Advertisement

Fractal Quintic Spline Solutions for Fourth-Order Boundary-Value Problems

  • N. BalasubramaniEmail author
  • M. Guru Prem Prasad
  • S. Natesan
Original Paper
  • 44 Downloads

Abstract

A new numerical method is developed for approximating the solution of the fourth-order boundary-value problems with the help fractal quintic spline. The proposed method has second-order convergence. Numerical examples are experimented for the numerical illustration of the proposed method. It is shown that the method developed in this paper is more efficient than the method developed by the quintic spline.

Keywords

Fractal quintic spline Boundary-value problem Truncation error 

Mathematics Subject Classification

28A80 65D07 34B05 

Notes

Acknowledgements

The first author is grateful to the Ministry of Human Resource Development, India for providing the research fellowship and Indian Institute of Technology Guwahati, India for the support provided during the period of this work. The authors wish to acknowledge the referees for their valuable comments.

References

  1. 1.
    Al-Said, E.A., Noor, M.A.: Quartic spline method for solving fourth-order obstacle boundary-value problems. J. Comput. Appl. Math. 143(1), 107–116 (2002)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Al-Said, E.A., Noor, M.A., Rassias, T.M.: Cubic splines method for solving fourth-order obstacle problems. Appl. Math. Comput. 174(1), 180–187 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2(1), 303–329 (1986)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. J. Approx. Theory 57(1), 14–34 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chand, A.K.B., Kapoor, G.P.: Generalized cubic spline fractal interpolation functions. SIAM J. Numer. Anal. 44(2), 655–676 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chand, A.K.B., Viswanathan, P.: A constructive approach to cubic hermite fractal interpolation function and its constrained aspects. BIT 53(4), 841–865 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Loghmani, G.B., Alavizadeh, S.R.: Numerical solution of fourth-order problems with separated boundary conditions. Appl. Math. Comput. 191(2), 571–581 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ramadan, M.A., Lashien, I.F., Zahra, W.K.: Quintic nonpolynomial spline solutions for fourth-order two-point boundary-value problem. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1105–1114 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ramadan, M.A., Lashien, I.F., Zahra, W.K.: High order accuracy nonpolynomial spline solutions for 2\(\mu \)th order two-point boundary-value problems. Appl. Math. Comput. 204(2), 920–927 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rashidinia, J., Golbabaee, A.: Convergence of numerical solution of a fourth-order boundary-value problem. Appl. Math. Comput. 171(2), 1296–1305 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Rashidinia, J., Jalilian, R.: Non-polynomial spline for solution of boundary-value problems in plate deflection theory. Int. J. Comput. Math. 84(10), 1483–1494 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Siddiqi, S.S., Akram, G.: Solution of the system of fourth-order boundary-value problems using non-polynomial spline technique. Appl. Math. Comput. 185(1), 128–135 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Siddiqi, S.S., Akram, G.: Quintic spline solutions of fourth-order boundary-value problems. Int. J. Numer. Anal. Model. 5(1), 101–111 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Siraj-ul-Islam, Tirmizi, I.A., Ashraf, S.: A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications. Appl. Math. Comput. 174(2), 1169–1180 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Usmani, R.A.: Smooth spline approximations for the solution of a boundary-value problem with engineering applications. J. Comput. Appl. Math. 6(2), 93–98 (1980)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Usmani, R.A.: The use of quartic splines in the numerical solution of a fourth-order boundary-value problem. J. Comput. Appl. Math. 44(2), 187–200 (1992)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Usmani, R.A., Warsi, S.A.: Smooth spline solutions for boundary-value problems in plate deflection theory. Comput. Math. Appl. 6(2), 205–211 (1980)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Van, D.M., Berghe, G.V., De Meyer, H.: A smooth approximation for the solution of a fourth-order boundary-value problem based on nonpolynomial splines. J. Comput. Appl. Math. 51(3), 383–394 (1994)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology GuwahatiGuwahatiIndia

Personalised recommendations