Skip to main content
SpringerLink
Log in

Numerical solution of singular two-point boundary value problems by the collocation method with the Chebyshev bases

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

This paper presents a study of the performance of the collocation method using Chebychev basis functions for solving linear and nonlinear singular boundary value problems. The equations are reduced to systems of linear and nonlinear algebraic equations. The numerical experiments show that the proposed method produces high accurate results.

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

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Akyz, A., Sezer, M.: A Chebyshev collocation method for the solution of linear integro-differential equations. Int. J. Comput. Math. 72, 491–507 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akyuz-Dascio\(\breve{g}\)lu A, Cerdik-Yaslan H (2011) The solution of high-order nonlinear ordinary differential equations by Chebyshev Series. Appl Math Comput 217:5658–5666

  4. Al-Khaled, K.: Theory and computation in singular boundary value problems. Chaos, Solitons Fractals 33, 678–684 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anderson, N., Arthurs, A.: Complementary variational principles for diffusion problems with Michaelis-Menten kinetics. Bull. Math. Biol. 42, 131–135 (1980)

    Article  MATH  Google Scholar 

  6. Bataineh, A., Noorani, M., Hashim, I.: Approximate solutions of singular two-point BVPs by modified homotopy analysis method. Phys. Lett. A 372, 4062–4066 (2008)

    Article  MATH  Google Scholar 

  7. Boyd, J.: Chebyshev and Fourier spectral methods. Dover Publications, New York (2000)

    Google Scholar 

  8. Caglar, N., Caglar, H.: B-spline solution of singular boundary value problems. Appl. Math. Comput. 182(2), 1509–1513 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Caglar, H., Caglar, S., Twizell, E.: The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions. Appl. Math. Lett. 12, 25–30 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caglar, H., Caglar, N., Ozer, M.: B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Solitons Fractals 39, 1232–1237 (2009)

    Article  MATH  Google Scholar 

  11. Cen, Z.: Numerical study for a class of singular two-point boundary value problems using Greens functions. Appl. Math. Comput. 183, 10–16 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Chun, C., Ebaid, A., Lee, M., Aly, E.: An approach for solving singular two-point boundary value problems: analytical and numerical treatment. ANZIAM J. 53E, 21–43 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Clenshaw, C., Norton, H.: The solution of nonlinear ordinary differential equations in Chebyshev series. Comput. J. 6, 88–92 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  14. De Hoog, F., Weiss, R.: Collocation methods for singular boundary value problems. SIAM J. Numer. Anal. 15, 198–217 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ebaid, A.: A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method. J. Comput Appl. Math. 235, 1914–1924 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. El-Sayed, S.: Integral methods for computing solutions of a class of singular two-point boundary value problems. Appl. Math. Comput. 130, 235–241 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Ebaid, A.: Exact solutions for a class of nonlinear singular two-point boundary value problems: The decomposition method. Zeitschrift fr Naturforschung A 65, 145–150 (2010)

    Google Scholar 

  18. El-Gamel, M., Sameeh, M.: An efficient technique for finding the eigenvalues of fourth-order Sturm-Liouville problems. Appl. Math. 3, 920–925 (2012)

    Article  Google Scholar 

  19. El-Gamel, M., Sameeh, M.: Generalized Sturm-Liouville problems and Chebychev collocation method. British J. Math. Computer Sci. 4, 1124–1133 (2014)

    Article  Google Scholar 

  20. El-Gamel, M.: Chebychev polynomial solutions of twelfth-order boundary-value problems. British J. Math. Comp. Sci. 6, 13–23 (2015)

    Article  Google Scholar 

  21. El-Gamel, M.: Numerical comparison of sinc-collocation and Chebychev-collocation methods for determining the eigenvalues of Sturm-Liouville problems with parameter-dependent boundary conditions. SeMA 66, 29–42 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Flesch, U.: The distribution of heat sources in the human head: a theoretical consideration. J. Theor. Biol. 54, 285–287 (1975)

    Article  Google Scholar 

  23. Fox, L., Parker, I.: Chebyshev polynomials in numerical analysis. Oxford University Press, London (1968)

    MATH  Google Scholar 

  24. Garner, J., Shivaji, R.: Diffusion problems with mixed nonlinear boundary condition. J. Math. Anal. Appl. 148, 422–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jamet, P.: On the convergence of finite difference approximations to one-dimensional singular boundary- value problems. Numer. Math. 14, 355–378 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lima, P., Carpentier, M.: Iterative methods for a singular boundary-value problem. J. Comput. Appl. Math. 111, 173–186 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kadalbajoo, M., Aggarwal, V.: Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline. Appl. Math. Comput. 160, 851–863 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Kanth, A., Reddy, Y.: Cubic spline for a class of singular two-point boundary value problems. Appl. Math. Comput. 170(2), 733–740 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Kanth, A., Bhattacharya, V.: Cubic spline for a class of non-linear singular boundary value problems arising in physiology. Appl. Math. Comput. 174(1), 768–774 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Kanth, A., Aruna, K.: Solution of singular two-point boundary value problems using differential transformation method. Phys. Lett. A 372, 4671–4673 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kanth, A., Reddy, Y.: Higher order finite difference method for a class of singular boundary value problems. Appl. Math. Comput. 155, 249–258 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Kanth, A., Aruna, K.: He’s variational iteration method for treating nonlinear singular boundary value problems. Comp. Math. Appl. 60, 821–829 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Mason, J., Handscomb, D.: Chebyshev polynomials. CRC Press, Boca Raton (2003)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  36. Norton, H.: The iterative solution of non-linear ordinary differential equations in Chebyshev series. Comput. J. 7, 76–85 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  37. Narierer, F.: A generalized spline method for singular boundary valute problems in ordinary differential equations. Linear Algebra Appl. 7, 189–216 (1973)

    Article  MathSciNet  Google Scholar 

  38. Onumanyi, P., Ortiz, E.: Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the tau method. Math. Comp. 43, 189–203 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Parter S, Stein M, Stein P (1973) On the multiplicity of solutions of a differential equation arising in chemical reactor theory. Technical report 194, Computer Sciences. University of Wisconsin, Madison

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

    MathSciNet  MATH  Google Scholar 

  41. Rivlin, T.: An introduction to the approximation of functions. Dover Publications Inc, New York (1969)

    MATH  Google Scholar 

  42. Russel, R., Shampine, L.: Numerical methods for singular boundary value problems. SIAM J. Numer. Anal. 12, 13–36 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  43. Russell, R., Shampine, L.: Numerical methods for singular boundary value problems. Numer. Math. 12, 13–35 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sezer, M., Kaynak, M.: Chebyshev polynomial solutions of linear differential equations. Int. Math. Educ. Sci. Technol. 27, 607–618 (1996)

    Article  MATH  Google Scholar 

  45. Wang, W., Cui, M., Han, B.: A new method for solving a class of singular two-point boundary value problems. J. Math. Anal. Appl. 206, 721–727 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Weinmuller, E.: On the boundary value problem for systems of ordinary second order differential equations with a singularity of the first kind. SIAM J. Math. Anal. 15, 287–307 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  47. Yildirim, A., Agirseven, D.: The homotopy perturbation method for solving singular initial value problems. Int. J. Nonlinear Sci. Numer. Simul. 10(2), 235–238 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

The author is extremely grateful to Dr. Amgad Abdrabou for programming the examples and to the referees for their helpful suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Mansoura, Egypt

    Mohamed El-Gamel & Mona Sameeh

Authors
  1. Mohamed El-Gamel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mona Sameeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed El-Gamel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Gamel, M., Sameeh, M. Numerical solution of singular two-point boundary value problems by the collocation method with the Chebyshev bases . SeMA 74, 627–641 (2017). https://doi.org/10.1007/s40324-016-0107-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-016-0107-8

Keywords

Mathematics Subject Classification

Search

Navigation