An efficient numerical technique for Lane–Emden–Fowler boundary value problems: Bernstein collocation method

Abstract

In this paper, we propose an efficient numerical technique for numerical solutions of the equivalent integral form of Emden–Fowler type boundary value problems (BVPs), which model many phenomena in mathematical physics and astrophysics. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. The iterative method is applied to solve the system numerically. The error analysis of the proposed method is provided. Several examples are provided to demonstrate the accuracy, applicability, and efficiency of the present method.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    S. Chandrasekhar, An introduction to the study of stellar structure. Ciel et Terre 55, 412 (1939)

    ADS  MATH  Google Scholar 

  2. 2.

    D. McElwain, A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. J. Theor. Biol. 71, 255–263 (1978)

    Article  Google Scholar 

  3. 3.

    B. Gray, The distribution of heat sources in the human head—theoretical considerations. J. Theor. Biol. 82(3), 473–476 (1980)

    Article  Google Scholar 

  4. 4.

    I. Rachnková, O. Koch, G. Pulverer, E. Weinmuller, On a singular boundary value problem arising in the theory of shallow membrane caps. J. Math. Anal. Appl. 332(1), 523–541 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    P. Chambre, On the solution of the Poisson–Boltzmann equation with application to the theory of thermal explosions. J. Chem. Phys. 20, 1795 (1952)

    ADS  Article  Google Scholar 

  6. 6.

    R. Russell, L. Shampine, M. Chawla, C. Katti, Numerical methods for singular boundary value problems. SIAM J. Numer. Anal. 12(1), 13–36 (1975)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    M. Chawla, C. Katti, Finite difference methods and their convergence for a class of singular two point boundary value problems. Numer. Math. 39(3), 341–350 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    M. Chawla, S. McKee, G. Shaw, Order \(h^2\) method for a singular two-point boundary value problem. BIT Numer. Math. 26(3), 318–326 (1986)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

    S. Iyengar, P. Jain, Spline finite difference methods for singular two point boundary value problems. Numer. Math. 50(3), 363–376 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    M. Sakai, R.A. Usmani, Non polynomial splines and weakly singular two-point boundary value problems. BIT Numer. Math. 28(4), 867–876 (1988)

    MATH  Article  Google Scholar 

  11. 11.

    M. Kumar, A three-point finite difference method for a class of singular two-point boundary value problems. J. Comput. Appl. Math. 145(1), 89–97 (2002)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    J. Rashidinia, Z. Mahmoodi, M. Ghasemi, Parametric spline method for a class of singular two-point boundary value problems. Appl. Math. Comput. 188(1), 58–63 (2007)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    A.R. Kanth, Cubic spline polynomial for non-linear singular two-point boundary value problems. Appl. Math. Comput. 189(2), 2017–2022 (2007)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    M. Ben-Romdhane, H. Temimi, An iterative numerical method for solving the Lane-Emden initial and boundary value problems. Int. J. Comput. Methods 15(04), 1850020 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    J. Niu, M. Xu, Y. Lin, Q. Xue, Numerical solution of nonlinear singular boundary value problems. J. Comput. Appl. Math. 331, 42–51 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    A.K. Verma, S. Kayenat, On the convergence of Mickens’ type nonstandard finite difference schemes on Lane–Emden type equations. J. Math. Chem. 56(6), 1667–1706 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    A.M. Wazwaz, The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models. Commun. Nonlinear Sci. Numer. Simul. 16(10), 3881–3886 (2011)

    ADS  MATH  Article  Google Scholar 

  18. 18.

    R. Singh, N. Das, J. Kumar, The optimal modified variational iteration method for the Lane–Emden equations with Neumann and Robin boundary conditions. Eur. Phys. J. Plus 132(6), 251 (2017)

    Article  Google Scholar 

  19. 19.

    R. Singh, J. Kumar, G. Nelakanti, Numerical solution of singular boundary value problems using Green’s function and improved decomposition method. J. Appl. Math. Comput. 43(1–2), 409–425 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    R. Singh, J. Kumar, Solving a class of singular two-point boundary value problems using new modified decomposition method, ISRN Computational Mathematics 2013

  21. 21.

    R. Singh, J. Kumar, G. Nelakanti, Approximate series solution of singular boundary value problems with derivative dependence using Green’s function technique. Comput. Appl. Math. 33(2), 451–467 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    R. Singh, J. Kumar, An efficient numerical technique for the solution of nonlinear singular boundary value problems. Comput. Phys. Commun. 185(4), 1282–1289 (2014)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    R. Singh, Optimal homotopy analysis method for the non-isothermal reaction-diffusion model equations in a spherical catalyst. J. Math. Chem. 56(9), 2579–2590 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    R. Singh, A modified homotopy perturbation method for nonlinear singular Lane–Emden equations arising in various physical models. Int. J. Appl. Comput. Math. 5(3), 64 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    S.A. Khuri, A. Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physiology. Math. Comput. Modell. 52(3–4), 626–636 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    H. Kaur, R. Mittal, V. Mishra, Haar wavelet approximate solutions for the generalized Lane-Emden equations arising in astrophysics. Comput. Phys. Commun. 184(9), 2169–2177 (2013)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    A.K. Verma, D. Tiwari, Higher resolution methods based on quasilinearization and Haar wavelets on Lane–Emden equations. Int. J. Wavelets Multiresol. Inf. Process. 17(03), 1950005 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    F. Zhou, X. Xu, Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets. Adv. Differ. Equ. 2016(1), 17 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    R. Singh, H. Garg, V. Guleria, Haar wavelet collocation method for Lane–Emden equations with Dirichlet, Neumann and Neumann–Robin boundary conditions. J. Comput. Appl. Math. 346, 150–161 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    R. Singh, J. Shahni, H. Garg, A. Garg, Haar wavelet collocation approach for Lane–Emden equations arising in mathematical physics and astrophysics. Eur. Phys. J. Plus 134(11), 548 (2019)

    Article  Google Scholar 

  31. 31.

    L. Bobisud, Existence of solutions for nonlinear singular boundary value problems. Appl. Anal. 35(1–4), 43–57 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    R.K. Pandey, A.K. Verma, Existence-uniqueness results for a class of singular boundary value problems-II. J. Math. Anal. Appl. 338, 1387–1396 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    R.K. Pandey, A.K. Verma, Existence-uniqueness results for a class of singular boundary value problems arising in physiology. Nonlinear Anal.: Real World Appl. 9(1), 40–52 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    R. Singh, J. Kumar, G. Nelakanti, New approach for solving a class of doubly singular two-point boundary value problems using Adomian decomposition method. Adv. Numer. Anal. 2012, 541083 (2012). https://doi.org/10.1155/2012/541083

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    L.H. Thomas, The calculation of atomic fields. Proc. Camb. Philos. Soc. 23, 542 (1927)

    ADS  MATH  Article  Google Scholar 

  36. 36.

    E. Fermi, Un metodo statistico per la determinazione di alcune priorieta dell’atome. Rend. Accad. Naz. Lincei 6(602–607), 32 (1927)

    Google Scholar 

  37. 37.

    C. Chan, Y. Hon, A constructive solution for a generalized Thomas–Fermi theory of ionized atoms. Q. Appl. Math. 45(3), 591–599 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    M. Desaix, D. Anderson, M. Lisak, Variational approach to the Thomas–Fermi equation. Eur. J. Phys. 25(6), 699 (2004)

    Article  Google Scholar 

  39. 39.

    V. Marinca, N. Herişanu, An optimal iteration method with application to the Thomas–Fermi equation. Open Phys. 9(3), 891–895 (2011)

    ADS  Article  Google Scholar 

  40. 40.

    K. Parand, H. Yousefi, M. Delkhosh, A. Ghaderi, A novel numerical technique to obtain an accurate solution to the Thomas–Fermi equation. Eur. Phys. J. Plus 131(7), 228 (2016)

    Article  Google Scholar 

  41. 41.

    K. Parand, P. Mazaheri, H. Yousefi, M. Delkhosh, Fractional order of rational Jacobi functions for solving the non-linear singular Thomas–Fermi equation. Eur. Phys. J. Plus 132(2), 77 (2017)

    Article  Google Scholar 

  42. 42.

    R. Pandey, On the convergence of a finite difference method for a class of singular two point boundary value problems. Int. J. Comput. Math. 42(3–4), 237–241 (1992)

    MATH  Article  Google Scholar 

  43. 43.

    R. Pandey, A finite difference method for a class of singular two point boundary value problems arising in physiology. Int. J. Comput. Math. 65(1–2), 131–140 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    M. El-Gebeily, I. Abu-Zaid, On a finite difference method for singular two-point boundary value problems. IMA J. Numer. Anal. 18(2), 179–190 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    S. Ha, C. Lee, Numerical study for two-point boundary value problems using Green’s functions. Comput. Math. Appl. 44(12), 1599–1608 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    R. Pandey, A.K. Singh, On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology. J. Comput. Appl. Math. 166(2), 553–564 (2004)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    R. Singh, A.-M. Wazwaz, J. Kumar, An efficient semi-numerical technique for solving nonlinear singular boundary value problems arising in various physical models. Int. J. Comput. Math. 93(8), 1330–1346 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    P. Roul, A new efficient recursive technique for solving singular boundary value problems arising in various physical models. Eur. Phys. J. Plus 131(4), 105 (2016)

    Article  Google Scholar 

  49. 49.

    P. Roul, An improved iterative technique for solving nonlinear doubly singular two-point boundary value problems. Eur. Phys. J. Plus 131(6), 209 (2016)

    MathSciNet  Article  Google Scholar 

  50. 50.

    D.D. Bhatta, M.I. Bhatti, Numerical solution of Kdv equation using modified Bernstein polynomials. Appl. Math. Comput. 174(2), 1255–1268 (2006)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    M.I. Bhatti, P. Bracken, Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 205(1), 272–280 (2007)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    B.N. Mandal, S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials. Appl. Math. Comput. 190(2), 1707–1716 (2007)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    K. Maleknejad, E. Hashemizadeh, R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation. Commun. Nonlinear Sci. Numer. Simul. 16(2), 647–655 (2011)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    H. Ahmed, Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis. J. Egypt. Math. Soc. 22(2), 227–237 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    P. Pirabaharan, R.D. Chandrakumar, A computational method for solving a class of singular boundary value problems arising in science and engineering. Egypt. J. Basic Appl. Sci. 3(4), 383–391 (2016)

    Article  Google Scholar 

  56. 56.

    E. Hosseini, G. Loghmani, M. Heydari, M. Rashidi, Numerical investigation of velocity slip and temperature jump effects on unsteady flow over a stretching permeable surface. Eur. Phys. J. Plus 132(2), 96 (2017)

    Article  Google Scholar 

  57. 57.

    E. Hosseini, G. Loghmani, M. Heydari, M. Rashidi, Investigation of magneto-hemodynamic flow in a semi-porous channel using orthonormal Bernstein polynomials. Eur. Phys. J. Plus 132(7), 326 (2017)

    Article  Google Scholar 

  58. 58.

    E. Hesameddini, M. Shahbazi, Solving system of Volterra–Fredholm integral equations with Bernstein polynomials and hybrid Bernstein block-pulse functions. J. Comput. Appl. Math. 315, 182–194 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    A. Babaaghaie, K. Maleknejad, A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis. J. Comput. Appl. Math. 344, 482–494 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    S.S. Sastry, Introductory Methods of Numerical Analysis (PHI Learning Pvt. Ltd., New Delhi, 2005)

    Google Scholar 

  61. 61.

    M.J.D. Powell, Approximation Theory and Methods (Cambridge University Press, Cambridge, 1981)

    Google Scholar 

  62. 62.

    G. Lorentz, R. DeVore, Constructive Approximation, Polynomials and Splines Approximation (Springer, Berlin, 1993)

    Google Scholar 

  63. 63.

    R. Duggan, A. Goodman, Pointwise bounds for a nonlinear heat conduction model of the human head. Bull. Math. Biol. 48(2), 229–236 (1986)

    MATH  Article  Google Scholar 

  64. 64.

    R. Singh, Analytic solution of singular Emden-Fowler-type equations by Green’s function and homotopy analysis method. Eur. Phys. J. Plus 134(11), 583 (2019)

    Article  Google Scholar 

Download references

Acknowledgements

One of the author, Julee Shahni, would like to acknowledge the financial assistance provided by Department of Science and Technology (DST) under the scheme of INSPIRE Fellowship, New Delhi, India.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Randhir Singh.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shahni, J., Singh, R. An efficient numerical technique for Lane–Emden–Fowler boundary value problems: Bernstein collocation method. Eur. Phys. J. Plus 135, 475 (2020). https://doi.org/10.1140/epjp/s13360-020-00489-3

Download citation