos, J.I.: Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method. Chaos Solitons Fractals 38, 400–408 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Richardson, O.U.: Emission of Electricity from Hot Bodies. Longmans, Green, London (1921)

    Google Scholar 

  • Rismani, A.M., Monfared, H.: Numerical solution of singular IVPs of Lane–Emden type using a modified Legendre-spectral method. Appl. Math. Model. 36, 4830–4836 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Rosenau, P.: A note on integration of the Emden–Fowler equation. Int. J. Non-Linear Mech. 19, 303–308 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Sahu, P.K., Saha Ray, S.: A new Bernoulli wavelet method for numerical solutions of nonlinear weakly singular Volterra integro-differential equations. Int. J. Comput. Methods 14, 1750022 (2017). https://doi.org/10.1142/S0219876217500220

    Article  MathSciNet  Google Scholar 

  • Shawagfeh, N.T.: Nonperturbative approximate solution for Lane–Emden equation. J. Math. Phys. 34(9), 4364–4369 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.: A new algorithm for solving differential equations of Lane–Emden type. Appl. Math. Comput. 118, 287–310 (2001)

    MathSciNet  MATH  Google Scholar 

  • Yang, C., Hou, J.: A numerical method for Lane-Emden equations using hybrid functions and the collocation method. J. Appl. Math. 2012, 316534 (2012)

    MathSciNet  MATH  Google Scholar 

  • Yildirim, A., Öziş, T.: Solutions of singular IVPs of Lane–Emden type by the variational iteration method. Nonlinear Anal. 70, 2480–2484 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Yousefi, S.A.: Legendre wavelets method for solving differential equations of Lane-Emden type. Appl. Math. Comput. 181, 1417–1422 (2006)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the DST-SERB Project, Government of India (Project No. SB/FTP/MS-012/2013). Our hearty thanks are due to prof. R. Sethuraman, Vice-Chancellor, SASTRA University, Dr. S. Vaidhyasubramaniam, Dean/Planning and Development and Dr. S. Swaminathan, Dean/Sponsored Research for their kind encouragement and for providing good research environment.

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  1. Department of Mathematics, SASTRA University, Thanjavur, 613 401, India

    S. Balaji & G. Hariharan

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  1. S. Balaji
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  2. G. Hariharan
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Correspondence to G. Hariharan.

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Balaji, S., Hariharan, G. A novel wavelet approximation method for the solution of nonlinear differential equations with variable coefficients arising in astrophysics. Astrophys Space Sci 363, 16 (2018). https://doi.org/10.1007/s10509-017-3236-3

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