THE SHOOTING METHOD FOR SOLVING SECOND ORDER
FUZZY TWO-POINT BOUNDARY VALUE PROBLEMS
Basem S. Attili
University of Sharjah
College of Sciences - Department of Mathematics
Sharjah - P. O. Box 27272
UNITED ARAB EMIRATES

Abstract

We consider the fuzzy two-point boundary value problem (FBVP) subject to some fuzzy boundary conditions on an interval $[a,\ b]$. Numerically, we start by transforming the two-point boundary value problem into a system of fuzzy initial value problems (FIVP). To solve the resulting system, we use an improved $s-$stage Runge-Kutta Nystrom 4th order method adopted to handle fuzzy problems. Numerical results will be presented to give the numerical details and to show the efficiency of the method.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 4
Year: 2019

DOI: 10.12732/ijam.v32i4.9

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] T. Allahviranloo, N. Ahmady and E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method, Inform Sci., 177 (2007), 1633-1647.
  2. [2] B. Attili, On the numerical implementation of the shooting methods to the one-dimensional singular boundary value problems, Intern. J. Comput. Math., 47 (1993), 65-75.
  3. [3] B. Bede, I. Rudas and A. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inform. Sci., 177 (2007), 16481662.
  4. [4] Y. Chalco-Cano, and H. Roman-Flores, On new solution of fuzzy differential equations, Chaos, Solit. Fract., 38 (2008), 112-119.
  5. [5] O. Fard, T. Bidgoli and A. Rivaz, On existence and uniqueness of solutions to the fuzzy dynamic equations on time scales, Math. Comput. Appl., 22 (2017), 1-16.
  6. [6] A. G. Fatullayev, E. Can and C. Koroglu, Numerical solution of a boundary value problem for a second order fuzzy differential equation, TWMS J. Pure Appl. Math., 4 (2013), 169-176.
  7. [7] X. Guo, D. Shang and X. Lu, Fuzzy approximate solutions of second order fuzzy linear boundary value problems, Boundary Value Problems, 2013 (2013), Art. 212; DOI: 10.1186/1687-2770-2013-212.
  8. [8] E. Hullermeier, An approach to modeling and simulation of uncertain dynamical systems, Inter. J. Uncertain. Fuzz. and Knowledge Based Syst., 5 (1997), 117-137.
  9. [9] R. Jafari, W. Yu, Xiaoou Li and S. Razvarz, Numerical solution of fuzzy differential equations with Z-numbers using Bernstein neural networks, Intern. J. Comput. Intelli. Sys., 10 (2017) 1226-1237.
  10. [10] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
  11. [11] H.B. Keller, Numerical Solution of Two Point Boundary Value Problems, SIAM (1976).
  12. [12] A. Khastan and K. Ivaz, Numerical solution of fuzzy differential equations by Nystrom method, Chaos, Solit. Fract., 41 (2009), 859-868.
  13. [13] A. Khastan, and J. Nieto, A boundary value problem for second order fuzzy differential equations, Nonlinear Anal., 72 (2010), 3583-3593.
  14. [14] V. Lakshmikantham, K. Murty and J. Turner, Two point boundary value problems associated with nonlinear fuzzy differential equations, Math. Inequal. and Applic., 4 (2001), 527-533.
  15. [15] D. Li, M. Chen and X. Xue, Two point boundary value problems of uncertain dynamical systems, Fuzzy Sets and Systems, 179 (2011), 50-61.
  16. [16] D. O’Regan, V. Lakshmikantham and J. Nieto, Initial and boundary value problems for fuzzy differential equations, Nonlin. Anal., 54 (2003), 405415.
  17. [17] S. Palligkinis, G. Papageorgiou and I. Famelis, Runge-Kutta methods for fuzzy differential equations, Appl. Math.and Comput. 209 (2009), 97-105.
  18. [18] F. Rabiei, F. Ismail, N. Senu and N. Abasi, Construction of improved Runge-Kutta Nystrom method for solving second order ordinary differential equations, World Appl. Sci. J. 20 (2012), 1685-1695.
  19. [19] F. Rabiei, F. Abd Hamid, M. Rashidi and F. Ismail, Numerical simulation of fuzzy differential equations using general linear method and B-series, Advanc. in Mech. Eng., 9 (2017), 1–16.
  20. [20] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, SpringerVerlag, New York (1980).
  21. [21] L.A. Zadeh, The concept of a linguistic variable and its applications in approximate reasoning, Inform. Sci., 8 (1975), 199-249.