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Haar wavelet collocation approach for Lane-Emden equations arising in mathematical physics and astrophysics

  • Randhir SinghEmail author
  • Julee Shahni
  • Himanshu Garg
  • Apoorv Garg
Regular Article
  • 12 Downloads

Abstract.

In this paper, we propose the Haar wavelet collocation method for the numerical solution of the Lane-Emden equation with boundary conditions. To overcome the singular behavior at the origin, we first transform the Lane-Emden equation into an integral equation. The Haar wavelet collocation method is utilized to reduce the integral equation into a system of algebraic equations in the unknown expansion coefficients, then Newton’s iterative process is employed for numerical solutions. We also discuss the convergence and error analysis of the technique. Six singular problems are presented to demonstrate the accuracy and applicability of the method, including; thermal explosion, oxygen-diffusion in a spherical cell, heat conduction through a solid with heat generation and shallow membrane caps problems. The numerical results are compared with existing numerical and exact solutions. The use of the Haar wavelet is found to be accurate, fast, flexible and convenient.

References

  1. 1.
    C.-H. Hsiao, W.-J. Wang, Math. Comput. Simul. 57, 347 (2001)CrossRefGoogle Scholar
  2. 2.
    C. Hsiao, Math. Comput. Simul. 64, 561 (2004)CrossRefGoogle Scholar
  3. 3.
    Ü. Lepik, Math. Comput. Simul. 68, 127 (2005)CrossRefGoogle Scholar
  4. 4.
    Ü. Lepik, Appl. Math. Comput. 185, 695 (2007)MathSciNetGoogle Scholar
  5. 5.
    I. Aziz, B. Sarler et al., Math. Comput. Modell. 52, 1577 (2010)CrossRefGoogle Scholar
  6. 6.
    I. Aziz, F. Khan et al., J. Comput. Appl. Math. 272, 70 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. ur Rehman, R.A. Khan, Appl. Math. Modell. 36, 894 (2012)CrossRefGoogle Scholar
  8. 8.
    I. Singh, S. Kumar, J. Comput. Appl. Math. 292, 541 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Babaaghaie, K. Maleknejad, J. Comput. Appl. Math. 317, 643 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Yüzbaşi, Math. Methods Appl. Sci. 34, 2218 (2011)MathSciNetGoogle Scholar
  11. 11.
    R. Singh, J. Math. Chem. 56, 2579 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Y.Y.E.H. Doha, Nonlinear Eng. 8, 318 (2019)ADSCrossRefGoogle Scholar
  13. 13.
    S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, New York, 1967)Google Scholar
  14. 14.
    A.-M. Wazwaz, Appl. Math. Comput. 118, 287 (2001)MathSciNetGoogle Scholar
  15. 15.
    R. Duggan, A. Goodman, Bull. Math. Biol. 48, 229 (1986)CrossRefGoogle Scholar
  16. 16.
    R. Russell, L. Shampine, SIAM J. Numer. Anal. 12, 13 (1975)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Jamet, Numer. Math. 14, 355 (1970)CrossRefGoogle Scholar
  18. 18.
    M. Chawla, C. Katti, Numer. Math. 39, 341 (1982)MathSciNetCrossRefGoogle Scholar
  19. 19.
    S. Iyengar, P. Jain, Numer. Math. 50, 363 (1986)CrossRefGoogle Scholar
  20. 20.
    M.K. Kadalbajoo, V. Kumar, Appl. Math. Comput. 188, 1856 (2007)MathSciNetGoogle Scholar
  21. 21.
    A.R. Kanth, Y. Reddy, Appl. Math. Comput. 170, 733 (2005)MathSciNetGoogle Scholar
  22. 22.
    A.R. Kanth, Appl. Math. Comput. 189, 2017 (2007)MathSciNetGoogle Scholar
  23. 23.
    S. Khuri, A. Sayfy, Math. Comput. Modell. 52, 626 (2010)CrossRefGoogle Scholar
  24. 24.
    A. Ebaid, J. Comput. Appl. Math. 235, 1914 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    R. Singh, J. Kumar, G. Nelakanti, J. Appl. Math. Comput. 43, 409 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    R. Singh, J. Kumar, Comput. Phys. Commun. 185, 1282 (2014)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    A.-M. Wazwaz, Int. J. Comput. Math. 88, 3406 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    R. Singh, N. Das, J. Kumar, Eur. Phys. J. Plus 132, 251 (2017)CrossRefGoogle Scholar
  29. 29.
    N. Das, R. Singh, A.-M. Wazwaz, J. Kumar, J. Math. Chem. 54, 527 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    M. Danish, S. Kumar, S. Kumar, Comput. Chem. Eng. 36, 57 (2012)CrossRefGoogle Scholar
  31. 31.
    R. Singh, H. Garg, V. Guleria, J. Comput. Appl. Math. 346, 150 (2019)MathSciNetCrossRefGoogle Scholar
  32. 32.
    E. Doha, W. Abd-Elhameed, Y. Youssri, New Astron. 23, 113 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    W. Abd-Elhameed, E. Doha, A. Saad, M. Bassuony, Rev. Mex. Astron. Astrofís. 52, 83 (2016)ADSGoogle Scholar
  34. 34.
    W. Abd-Elhameed, Y. Youssri, E. Doha, Math. Sci. 9, 93 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    W. Abd-Elhameed, Eur. Phys. J. Plus 130, 52 (2015)CrossRefGoogle Scholar
  36. 36.
    R. Singh, A.-M. Wazwaz, MATCH Commun. Math. Comput. Chem. 80, 369 (2018)ADSMathSciNetGoogle Scholar
  37. 37.
    R. Singh, Int. J. Appl. Comput. Math. 5, 64 (2019)CrossRefGoogle Scholar
  38. 38.
    Y.H. Youssri, Adv. Differ. Eq. 2017, 73 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    W. Abd-Elhameed, Y. Youssri, Nonlinear Dyn. 89, 1341 (2017)CrossRefGoogle Scholar
  40. 40.
    W. Abd-Elhameed, Y. Youssri, Comput. Appl. Math. 37, 2897 (2018)MathSciNetCrossRefGoogle Scholar
  41. 41.
    A. Atta, G. Moatimid, Y. Youssri, Int. J. Appl. Comput. Math. 5, 9 (2019)CrossRefGoogle Scholar
  42. 42.
    W. Abd-Elhameed, Y. Youssri, Int. J. Nonlinear Sci. Numer. Simul. 20, 191 (2019)MathSciNetCrossRefGoogle Scholar
  43. 43.
    E. Babolian, A. Shahsavaran, J. Comput. Appl. Math. 225, 87 (2009)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Y. Youssri, W. Abd-Elhameed, E. Doha, Rom. J. Phys. 60, 1298 (2015)Google Scholar
  45. 45.
    W.M. Abd-Elhameed, Y. Youssri, E.H. Doha, Comput. Methods Differ. Equ. 2, 171 (2014)MathSciNetGoogle Scholar
  46. 46.
    S. Iqbal, A. Javed, Appl. Math. Comput. 217, 7753 (2011)MathSciNetGoogle Scholar
  47. 47.
    R. Pandey, Int. J. Comput. Math. 65, 131 (1997)MathSciNetCrossRefGoogle Scholar
  48. 48.
    A. Ravi Kanth, V. Bhattacharya, Appl. Math. Comput. 174, 768 (2006)MathSciNetGoogle Scholar
  49. 49.
    A. Ravi Kanth, K. Aruna, Comput. Math. Appl. 60, 821 (2010)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsBirla Institute of TechnologyMesraIndia
  2. 2.Department of Computer Science EngineeringBirla Institute of TechnologyMesraIndia

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