Intuitionistic fuzzy initial value problems - an application

Abstract

In this paper, by using the properties of $\alpha$ and $\beta$ cuts of intuitionistic fuzzy numbers, we have firstly proposed a method to find the general solution of the second order initial value problem with intuitionistic fuzzy initial values under intuitionistic Zadeh's extension principle interpretation. Then we have given some numerical examples for the proposed method.

Keywords

• intuitionistic fuzzy set and numbers,intuitionistic characterization and stacking theorems,intuitionistic initial value problems,intuitionistic Zadeh's extension principle

References

1. [1] O. Akin and O. Oruc, A prey predator model with fuzzy initial values, Hacet. J. Math. Stat. 41 (3), 387–395, 2012.
2. [2] O. Akin, T. Khaniyev, S. Bayeg and I.B. Turksen, Solving a Second Order Fuzzy Initial Value Problem Using The Heaviside Function, Turk. J. Math. Comput. Sci. 4, 16–25, 2016.
3. [3] O. Akin, T. Khaniyev, O. Oruc and I.B. Turksen, An algorithm for the solution of second order fuzzy initial value problems, Expert Syst. Appl. 40, 953–957, 2013.
4. [4] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20, 87–96, 1986.
5. [5] K.T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets Syst. 33, 37–45, 1989.
6. [6] K.T. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 64, 159–174, 1994.
7. [7] K.T. Atanassov, Remarks on the intuitionistic fuzzy sets — III, Fuzzy Sets Syst. 75, 401–402, 1995.
8. [8] K.T. Atanassov, Remark on the intuitionistic fuzzy logics, Fuzzy Sets Syst. 95, 127– 129, 1998.

9. [9] K.T. Atanassov, Two theorems for intuitionistic fuzzy sets, Fuzzy Sets Syst. 110, 267–269, 2000.
10. [10] K.T. Atanassov, Answer to D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade’s paper "Terminological difficulties in fuzzy set theory-the case of "Intuitionistic Fuzzy Sets"", Fuzzy Sets and Syst. 156 (3), 496–499, 2005.
11. [11] L. Atanassova, On Intuitionistic Fuzzy Versions of L. Zadeh’s Extension Principle, NIFS, 13, 33–36, 2007.
12. [12] K.T. Atanassov and C. Georgiev, Intuitionistic fuzzy prolog, Fuzzy Sets Syst. 53, 121–128, 1993.
13. [13] G.A. Banini and R.A. Bearman, Application of fuzzy cognitive maps to factors affecting slurry rheology, Int. J. Miner. Process. 52 (4), 233–244, 1998.
14. [14] L.C. Barros, R.C. Bassanezi and P.A. Tonelli, Fuzzy modelling in population dynamics, Ecol. Model, 128 (2000), 27–33, 2000.
15. [15] B. Bede, Mathematics of Fuzzy Sets and Fuzzy Logic, Springer-Verlag, Berlin Heidelberg, 2013.
16. [16] B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230, 119–141, 2013.
17. [17] B. Bede, I.J. Rudas and A.L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inform. Sci. 177 (7), 2007.
18. [18] J.J. Buckley and T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems, 110 (1), 43–54, 2000.
19. [19] J.J. Buckley and T. Feuring, Fuzzy initial value problem for N th-order linear differential equations, Fuzzy Sets and Systems, 121, 247–255, 2001.
20. [20] J.J. Buckley, T. Feuring and Y. Hayashi, Linear systems of first order ordinary differential equations: fuzzy initial conditions, Soft Comput. 6, 415–421, 2002.
21. [21] J. Casasnovas and F. Rossell, Averaging fuzzy bio polymers, Fuzzy Sets and Systems, 152, 139–158, 2005.
22. [22] P. Diamond And P. Kloeden, Metric Spaces of Fuzzy Sets, Fuzzy Sets and Systems, 37–45, 1994.
23. [23] O. Duman, Statistical fuzzy approximation to fuzzy differentiable functions by fuzzy linear operators, Hacet. J. Math. Stat. 39 (4), 497–514, 2010.
24. [24] M.S. El Naschie, From experimental quantum optics to quantum gravity via a fuzzy Khler manifold, Chaos Solitons Fractals, 25, 969–977, 2005.
25. [25] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145–174, 1967.
26. [26] Z. Hassan,V.K. Hali and H. Akbar, Fuzzy Modeling and Control of HIV infection, Comput. Math. Methods Med. 17, Article ID: 893474, 1–17, 2012.
27. [27] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (3), 301–317, 1987.
28. [28] J.M. Mendel, Advances in type-2 fuzzy sets and systems, Information Sci. 177, 84– 110, 2007.
29. [29] S.P. Mondal and T.K. Roy,First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number, Journal of Uncertainty in Mathematics Science, 1–17, 2014.
30. [30] S.P. Mondal and T.K. Roy,System of differential equation with initial value as triangular intuitionistic fuzzy number and its application, Int. J. Appl. Comput. Math 2015, 449–474 2015.
31. [31] S.P. Mondal, S. Banerjee and T.K. Roy, First Order Linear Homogeneous Ordinary Differential Equation in Fuzzy Environment, Int. J. Pure Appl. Sci. Technol. 14 (1), 16–26, 2013.
32. [32] V. Nirmala, Numerical Approach for Solving Intuitionistic Fuzzy Differential Equation under Generalised Differentiability Concept, Appl. Math. Sci. 9, 2015.
33. [33] M. Oberguggenberger and S. Pittschmann, Differential equations with fuzzy parameters, Math. Mod. Syst. 5 (1999) 181–202, 1999.
34. [34] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (3), 319–330, 1987.
35. [35] X. Wei-Chau, Differential Equations for Engineers, Cambridge University Press, 188- 232, New York, 2010.
36. [36] L.A. Zadeh, Fuzzy sets, Inf. Control, 338–353, 1965.