Skip to main content
SpringerLink
Log in

Spectral method for solving high order nonlinear boundary value problems via operational matrices

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A spectral method based on operational matrices of Bernstein polynomials using collocation method is elaborated and employed for solving nonlinear ordinary and partial differential equations with multi-point boundary conditions. First, properties of Bernstein polynomial, operational matrices of integration, differentiation and product are introduced and then utilized to reduce the given differential equation to the solution of a system of algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent integro-differential equation which implies all boundary condition. Approximate solution is achieved by expanding the desired function in terms of a Bernstein basis and employing operational matrices. Unknown coefficients are determined by collocation. The method is compared with modified Adomian decomposition method, Birkhoff-type interpolation method, reproducing kernel Hilbert space method, fixed point method, finite-difference Keller-box method, multilevel augmentation method and shooting method. Illustrative examples are included to demonstrate the high precision, validity and good performance of the new scheme even for solving nonlinear singular differential equations.

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. Abbasbandy, S.: Numerical study on gas flow through a micro-nano porous media. Acta Phys. Pol. A 121, 581–585 (2012)

    Article  Google Scholar 

  2. Agarwal, R.P.: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986)

    Book  MATH  Google Scholar 

  3. Agarwal, R.P., O’Regan, D.: Infinite interval problems modeling the flow of a gas through a semi-infinite porous medium. Stud. Appl. Math. 108, 245–257 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carnicer, J.M., Peña, J.M.: Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math. 1, 173–196 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, J.: Fast multilevel augmentation methods for nonlinear boundary value problems. Comput. Math. Appl. 61, 612–619 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciarlet, P.G., Schultz, M.H., Varga, R.S.: Numerical methods of high-order accuracy for nonlinear boundary value problems I. One dimensional problems. Numer. Math. 9, 394–430 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dehghan, M., Aryanmehr, S., Eslahchi, 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, 125–139 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dehghan, M., Tatari, M.: The use of Adomian decomposition method for solving problems in calculus of variations. Math. Probl. Eng. 2006, 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dehghan, M., Tatari, M.: Finding approximate solutions for a class of third-order non-linear boundary value problems via the decomposition method of Adomian. Int. J. Comput. Math. 87, 1256–1263 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duan, J.S., Rach, R.: A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations. Appl. Math. Comput. 218, 4090–4118 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duan, J.S., Rach, R., Wazwaz, A.M.: Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems. Int. J. Nonlinear Mech. 49, 159–169 (2013)

    Article  Google Scholar 

  12. Farouki, R.T.: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geom. D. 29, 379–419 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gámez, D., Garralda, A.I., Guillem, M., Ruiz Galán, M.: High-order nonlinear initial-value problems countably determined. J. Comput. Appl. Math. 228, 77–82 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geng, F.: A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. Appl. Math. Comput. 213, 163–169 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Geng, F., Cui, M., Zhang, B.: Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods. Nonlinear Anal. Real 11, 637–644 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grossinho, M.D.R., Minhós, F.M., Santos, A.I.: Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control. J. Math. Anal. Appl. 309, 271–283 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kidder, R.F.: Unsteady flow of gas through a semi-infinite porous medium. J. Appl. Mech. 27, 329–332 (1957)

    MathSciNet  MATH  Google Scholar 

  18. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)

  19. Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic Press, New York (1979)

    MATH  Google Scholar 

  20. Pelesko, J.A., Bernstein, D.H.: Modeling MEMS and NEMS. Chapman and Hall, Boca Raton (2003)

    MATH  Google Scholar 

  21. Scott, M.R., Vandevender, W.H.: A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems. Appl. Math. Comput. 1, 187–218 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tatari, M., Dehghan, M.: The use of the Adomian decomposition method for solving multipoint boundary value problems. Phys. Scripta 73, 672–676 (2006)

    Article  MATH  Google Scholar 

  23. Wazwaz, A.M.: The numerical solution of special fourth-order boundary value problems by the modified decomposition method. Int. J. Comput. Math. 79, 345–356 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wazwaz, A.M.: The modified decomposition method applied to unsteady flow of gas through a porous medium. Appl. Math. Comput. 118, 123–132 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, Y.T., Chang, C.C., Chen, C.K.: A double decomposition method for solving the annular hyperbolic profile fins with variable thermal conductivity. Heat Transf. Eng. 31, 1165–1172 (2010)

    Article  Google Scholar 

  26. Yang, Y.T., Chien, S.K., Chen, C.K.: A double decomposition method for solving the periodic base temperature in convective longitudinal fins. Energy Convers. Manag. 49, 2910–2916 (2008)

    Article  Google Scholar 

  27. Yousefi, S.A., Behroozifar, M.: Operational matrices of Bernstein polynomials and their applications. Inter. J. Syst. Sci. 41, 709–716 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yousefi, S.A., Behroozifar, M., Dehghan, M.: The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass. J. Comput. Appl. Math. 235, 5272–5283 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yousefi, S.A., Behroozifar, M., Dehghan, M.: Numerical solution of the nonlinear age structured population models by using the operational matrices of Bernstein polynomials. Appl. Math. Model. 36, 945–963 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to sincerely thank professor Mehdi Dehghan (Amirkabir University of Technology, Tehran, Iran) for his helpful discussions and significant suggestions which helped the author to improve the contents of this article. Also, author is very grateful to one of the referees for carefully reading this paper and for the comments which have enhanced the quality of paper.

Author information

Authors and Affiliations

  1. Faculty of Basic Sciences, Babol University of Technology, 47148-71167, Babol, Mazandaran, Iran

    Mahmoud Behroozifar

Authors
  1. Mahmoud Behroozifar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahmoud Behroozifar.

Additional information

Communicated by Jan Hesthaven.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Behroozifar, M. Spectral method for solving high order nonlinear boundary value problems via operational matrices. Bit Numer Math 55, 901–925 (2015). https://doi.org/10.1007/s10543-015-0544-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-015-0544-2

Keywords

Mathematics Subject Classification

Search

Navigation