Advertisement

Astrophysics and Space Science

, 363:264 | Cite as

A high-order numerical method for solving nonlinear Lane-Emden type equations arising in astrophysics

  • Soner Aydinlik
  • Ahmet Kiris
Original Article
  • 8 Downloads

Abstract

In this paper, some nonlinear Lane-Emden type equations arising in astrophysics are solved by Chebyshev Finite Difference Method. Convergence and error analysis of the method are examined. To show the applicability and efficiency, some astrophysics problems such as the isothermal gas spheres, standard Lane-Emden equation and white-dwarf equation are realized. Besides, the method carried out for some boundary value problems, and it is shown that the method also works with boundary conditions as well as with initial conditions without any modification. The results demonstrated that the proposed method is rather efficient and more accurate than the many methods given in the literature.

Keywords

Chebyshev finite difference method Lane-Emden type equation White-dwarf equation Collocation method 

References

  1. Aminikhah, H., Kazemi, S.: On the numerical solution of singular Lane–Emden type equations using cubic B-spline approximation. Int. J. Appl. Comput. Math. 3, 703–712 (2017) MathSciNetCrossRefGoogle Scholar
  2. 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) CrossRefGoogle Scholar
  3. 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) ADSMathSciNetCrossRefGoogle Scholar
  4. Bildik, N., Deniz, S.: Comparative study between optimal homotopy asymptotic method and perturbation-iteration technique for different types of nonlinear equations. Iran. J. Sci. Technol., Trans. A, Sci. 42, 647–654 (2018a) MathSciNetCrossRefGoogle Scholar
  5. Bildik, N., Deniz, S.: Solving the Burgers’ and regularized long wave equations using the new perturbation iteration technique. Numer. Methods Partial Differ. Equ. 34, 1489–1501 (2018b) MathSciNetCrossRefGoogle Scholar
  6. Canuto, C., Quarteroni, A., Hussaini, M.Y., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer, Berlin/Heidelberg (2006) zbMATHGoogle Scholar
  7. 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) ADSMathSciNetCrossRefGoogle Scholar
  8. Chandrasekhar, S.: Introduction to Study of Stellar Structure. Dover, New York (1967) Google Scholar
  9. Chowdhury, M.S.H., Hashim, I.: Solutions of Emden–Fowler equations by homotopy perturbation method. Nonlinear Anal., Real World Appl. 10, 104–115 (2009) MathSciNetCrossRefGoogle Scholar
  10. Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197 (1960) MathSciNetCrossRefGoogle Scholar
  11. Davis, H.T.: Introduction to Nonlinear Differential and Integral Equations. Dover, New York (1962) Google Scholar
  12. Dehghan, M., Aryanmehr, S., Eslahch, M.R.: A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method. J. Comput. Appl. Math. 244, 125 (2013) MathSciNetCrossRefGoogle Scholar
  13. Deniz, S., Bildik, N.: A new analytical technique for solving Lane–Emden type equations arising in astrophysics. Bull. Belg. Math. Soc. Simon Stevin 24, 305–320 (2017) MathSciNetzbMATHGoogle Scholar
  14. Duggan, R.C., Goodman, A.M.: Pointwise bounds for a nonlinear heat conduction model of the human head. Bull. Math. Biol. 48, 229–236 (1986) CrossRefGoogle Scholar
  15. Elbarbary, E.M.E., El-Kady, M.: Chebyshev finite difference approximation for the boundary value problems. Appl. Math. Comput. 139, 513–523 (2003) MathSciNetzbMATHGoogle Scholar
  16. Elbarbary, E.M.E., El-Sayed, S.M.: Higher order pseudospectral differentiation matrices. Appl. Numer. Math. 55, 425–438 (2005) MathSciNetCrossRefGoogle Scholar
  17. Emden, R.: Gaskugeln Anwendungen der Mechan. Warmtheorie. Teubner, Leipzig/Berlin (1907) Google Scholar
  18. Fox, L., Parker, I.B.: Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London (1968) zbMATHGoogle Scholar
  19. Ghorbani, A., Bakherad, M.: A variational iteration method for solving nonlinear Lane–Emden problems. New Astron. 54, 1 (2017) ADSCrossRefGoogle Scholar
  20. Gürbüz, B., Sezer, M.: Laguerre polynomial approach for solving Lane–Emden type functional differential equations. Appl. Math. Comput. 242, 255 (2014) MathSciNetzbMATHGoogle Scholar
  21. Horedt, G.P.: Polytropes: Applications in Astrophysics and Related Fields. Kluwer Academic Publishers, Dordrecht (2004) Google Scholar
  22. Kilic, M., Thorstensen, J., Kowalski, P.: Monthly Notices of the Royal Astronomical Society. Columbia University, New York (2012) Google Scholar
  23. Krivec, R., Mandelzweig, V.B.: Quasilinearization approach to computation with singular potentials. Comput. Phys. Commun. 179, 865–867 (2008) ADSMathSciNetCrossRefGoogle Scholar
  24. Kumar, N., Pandey, R.K., Cattani, C.: Solution of the Lane–Emden equation using the Bernstein operational matrix of integration. ISRN Astron. Astrophys. 2011, 351747.1–351747.7 (2011) CrossRefGoogle Scholar
  25. 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) MathSciNetCrossRefGoogle Scholar
  26. 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 2(50), 57–74 (1870) ADSCrossRefGoogle Scholar
  27. Liao, S.: A new analytic algorithm of Lane–Emden type equations. Appl. Math. Comput. 142, 1–16 (2003) MathSciNetzbMATHGoogle Scholar
  28. Merafina, M., Bisnovatyi-Kogan, G.S., Tarasov, S.O.: A brief analysis of self-graviating polytropic models with a non-zero cosmological constant. Astron. Astrophys. 541, A84 (2012) ADSCrossRefGoogle Scholar
  29. Pandey, R.K.: A finite difference method for a class of singular two-point boundary value problems arising in physiology. Int. J. Comput. Math. 65, 131–140 (1997) MathSciNetCrossRefGoogle Scholar
  30. Pandey, R.K., Kumar, N.: Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation. New Astron. 17, 303–308 (2012) ADSCrossRefGoogle Scholar
  31. 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) MathSciNetzbMATHGoogle Scholar
  32. Parand, K., Delkhosh, M.: An effective numerical method for solving the nonlinear singular Lane–Emden type equations of various orders. J. Teknol. 79, 25–36 (2017) Google Scholar
  33. 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) ADSMathSciNetCrossRefGoogle Scholar
  34. Ramos, J.I.: Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method. Chaos Solitons Fractals 38, 400–408 (2008) ADSMathSciNetCrossRefGoogle Scholar
  35. Ravi Kanth, A., Aruna, K.: He’s varitional iteration method for treating nonlinear singular boundary value problem. Comput. Math. Appl. 60(3), 821–829 (2010) CrossRefGoogle Scholar
  36. Reger, K., Van Gorder, R.A.: Lane–Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere. Appl. Math. Mech. 34, 1439–1452 (2013) MathSciNetCrossRefGoogle Scholar
  37. Richardson, O.U.: Emission of Electricity from Hot Bodies. Longmans, Green, London (1921) Google Scholar
  38. Rosenau, P.: A note on integration of the Emden–Fowler equation. Int. J. Non-Linear Mech. 19, 303–308 (1984) ADSMathSciNetCrossRefGoogle Scholar
  39. Roul, P.: Kiran, T.: A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems. Int. J. Appl. Comput. Math. (2017).  https://doi.org/10.1080/00207160.2017.1417592 CrossRefGoogle Scholar
  40. Saadatmandi, A., Dehghan, M.: The numerical solution of problems in calculus of variation using Chebyshev finite difference method. Phys. Lett. A 372, 4037–4040 (2008) ADSCrossRefGoogle Scholar
  41. Saadatmandi, A., Farsangi, J.A.: Chebyshev finite difference method for a nonlinear system of second-order boundary value problems. Appl. Math. Comput. 192, 586–591 (2007) MathSciNetzbMATHGoogle Scholar
  42. Sahu, P.K., Saha, R.: Chebyshev wavelet method for numerical solutions of integro differential form of Lane–Emden type differential equations. Int. J. Wavelets Multiresolut. Inf. Process. 15, 1750015 (2017) MathSciNetCrossRefGoogle Scholar
  43. Shawagfeh, N.T.: Nonperturbative approximate solution for Lane–Emden equation. J. Math. Phys. 34(9), 4364–4369 (1993) ADSMathSciNetCrossRefGoogle Scholar
  44. Singh, H.: An efficient computational method for the approximate solution of nonlinear Lane–Emden type equations arising in astrophysics. Astrophys. Space Sci. 363, 71 (2018) ADSMathSciNetCrossRefGoogle Scholar
  45. Singh, M., Verma, A.K.: An effective computational technique for a class of Lane–Emden equations. J. Math. Chem. 54, 231–251 (2016) ADSMathSciNetCrossRefGoogle Scholar
  46. Singh, O.P., Pandey, R.K., Singh, V.K.: An analytic algorithm of Lane–Emden type equations arising in astrophysics using modified homotopy analysis method. Comput. Phys. Commun. 180, 1116 (2009) ADSMathSciNetCrossRefGoogle Scholar
  47. Singh, R., Garg, H., Guleria, V.: Haar wavelet collocation method for Lane–Emden equations with Dirichlet, Neumann and Neumann–Robin boundary conditions. J. Comp. Appl. Math. 346, 150–161 (2019) MathSciNetCrossRefGoogle Scholar
  48. Wazwaz, A.: A new algorithm for solving differential equations of Lane–Emden type. Appl. Math. Comput. 118, 287–310 (2001) MathSciNetzbMATHGoogle Scholar
  49. Wazwaz, A.: Solving the non-isothermal reaction–diffusion model equations in a spherical catalyst by the variational iteration method. Chem. Phys. Lett. 679, 132–136 (2017) ADSCrossRefGoogle Scholar
  50. Yousefi, S.A.: Legendre wavelets method for solving differential equations of Lane–Emden type. Appl. Math. Comput. 181, 1417–1422 (2006) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematical Engineering, Faculty of Arts and SciencesIstanbul Technical UniversityIstanbulTurkey

Personalised recommendations