Skip to main content
  • Research Article
  • Open access
  • Published:

Generalized quasilinearization method and higher order of convergence for second-order boundary value problems

Abstract

The method of generalized quasilinearization for second-order boundary value problems has been extended when the forcing function is the sum of-hyperconvex and-hyperconcave functions. We develop two sequences under suitable conditions which converge to the unique solution of the boundary value problem. Furthermore, the convergence is of order. Finally, we provide numerical examples to show the application of the generalized quasilinearization method developed here for second-order boundary value problems.

[123456789101112131415]

References

  1. Bellman RE: Methods of Nonlinear Analysis. Vol. 1, Mathematics in Science and Engineering. Volume 61-I. Academic Press, New York; 1970:xx+340.

    Google Scholar 

  2. Bellman RE, Kalaba RE: Quasilinearization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics. Volume 3. American Elsevier, New York; 1965:ix+206.

    Google Scholar 

  3. Bernfeld SR, Lakshmikantham V: An Introduction to Nonlinear Boundary Value Problems, Mathematics in Science and Engineering. Volume 109. Academic Press, New York; 1974:xi+386.

    Google Scholar 

  4. Cabada A, Nieto JJ: Rapid convergence of the iterative technique for first order initial value problems. Applied Mathematics and Computation 1997,87(2-3):217-226. 10.1016/S0096-3003(96)00285-8

    Article  MathSciNet  MATH  Google Scholar 

  5. Cabada A, Nieto JJ: Quasilinearization and rate of convergence for higher-order nonlinear periodic boundary-value problems. Journal of Optimization Theory and Applications 2001,108(1):97-107. 10.1023/A:1026413921997

    Article  MathSciNet  MATH  Google Scholar 

  6. Heikkilä S, Lakshmikantham V: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 181. Marcel Dekker, New York; 1994:xii+514.

    MATH  Google Scholar 

  7. Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics. Volume 27. Pitman, Massachusetts; 1985:x+236.

    Google Scholar 

  8. Lakshmikantham V, Nieto JJ: Generalized quasilinearization iterative method for initial value problems. Nonlinear Studies 1995, 2: 1-9.

    MathSciNet  MATH  Google Scholar 

  9. Lakshmikantham V, Vatsala AS: Generalized Quasilinearization for Nonlinear Problems, Mathematics and Its Applications. Volume 440. Kluwer Academic, Dordrecht; 1998:x+276.

    Book  MATH  Google Scholar 

  10. Mandelzweig VB: Quasilinearization method and its verification on exactly solvable models in quantum mechanics. Journal of Mathematical Physics 1999,40(12):6266-6291. 10.1063/1.533092

    Article  MathSciNet  MATH  Google Scholar 

  11. Mandelzweig VB, Tabakin F: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Computer Physics Communications 2001,141(2):268-281. 10.1016/S0010-4655(01)00415-5

    Article  MathSciNet  MATH  Google Scholar 

  12. Melton T, Vatsala AS: Generalized quasilinearization and higher order of convergence for first order initial value problems. to appear in Dynamic Systems & Applications

  13. Mohapatra RN, Vajravelu K, Yin Y: Extension of the method of quasilinearization and rapid convergence. Journal of Optimization Theory and Applications 1998,96(3):667-682. 10.1023/A:1022620813436

    Article  MathSciNet  MATH  Google Scholar 

  14. Sokol M, Vatsala AS: A unified exhaustive study of monotone iterative method for initial value problems. Nonlinear Studies 2001,8(4):429-438.

    MathSciNet  MATH  Google Scholar 

  15. West IH, Vatsala AS: Generalized monotone iterative method for initial value problems. Applied Mathematics Letters 2004,17(11):1231-1237. 10.1016/j.aml.2004.03.003

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to AS Vatsala.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Melton, T.G., Vatsala, A. Generalized quasilinearization method and higher order of convergence for second-order boundary value problems. Bound Value Probl 2006, 25715 (2006). https://doi.org/10.1155/BVP/2006/25715

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/BVP/2006/25715

Keywords