Skip to main content

A Fourth Order Runge-Kutta Gill Method for the Numerical Solution of Intuitionistic Fuzzy Differential Equations

  • Chapter
  • First Online:
Recent Advances in Intuitionistic Fuzzy Logic Systems

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 372))

Abstract

In this paper we study the numerical methods for Intuitionistic Fuzzy Differential equations by an application of the Fourth Order Runge-Kutta Gill method for intuitionistic fuzzy differential equations. We give a numerical example to illustrate the theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Abbasbandy, T. Allahviranloo, Numerical solution of fuzzy differential equation by Runge-Kutta method and the intutionistic treatment. Notes IFS 8(3), 45–53 (2002)

    Google Scholar 

  2. A.K. Adak, M. Bhowmik, M. Pal, Intuitionistic fuzzy block matrix and its some properties. Ann. Pure Appl. Math. 1(1), 13–31 (2012)

    Google Scholar 

  3. K.T. Atanassov, Intuitionistic fuzzy sets. VII ITKRs session, Sofia (deposited in Central Science and Technical Library of the Bulgarian Academy of Sciences 1697/84) (1983)

    Google Scholar 

  4. K.T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)

    Article  Google Scholar 

  5. K.T. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 43–49 (1989)

    Article  MathSciNet  Google Scholar 

  6. K.T. Atanassov, More on intuitionistic fuzzy sets. Fuzzy Sets Syst. 33(1), 37–45 (1989)

    Article  MathSciNet  Google Scholar 

  7. K.T. Atanassov, Operators over interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 64(2), 159–174 (1994)

    Article  MathSciNet  Google Scholar 

  8. K.T. Atanassov, G. Gargov, Elements of intuitionistic fuzzy logic, part I. Fuzzy Sets Syst. 95(1), 39–52 (1998)

    Article  Google Scholar 

  9. K.T. Atanassov, Intuitionistic Fuzzy Sets (Physica-Verlag, Heidelberg, 1999)

    Book  Google Scholar 

  10. K.T. Atanassov, Two theorems for Intuitionistic fuzzy sets. Fuzzy Sets Syst. 110, 267–269 (2000)

    Article  MathSciNet  Google Scholar 

  11. T. Buhaesku, On the convexity of intuitionistic fuzzy sets, Itinerant Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca (1988), pp. 137–144

    Google Scholar 

  12. T. Buhaesku, Some observations on intuitionistic fuzzy relations, Itinerant Seminar of Functional Equations, Approximation and Convexity (1989), pp. 111–118

    Google Scholar 

  13. A.I. Ban, Nearest interval approximation of an intuitionistic fuzzy number, in Computational Intelligence, Theory and Applications (Springer, Berlin, 2006), pp. 229–240

    Google Scholar 

  14. B. Ben Amma, S. Melliani, L.S. Chadli, Numerical solution of intuitionistic fuzzy differential equations by Euler and Taylor methods. Notes Intuitionistic Fuzzy Sets 22(2), 71–86 (2016)

    Google Scholar 

  15. B. Ben Amma, S. Melliani, L.S. Chadli, Numerical solution of intuitionistic fuzzy differential equations by Adams three order predictor-corrector method. Notes Intuitionistic Fuzzy Sets 22(3), 47–69 (2016)

    Google Scholar 

  16. B. Ben Amma, L.S. Chadli, Numerical solution of intuitionistic fuzzy differential equations by Runge-Kutta Method of order four. Notes Intuitionistic Fuzzy Sets 22(4), 42–52 (2016)

    Google Scholar 

  17. B. Ben Amma, S. Melliani, L.S. Chadli, The Cauchy problem of intuitionistic fuzzy differential equations. Notes Intuitionistic Fuzzy Sets 24(1), 37–47 (2018)

    Article  Google Scholar 

  18. B. Ben Amma, S. Melliani, L.S. Chadli, Intuitionistic fuzzy functional differential equations, in Fuzzy Logic in Intelligent System Design: Theory and Applications, ed. by P. Melin, O. Castillo, J. Kacprzyk, M. Reformat, W. Melek (Springer International Publishing, Cham, 2018), pp. 335–357

    Google Scholar 

  19. C. Cornelis, G. Deschrijver, E.E. Kerre, Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, application. Int. J. Approximate Reasoning 35, 55–95 (2004)

    Article  MathSciNet  Google Scholar 

  20. S.K. De, R. Biswas, A.R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets Syst. 117, 209–213 (2001)

    Article  Google Scholar 

  21. G. Deschrijver, E.E. Kerre, On the relationship between intuitionistic fuzzy sets and some other extensions of fuzzy set theory. J. Fuzzy Math. 10(3), 711–724 (2002)

    MathSciNet  MATH  Google Scholar 

  22. T. Gerstenkorn, J. Manko, Correlation of intuitionistic fuzzy sets. Fuzzy Sets Syst. 44, 39–43 (1991)

    Article  MathSciNet  Google Scholar 

  23. A. Kharal, Homeopathic drug selection using intuitionistic fuzzy sets. Homeopathy 98, 35–39 (2009)

    Article  Google Scholar 

  24. D.F. Li, C.T. Cheng, New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognit Lett. 23, 221–225 (2002)

    Article  Google Scholar 

  25. D.F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets. J. Comput. Syst. Sci. 70, 73–85 (2005)

    Article  MathSciNet  Google Scholar 

  26. S. Melliani, L.S. Chadli, Intuitionistic fuzzy differential equation. Notes Intuitionistic Fuzzy Sets 6, 37–41 (2000)

    MathSciNet  MATH  Google Scholar 

  27. G.S. Mahapatra, T.K. Roy, Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations. Proc. World Acad. Sci. Eng. Technol. 38, 587–595 (2009)

    Google Scholar 

  28. S. Melliani, M. Elomari, L.S. Chadli, R. Ettoussi, Intuitionistic fuzzy metric space. Notes Intuitionistic Fuzzy sets 21(1), 43–53 (2015)

    Google Scholar 

  29. S. Melliani, M. Elomari, L.S. Chadli, R. Ettoussi, Intuitionistic fuzzy fractional equation. Notes Intuitionistic Fuzzy sets 21(4), 76–89 (2015)

    Google Scholar 

  30. S. Melliani, M. Atraoui Elomari, L.S. Chadli, Intuitionistic fuzzy differential equation with nonlocal condition. Notes Intuitionistic Fuzzy sets 21(4), 58–68 (2015)

    Google Scholar 

  31. M. Nikolova, N. Nikolov, C. Cornelis, G. Deschrijver, Survey of the research on intuitionistic fuzzy sets. Adv. Stud. Contemp. Math. 4(2), 127–157 (2002)

    MathSciNet  MATH  Google Scholar 

  32. V. Nirmala, Numerical approach for solving intuitionistic fuzzy differential equation under generalised differentiability concept. Appl. Math. Sci. 9(67), 3337–3346 (2015)

    Google Scholar 

  33. V. Parimala, P. Rajarajeswari, V. Nirmala, Numerical solution of intuitionistic fuzzy differential equation by Milne’s Predictor-Corrector Method under generalised differentiability. Int. J. Math. Appl. 5, 45–54 (2017)

    Google Scholar 

  34. E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst. 114(3), 505–518 (2000)

    Article  MathSciNet  Google Scholar 

  35. M.H. Shu, C.H. Cheng, J.R. Chang, Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly. Microelectron. Reliab. 46(12), 2139–2148 (2006)

    Article  Google Scholar 

  36. P.M. Sankar, T.K. Roy, First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number. J. Uncertainty Math. Sci. 2014, 1–17 (2014)

    Google Scholar 

  37. P.M. Sankar, T.K. Roy, System of differential equation with initial value as triangular intuitionistic fuzzy number and its application. Int. J. Appl. Comput. Math 1(3), 449–474 (2015)

    Article  MathSciNet  Google Scholar 

  38. Z. Wang, K.W. Li, W. Wang, An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights. Inform. Sci. 179(17), 3026–3040 (2009)

    Article  MathSciNet  Google Scholar 

  39. J. Ye, Multicriteria fuzzy decision-making method based on a novel accuracy function under interval valued intuitionistic fuzzy environment. Expert Syst. Appl. 36, 6899–6902 (2009)

    Article  Google Scholar 

  40. L.A. Zadeh, Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Said Melliani .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ben Amma, B., Melliani, S., Chadli, L.S. (2019). A Fourth Order Runge-Kutta Gill Method for the Numerical Solution of Intuitionistic Fuzzy Differential Equations. In: Melliani, S., Castillo, O. (eds) Recent Advances in Intuitionistic Fuzzy Logic Systems. Studies in Fuzziness and Soft Computing, vol 372. Springer, Cham. https://doi.org/10.1007/978-3-030-02155-9_5

Download citation

Publish with us

Policies and ethics