[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Runge-Kutta method of order four for solving fuzzy delay differential equations under generalized differentiability

Journal of Applied Nonlinear Dynamics 7(2) (2018) 131--146 | DOI:10.5890/JAND.2018.06.003

S. Indrakumar; K. Kanagarajan

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore 641020, India

Assistant professor in Kongu Engineering College, Erode, Tamilnadu 638052, India

Download Full Text PDF

 

Abstract

This paper portrays and interpret the fuzzy delay differential equations using the generalized differentiability concept by applying the Generalized Characterization Theorem. Subsequently we also investigate the problem of finding a numerical approximation of solutions. Moreover, the Runge-Kutta approximation methods is implemented and its error analysis are also discussed. The applicability of the theoretical results are illustrated with some examples.

References

  1. [1]  Zadeh, L.A. (1965), Fuzzy Sets, Information and Control, 8, 338-353.
  2. [2]  Dubois, D. and Prade, H. (1982), Towards fuzzy differential calculus part 3: differentiation, Fuzzy Sets and Systems, 8, 225-233.
  3. [3]  Seikkala, S. (1987), On the fuzzy initial value problem, Fuzzy Sets and Systems, 24(3), 319-330.
  4. [4]  Kaleva, O. (1987), Fuzzy differential equations, Fuzzy Sets and Systems, 24, 301-317.
  5. [5]  Friedman, M., Ma, M., and Kandel, A. (1999), Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets and Systems, 106 35-48.
  6. [6]  Ma, M., Friedman, M., and Kandel, A. (1999), Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105, 133-138.
  7. [7]  Abbasbandy, S. and Allahviranloo, T. (2002), Numerical solutions of fuzzy differential equations by Taylor Method, Computational Methods in Applied Mathematics, 2, 113-124.
  8. [8]  Abbasbandy, S. and Allahviranloo, T. (2004), Numerical solution of Fuzzy differential equation by Runge- Kutta method, Nonlinear Studies, 11, 117-129. 199-211.
  9. [9]  Allahviranloo, T., Ahmady, N., and Ahmady, E. (2007), Numerical solutions of fuzzy differential equations by predictor-corrector method, Information Sciences, 177(7), 1633-1647.
  10. [10]  Khastan, A. and Ivaz, K. (2009), Numerical solution of fuzzy differential equations by Nystrom method, Chaos, Solitons and Fractals, 41, 859-868.
  11. [11]  Bede, B. and Gal, S.G. (2004), Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems, 147, 385-403.
  12. [12]  Bede, B. and Gal, S.G. (2005), Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151, 581-599.
  13. [13]  Bede, B., Rudas, I.J., and Bencsik, A.L. (2007), First order linear fuzzy differential equations under general- ized differentiability, Information Sciences, 177, 1648-1662.
  14. [14]  Chalco-Cano, Y. and Roman-Flores, H. (2008), On new solutions of fuzzy differential equations, Chaos Solitons Fractals, 38, 112-119.
  15. [15]  Khastan, A., Bahrami, F., and Ivaz, K. (2009), New results on multiple solutions for Nth-order fuzzy differ- ential equations under generalized differentiability, Boundary value Problems, 13, Article ID 395714.
  16. [16]  Nieto, J.J., Khastan, A., and Ivaz, K. (2009), Numerical solution of fuzzy differential equations under gener- alized differentiability, Nonlinear Analysis: Hybrid system, 3, 700-707.
  17. [17]  Khastan, A., Nieto, J.J., and Rodriguez-Lopez, R. (2014), Fuzzy delay differential equations under generalized differentiability, Information Sciences, 275, 145-167.
  18. [18]  Bede, B. (2008), Note on “Numerical solutions of fuzzy differential equations by predictor corrector method”, Information Sciences, 178, 1917-1922.
  19. [19]  Diamond, P. and Kloeden, P. (1994), Metric Spaces of Fuzzy sets, World Scientific, Singapore.
  20. [20]  Puri, M.L. and Ralescu, D.A. (1983), Differentials of fuzzy functions, Journal of Mathematical Analysis Application, 91, 552-558.
  21. [21]  Lupulescu, V. (2009), On a class of fuzzy functional differential equations, Fuzzy Sets and Systems, 160, 1547-1562.
  22. [22]  Hale, J.K. (1997), Theory of Functional Differential Equations, Springer, New York.
  23. [23]  Khastan, A., Nieto, J.J., and Rodriguez-Lopez, R. (2011), Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets and Systems, 177, 20-33.
  24. [24]  Kuang, Y. (1993), Delay differential equations with Applications in population dynamics, Academic Press, Boston.
  25. [25]  Schuster, R. and Schuster, H. (1995), Reconstruction models for the Ehrlich ascites tumor of the mouse in O. Arino, D. Axelrod, M. Kimmel, (Eds), Mathematical Population Dynamics, Volume 2, Wuertz, Winnipeg, Canada, 335-348.
  26. [26]  Mackey, M.C. (1977), Leon Glass, Oscillation and chaos in physiological control systems, Science, 197 287- 289.
  27. [27]  Akira Shibata. and Nobuhiko Saito. (1980), Time delays and chaos in two competing species, Mathematical Biosciences, 51, 199-211.
  28. [28]  May, R.M. (1980), Nonlinear phenomena in ecology and epidemiology, The New York Academy of Sciences, 357 267-281.
  29. [29]  Doney Farmer, J. (1982), Choatic attractors of an infinite-dimensional dynamical system, Physica, 4D, 366- 393.
  30. [30]  Grassberger, P. (1984), Dimensions and entropies of strange attractors from a fluctuating dynamic approach, Physica, 13D, 34-54.
  31. [31]  Lakshmikantham, V. and Mohapatra, R.N. (2003), Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, United Kingdom.
  32. [32]  Lupulescu, V. and Abbas, U. (2012), Fuzzy delay differential equations, Fuzzy Optimum Desis Making, 11(1), 99-111.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates