In this paper, we define the concept of fuzzy derivatives for perfect and semi-perfect interval-valued fuzzy mappings. Based on this definition, we then present a method for solving interval-valued fuzzy differential equations using the alpha-cut extension principle. Finally, several examples are provided to demonstrate the effectiveness of the proposed method.
BencsikA., BedeB., TarJ. and FodorJ., Fuzzy differential equations in modeling hydraulic differential servo cylinders, In: Third Romanian-Hungarian joint symposiumon applied computational intelligence (SACI) Timisoara, Romania, 2006.
2.
KhastananA., NietoaJ.J. and LópezR.R., Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets Syst17 (2011), 20–33.
3.
WangC., ChengC. and LeeT., Dynamical optimal training for interval type-2 fuzzy neural network (T2FNN), IEEE Trans Syst, Man, Cybern34 (2004), 1462–1477.
4.
JuangC.F. and HsuC.H., Reinforcement interval type-2 fuzzy controller design by online rule generation and Q-value-aided ant colony optimization, IEEE Trans Syst, Man, Cybern39 (2009), 1528–1542.
5.
LiD.F., Mathematical-programming approach to matrix games with payoffs represented by Atanassov’s interval-valued intuitionistic fuzzy sets, IEEE Trans Fuzzy Syst18 (2010), 1112–1128.
6.
BabolianE., SadeghiH. and JavadiSh., Numerical solution of fuzzy differential equations by Adomian method, Appl Math Comput149 (2004), 547–557.
DeschrijverG., Arithmetic operators in interval-valued fuzzy set theory, Inf Sci177 (2007), 2906–2924.
9.
DeschrijverG. and KerreE.E., Implicators based on binary aggregation operators in interval-valued fuzzy set theory, Fuzzy Sets Syst153 (2005), 229–248.
10.
BustinceH., BarrenecheaE. and PagolaM., Generation of interval-valued fuzzy and Atanassovs intuitionistic fuzzy connectives from fuzzy connectives and from kα operators: Laws for conjunctions and disjunctions, amplitude, Int J Intell Syst23 (2008), 680–714.
11.
BustinceH., MonteroJ., PagolaM., BarrenecheaE. and GomezD., A survey of interval-valued fuzzy sets, in Handbook of Granular Computing, New York: Wiley, (2008), 489–515.
12.
HamrawiH., CouplandS. and JohnR., Extending operations on type-2 fuzzy sets, in Computational Intelligence (UKCI), 2009.
13.
HamrawiH. and CouplandS., Measures of uncertainty for type-2 fuzzy sets, in Computational Intelligence (UKCI) (2010), pp. 1–6.
14.
HamrawiH. and CouplandS., Type-2 fuzzy arithmetic using alpha-planes, in Fuzzy Systems, IFSA/EUSFLAT (2009b), pp. 606–611.
15.
HamrawiH. and CouplandS., Non-specificity measures for type-2 fuzzy sets, in Fuzzy Systems, FUZZ-IEEE (2009a), pp. 732–737.
16.
HamrawiH., CouplandS. and JohnR., A novel alpha-cut representation for type-2 fuzzy sets, in Fuzzy Systems (FUZZ) (2010), pp. 1–8.
17.
HamrawiH., Type-2 Fuzzy Alpha-cuts Ph.D. Dissertation, De Montfort University, 2011.
HagrasH.A., A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots, IEEE Transactions Fuzzy Syst12 (2004), 524–539.
20.
CasasnovasJ. and RossellF., Averaging fuzzy biopolymers, Fuzzy Sets and Systems152 (2005), 139–158.
21.
GoguenJ., L-fuzzy sets, J Math Anal Appl18 (1967), 143–174.
22.
ZhanJ., LiuQ. and DavvazB., A new rough set theory: Rough soft hemirings, Journal of Intelligent & Fuzzy Systems28 (2015), 1687–1697.
23.
MendelJ.M., JohnR.I. and LiuF., Interval type-2 fuzzy logic systems made simple, IEEE Trans Fuzzy Syst14 (2006), 808–821.
24.
MendelJ.M., Uncertainty, fuzzy logic, and signal processing, Signal Processing80 (2000), 913–933.
25.
YaoK. and ChenX., A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems25 (2013), 825–832.
26.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
27.
ZadehL.A., Is there a need for fuzzy logic?Inf Sci178 (2008), 2751–2779.
28.
ZadehL.A., Outline of a new approach to the analysis of complex systems and decision processes interval-valued fuzzy sets, IEEE Trans Syst, Man, Cybern3 (1973), 28–44.
29.
ZadehL.A., Toward extended fuzzy logic— A first step, Fuzzy Sets Syst160 (2009), 3175–3181.
30.
BiglarbegianM., MelekW. and MendelJ., Design of novel interval type-2 fuzzy controllers for modular and reconfigurable robots: Theory and experiments, Industrial Electronics IEEE Transactions on58 (2011), 1371–1384.
31.
FriedmanM., MaM. and KandelA., Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets Syst106 (1999), 35–48.
32.
GhaemiM. and AkbarzadehM.R., Indirect adaptive interval type-2 fuzzy PI sliding mode control for a class of uncertain nonlinear systems, Iranian Journal of Fuzzy Systems11 (2014), 1–21.
33.
GuoM., XueX. and LiR., Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems138 (2003), 601–615.
34.
OberguggenbergerM. and PittschmannS., Differential equations with fuzzy parameters, Math Mod Syst5 (1999), 181–202.
35.
PuriM.L. and RalescuD.A., Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications91 (1983), 552–558.
36.
NaschieM.S.E., From experimental quantum optics to quantum gravity via a fuzzy Khler manifold, Chaos, Solitons and Fractals25 (2005), 969–977.
37.
HsiaoM.Y., LiT.H.S., LeeJ.Z., ChaoC.H. and TsaiS.H., Design of interval type-2 fuzzy sliding-mode controller, Inf Sci178 (2008), 1696–1716.
38.
BatenM. and AbdulbasahA., A Fuzzy optimal control with application to discounted profit advertising problem, Journal of Intelligent and Fuzzy Systems23 (2012), 187–192.
39.
KarnikandN. and MendelJ., Applications of type-2 fuzzy logic systems to forecasting of time-series, Inf Sci120 (1999), 89–111.
40.
CastilloO. and MelinP., Type-2 Fuzzy Logic Theory and Applications, Berlin Germany: Springer-Verlag, 2008.
41.
KalevaO., Fuzzy differential equations, Fuzzy Set Syst24 (1987), 301–317.
42.
LindaO. and ManicM., Uncertainty-robust design of interval type-2 fuzzy logic controller for delta parallel robot, IEEE Trans Ind Inf7 (2011), 661–671.
43.
AbbasbandyS. and AllahviranlooT., Numerical solutions of fuzzy differential equations by Taylor method, Journal of Computational Methods in Applied Mathematics2 (2002), 113–124.
44.
ChangS.L. and ZadehL.A., On fuzzy mapping and control, IEEE Trans Syst, Man, Cybern2 (1972), 30–34.
AllahviranlooT. and HooshangianL., Fuzzy generalized H-differential and applications to fuzzy differential equations of second-order, Journal of Intelligent and Fuzzy Systems26 (2014), 1951–1967.
47.
AllahviranlooT. and AfsharM., Numerical methods for fuzzy linear partial differential equations under new definition for derivative, in Journal of Iranian Journal of Fuzzy Systems7 (2010), 33–50.
48.
AllahviranlooT., AhmadyN. and AhmadyE., Improved predictor–corrector method Numerical for solving fuzzy initial value problems, differential equations, Inf Sci179 (2009), 945–955.
49.
AllahviranlooT., AhmadyN. and AhmadyE., Numerical solution of fuzzy differential equations by predictor–corrector method, Inf Sci177 (2007), 1633–1647.
50.
AllahviranlooT. and TaheriN., An analytic approximation to the solution of fuzzy heat equation by Adomian decomposition method, in Journal of Int J Contemp Math Sciences4 (2009), 105–114.
51.
AllahviranlooT., AbbasbandyS. and BehzadiSh., Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method, Soft Computing18 (2014), 2191–2200.
52.
AllahviranlooT., GouyandehZ., ArmandA. and HasanogluA., On fuzzy solutions for heat equation based on generalized Hukuhara differentiability, Fuzzy Sets and Systems265 (2015), 1–23.
53.
AllahviranlooT., GouyandehZ. and ArmandA., Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems26 (2014), 1481–1490.
54.
FartashT.S., AkbarzadehM.R., AkbarzadehA. and Jalaeian-FM., Position tracking of a 3-PSP parallel robot using dynamic growing interval type-2 fuzzy neural control, Applied Soft Computing37 (2015), 1–14.
55.
Chalco-CanoY. and Román-FloresH., On new solutions of fuzzy differential equations, Chaos, Solitons & Fractals38 (2008), 112–119.
56.
PawlakZ., Rough sets, International J Comput Inform Sci11 (1982), 341–356.
