DOI QR코드

DOI QR Code

NUMERICAL METHOD FOR A 2NTH-ORDER BOUNDARY VALUE PROBLEM

  • Xu, Chenmei (College of Mathematics and Information Science, Henan University) ;
  • Jian, Shuai (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) ;
  • Wang, Bo (College of Mathematics and Information Science, Henan University)
  • Received : 2010.04.12
  • Published : 2013.07.01

Abstract

In this paper, a finite difference scheme for a two-point boundary value problem of 2nth-order ordinary differential equations is presented. The convergence and uniqueness of the solution for the scheme are proved by means of theories on matrix eigenvalues and norm. Numerical examples show that our method is very simple and effective, and that this method can be used effectively for other types of boundary value problems.

Keywords

boundary valve problem;finite difference scheme;2nth-order ordinary differential equation;total truncation error

Acknowledgement

Supported by : Natural Science Foundation of the Education Department of Henan Province, National Natural Science Foundation of China

References

  1. A. Barari and M. Omidvar, An apprioximate solution for boundary value problems instructural engineering and mechanics, Math. Prom. Eng. 2008 (2008), no. 10, 1-10.
  2. M. K. Jain, S. R. K. Iyengar, and J. S. V. Saldanha, Numerical solution of a fourth-order ordinary differential equation, J. Engrg. Math. 11 (1977), no. 4, 373-380. https://doi.org/10.1007/BF01537095
  3. M. A. Noor and S. T. Molyud-Din, A reliable approach for solving linear and nonlinear sixth-order boundary value problems, Int. J. Comput. Appl. Math. 2 (2007), no. 2, 163-172.
  4. M. A. Noor, K. I. Noor, and S. T. Mohyud-Din, Variational iteration method for solving sixth-order boundary value problems, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 6, 2571-2580. https://doi.org/10.1016/j.cnsns.2008.10.013
  5. M. H. Pei, Existence and uniqueness of solutions to two-point boundary value problems for ordinary differential equations of order 2n, J. Systems Sci. Math. Sci. 17 (1997), no. 2, 165-172.
  6. R. A. Usmani, An O($h^6$) finite difference analogue for the solution of some differential equations occurring in plate deflection theory, J. Inst. Math. Appl. 20 (1977), no. 3, 331-333. https://doi.org/10.1093/imamat/20.3.331
  7. R. A. Usmani, A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc. 77 (1979), no. 3, 329-335. https://doi.org/10.1090/S0002-9939-1979-0545591-4
  8. R. A. Usmani, Discrete methods for boundary value problems with applications in plate deflection theory, Z. Angew. Math. Phys. 30 (1979), no. 1, 87-99. https://doi.org/10.1007/BF01597483
  9. R. A. Usmani and M. J. Marsden, Numerical solution of some ordinary differential equations occurring in plate deflection thery, J. Engrg. Math. 9 (1975), no. 1, 1-10. https://doi.org/10.1007/BF01535492
  10. R. A. Usmani and M. J. Marsden, Convergence of a numerical procedure for the solution of a fourth-order boundary value problem , Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 1, 21-30. https://doi.org/10.1007/BF02898331
  11. X. Q. Wang, On a boundary value problem arising in elastic deflection theory, Bull. Austral. Math. Soc. 74 (2006), no. 3, 337-345. https://doi.org/10.1017/S0004972700040405
  12. A. M. Wazwaz, The numerical solution of sixth-order boundary value problems by the modified decomposition method, Apple. Math. Comput. 118 (2001), no. 2-3, 311-325. https://doi.org/10.1016/S0096-3003(99)00224-6
  13. C. Xu and S. Sun, Introduction to Computational Methods, Higher Education Press, Beijing, 2002.
  14. K. Zhang and Y. Zhao, Algorithm and Analysis of Numerical Computation, Science Press, Beijing, 2003.