Skip to main content
SpringerLink
Log in

Laguerre wavelet method for solving Thomas–Fermi type equations

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

An efficient numerical algorithm based on the Laguerre wavelets collocation technique for numerical solutions of a class of Thomas–Fermi boundary value problems, which arises in the study the charge densities and potential in many scientific models like in atoms, molecules, metals and crystals is presented. This technique includes the theory of Laguerre wavelets and the collocation technique. To avoid singular behavior at \(x=0\), we consider the equivalent integral form of the generalized Thomas–Fermi equation. We apply the Laguerre wavelets collocation technique to get a nonlinear system of equations having unknown Laguerre wavelet coefficients. The Newton–Raphson method is applied to analyze this nonlinear system of equations. We provide the error analysis of the present method. Several numerical problems are provided to test the effectiveness of the proposed method. The obtained results are compared by the exact solution and the results obtained by the other known techniques. The numerical results reveal that the current analysis provides a better approximation than existing methods.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Thomas LH (1927) The calculation of atomic fields. Proc Cambr Philos Soc 23:542

    MATH  Google Scholar 

  2. Fermi E (1927) Un metodo statistico per la determinazione di alcune priorieta dell’atome. Rendiconti Accademia Nazionale Lincei 32(6):602–607

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Banerjee B, Constantinescu D, Rehak P (1974) Thomas–Fermi and Thomas–Fermi–Dirac calculations for atoms in a very strong magnetic field. Phys Rev D 10(8):2384

    Google Scholar 

  6. Coulson CA, March NH (1950) Momenta in atoms using the Thomas–Fermi method. Proc Phys Soc Lond Sect A 63(4):367

    MATH  Google Scholar 

  7. March N (1952) Thomas–Fermi fields for molecules with tetrahedral and octahedral symmetry. Math Proc Cambr Philos Soc 48(4):665–682

    MATH  Google Scholar 

  8. March N, Tomishima Y (1979) Behavior of positive ions in extremely strong magnetic fields. Phys Rev D 19(2):449

    Google Scholar 

  9. Umeda K, Tomishima Y (1955) On the influence of the packing on the atomic scattering factor based on the Thomas–Fermi theory. J Phys Soc Jpn 10(9):753–758

    Google Scholar 

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

    MATH  Google Scholar 

  11. McElwain D (1978) A re-examination of oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics. J Theor Biol 71(2):255–263

    Google Scholar 

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

    Google Scholar 

  13. Adomian G (1998) Solution of the Thomas–Fermi equation. Appl Math Lett 11(3):131–133

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Zaitsev N, Matyushkin I, Shamonov D (2004) Numerical solution of the Thomas–Fermi equation for the centrally symmetric atom. Russ Microlectron 33(5):303–309

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  21. Singh R, Kumar J, Nelakanti G (2012) New approach for solving a class of doubly singular two-point boundary value problems using Adomian decomposition method. In: Advances in numerical analysis

  22. Taghavi A, Pearce S (2013) A solution to the Lane–Emden equation in the theory of stellar structure utilizing the Tau method. Math Methods Appl Sci 36(10):1240–1247

    MathSciNet  MATH  Google Scholar 

  23. Lakestani M, Dehghan M (2013) 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 36(16):2243–2253

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Mohammadzadeh R, Lakestani M, Dehghan M (2014) Collocation method for the numerical solutions of Lane–Emden type equations using cubic Hermite spline functions. Math Methods Appl Sci 37(9):1303–1717

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. Singh R, Kumar J (2014) The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems. J Appl Math Comput 44(1–2):397–416

    MathSciNet  MATH  Google Scholar 

  29. Singh R, Wazwaz A-M, Kumar J (2016) 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

    MathSciNet  MATH  Google Scholar 

  30. Raja MAZ, Zameer A, Khan AU, Wazwaz AM (2016) A new numerical approach to solve Thomas–Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming. Springer Plus 5(1):1400

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  34. Rosu HC, Mancas SC (2017) Generalized Thomas–Fermi equations as the Lampariello class of Emden–Fowler equations. Phys A 471:212–218

    MathSciNet  MATH  Google Scholar 

  35. Turkyilmazoglu M (2017) Solution of initial and boundary value problems by an effective accurate method. Int J Comput Methods 14(06):1750069

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  38. Singh R (2018) Analytical approach for computation of exact and analytic approximate solutions to the system of Lane–Emden–Fowler type equations arising in astrophysics. Eur Phys J Plus 133(8):320

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  44. Raja MAZ, Mehmood J, Sabir Z, Nasab AK, Manzar MA (2019) Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput Appl 31(3):793–812

    Google Scholar 

  45. Verma AK, Tiwari D (2019) Higher resolution methods based on quasilinearization and Haar wavelets on Lane-Emden equations. Int J Wavelets Multiresolut Inf Process 17(03):1950005

    MathSciNet  MATH  Google Scholar 

  46. Singh R, Guleria V, Singh M (2020) Haar wavelet quasilinearization method for numerical solution of Emden–Fowler type equations. Math Comput Simul 174:123–133

    MathSciNet  MATH  Google Scholar 

  47. Chapwanya M, Dozva R, Muchatibaya G (2019) A nonstandard finite difference technique for singular Lane–Emden type equations. Eng Comput 36(5):1566–1578

    Google Scholar 

  48. Umesh KM (2020) Numerical solution of singular boundary value problems using advanced Adomian decomposition method. Eng Comput. https://doi.org/10.1007/s00366-020-00972-6

    Article  Google Scholar 

  49. Shahni J, Singh R (2020) An efficient numerical technique for Lane–Emden–Fowler boundary value problems: Bernstein collocation method. Eur Phys J Plus 135(06):1–21

    Google Scholar 

  50. Shahni J, Singh R (2020) Numerical results of Emden–Fowler boundary value problems with derivative dependence using the Bernstein collocation method. Eng Comput. https://doi.org/10.1007/s00366-020-01155-z

    Article  Google Scholar 

  51. Iqbal MA, Saeed U, Mohyud-Din ST (2015) Modified Laguerre wavelets method for delay differential equations of fractional-order. Egypt J Basic Appl Sci 2(1):50–54

    Google Scholar 

  52. Satyanarayan B, Kumar YP, Abdulelah A (2017) Laguerre wavelet and its programming. Int J Math Trends Technol 49(2):129–137

    Google Scholar 

  53. Shiralashetti S, Kumbinarasaiah S, Naregal S (2017) Laguerre wavelet based numerical method for the solution of differential equations with variable coefficients. Int J Eng Sci Math 6:40–48

    Google Scholar 

  54. Bavanari S, Abdulrahman AA (2018) Mathematical aspects of Laguerre wavelets transformation. Ann Pure Appl Math 16(1):53–61

    Google Scholar 

  55. Shiralashetti S, Angadi L, Kumbinarasaiah S (2018) Laguerre wavelet-Galerkin method for the numerical solution of one dimensional partial differential equations. Int J Math Appl 55(1):939–949

    Google Scholar 

  56. Shiralashetti S, Kumbinarasaiah S (2019) Laguerre wavelets collocation method for the numerical solution of the Benjamina–Bona–Mohany equations. J Taibah Univ Sci 13(1):9–15

    Google Scholar 

  57. Shiralashetti S, Kumbinarasaiah S (2020) Laguerre wavelets exact parseval frame-based numerical method for the solution of system of differential equations. Int J Appl Comput Math 6(4):1–16

    MathSciNet  MATH  Google Scholar 

  58. Grossmann A, Morlet J (1984) Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J Math Anal 15(4):723–736

    MathSciNet  MATH  Google Scholar 

  59. Singh R, Kumar J (2013) Solving a class of singular two-point boundary value problems using new modified decomposition method. ISRN Comput Math 2013:1–11

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  61. Lin S (1976) Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. J Theor Biol 60(2):449–457

    Google Scholar 

Download references

Acknowledgements

One of the authors, 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

Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology Mesra, Ranchi, 835215, India

    Julee Shahni & Randhir Singh

Authors
  1. Julee Shahni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Randhir Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Randhir Singh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shahni, J., Singh, R. Laguerre wavelet method for solving Thomas–Fermi type equations. Engineering with Computers 38, 2925–2935 (2022). https://doi.org/10.1007/s00366-021-01309-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01309-7

Keywords

Mathematics Subject Classification

Search

Navigation