SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2020 / Volume: 49 Issue: 6

Fractal quintic spline method for nonlinear boundary-value problems

N Balasubramani(National Institute of Technology Puducherry)0000-0001-8281-8997
M. Guru Prem Prasad(Indian Institute of Technology Guwahati)0000-0002-4484-231X
S Natesan(Indian Institute of Technology Guwahati)0000-0001-7527-1989
Pages
—
Publication Date
2020-12-08
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Numerical examples are provided to show practical usefulness of the method and numerical results are compared with the existing numerical methods.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this article, numerical solutions of nonlinear boundary-value problems are obtained using fractal quintic spline.
📊
Key Findings & Data:Convergence analysis of the proposed method is also established.
💡
Conclusion & Impact:Numerical examples are provided to show practical usefulness of the method and numerical results are compared with the existing numerical methods.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:fractal quintic spline · nonlinear boundary-value problem · quasilinearization

Abstract

In this article, numerical solutions of nonlinear boundary-value problems are obtained using fractal quintic spline. Convergence analysis of the proposed method is also established. Proposed method has fourth-order convergence. Numerical examples are provided to show practical usefulness of the method and numerical results are compared with the existing numerical methods.

🔖

Cite This Article

Balasubramani, N., Prasad, M. G. P., Natesan, S. (2020). Fractal quintic spline method for nonlinear boundary-value problems. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 6. https://doi.org/10.15672/hujms.452998

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (26)

  1. M. Baccouch, A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems, Numer. Algor. 79 (3), 697–718, 2018.
  2. M. Baccouch, An adaptive local discontinuous Galerkin method for nonlinear twopoint boundary-value problems, Numer. Algor. 2019, doi:10.1007/s11075-019-00794-8.
  3. M.F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2 (1), 303–329, 1986.
  4. M.F. Barnsley and A.N. Harrington, The calculus of fractal interpolation functions, J. Approx. Theory 57 (1), 14–34, 1989.
  5. R.E. Bellman and R.E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, American Elsevier, New York, 1965.
  6. R. Bhatia, L. Elsner, and G. Krause, Bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix, Linear Algebra Appl. 142, 195–209, 1990.
  7. S.K. Bhatta and K.S. Sastri, A sixth order spline procedure for a class of nonlinear boundary-value problems, Int. J. Comput. Math. 49 (3–4), 255–271, 1993.
  8. S.K. Bhatta and K.S. Sastri, Symmetric spline procedures for boundary-value problems with mixed boundary conditions, J. Comput. Appl. Math. 45 (3), 237–250, 1993.
  9. A.K.B. Chand and G.P. Kapoor, Generalized cubic spline fractal interpolation functions, SIAM J. Numer. Anal. 44 (2), 655–676, 2006.
  10. A.K.B. Chand and P. Viswanathan, A constructive approach to cubic hermite fractal interpolation function and its constrained aspects, BIT 53 (4), 841–865, 2013.
  11. M.M. Chawla, A sixth order tridiagonal finite difference method for nonlinear twopoint boundary-value problems, BIT 17 (2), 128–133, 1977.
  12. M.M. Chawla, An eighth order tridiagonal finite difference method for nonlinear twopoint boundary-value problems, BIT 17 (3), 281–285, 1977.
  13. M.M. Chawla and P.N. Shivakumar, Numerov’s method for nonlinear two-point boundary-value problems, Int. J. Comput. Math. 17 (2), 167–176, 1985.
  14. M.M. Chawla and R. Subramanian, A new fourth-order cubic spline method for nonlinear two-point boundary-value problems, Int. J. Comput. Math. 22 (3-4), 321–341, 1987.
  15. M.M. Chawla and R. Subramanian, A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems, J. Comput. Appl. Math. 23 (1), 1–10, 1988.
  16. M.M. Chawla, R. Subramanian, and P.N. Shivakumar, Numerov’s method for nonlinear two-point boundary-value problems II. monotone approximations, Int. J. Comput. Math. 26 (3-4), 219–227, 1989.
  17. U. Erdogan and T. Ozis, A smart nonstandard finite difference scheme for second order nonlinear boundary-value problems, J. Comput. Phys. 230 (17), 6464–6474, 2011.
  18. P. Henrici, Discrete variable methods in ordinary differential equations, John Wiley and Sons, New York, 1962.
  19. M.K. Kadalbajoo and K.C. Patidar, Spline techniques for solving singularly-perturbed nonlinear problems on nonuniform grids, J. Optim. Theory Appl. 114 (3), 573–591, 2002.
  20. M. Lees, Discrete method for nonlinear two-point boundary-value problems, in: Numerical Solution of Partial Differential Equations, ed. J.H. Bramble, Academic Press, New York, 1966.
  21. L.B. Liu, H.W. Liu, and Y. Chen, Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions, Appl. Math. Comput. 217 (16), 6872–6882, 2011.
  22. J. Rashidinia, R. Mohammadi, and R. Jalilian, Spline solution of nonlinear singular boundary-value problems, Int. J. Comput. Math. 85 (1), 39–52, 2008.
  23. A.S.V. Ravikanth and V. Bhattacharya, Cubic spline for a class of nonlinear singular boundary-value problems arising in physiology, Appl. Math. Comput. 174 (1), 768– 774, 2006.
  24. S.B.G. Karakoç, N.M. Yagmurlu, and Y. Ucar, Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines, Bound. Value Probl. 2013 (1), 2013.
  25. I.A. Tirmizi and E.H. Twizell, Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems, Appl. Math. Lett. 15 (7), 897–902, 2002.
  26. H. Zeybek and S.B.G. Karakoç, Application of the collocation method with B-splines to the GEW equation, Electron. Trans. Numer. Anal. 46, 71–88, 2017.