Fuzzy differential inclusion
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Fuzzy differential inclusion is tha culmination of Fuzzy concept and Differential inclusion introduced by Lotfi A. Zadeh which became popular.[1][2]
,
![{\displaystyle x'(t)\epsilon [f(t,x(t)]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d6a7ebebbe1a8bc61fccd1671393fea60bf366)
![{\displaystyle x(0)\epsilon [x_{0}]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bc6405efb8196eff2b97e118117de6360604ae)
f(t,x(t)] is a fuzzy valued continuous function on euclidian space which is collection of all normal, upper semi-continuous, Convex set
,Compact space, supported fuzzy subsets of .
Second order differential
The second order differential is
where
![{\displaystyle x''(t)\epsilon [kx]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbac732d7ad9b686da83639d2a2efcda0cb87ca)
![{\displaystyle k\epsilon [K]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e6ec703efce876b9f052f7381c9f9b4d713832)
K is trapezoidal fuzzy number (-1,-1/2,0,1/2)
is a trianglular fuzzy number (-1,0,1) .
Applications
Fuzzy differential inclusion (FDI) has applications in
- Cybernetics[3]
- Artificial intelligence, Neural network,[4][5]
- Medical imaging
- Robotics
- Atmospheric dispersion modeling
- Weather forecasting
- Cyclone
- Pattern recognition
- Population biology[6]
- Stochastic process, Probability theory[citation needed]
References
- ^ Lakshmikantham, V.; Mohapatra, Ram N. (11 September 2019). Theory of Fuzzy Differential Equations and Inclusions. ISBN 978-0-367-39532-2.
- ^ Min, Chao; Liu, Zhi-bin; Zhang, Lie-hui; Huang, Nan-jing (2015). "On a System of Fuzzy Differential Inclusions". Filomat. 29 (6): 1231–1244. doi:10.2298/FIL1506231M. ISSN 0354-5180. JSTOR 24898205.
- ^ "Fuzzy differential inclusion in atmospheric and medical cybernetics" (PDF).
- ^ Tafazoli, Sina; Menhaj, Mohammad Bagher (March 2009). "Fuzzy differential inclusion in neural modeling". 2009 IEEE Symposium on Computational Intelligence in Control and Automation. pp. 70–77. doi:10.1109/CICA.2009.4982785. ISBN 978-1-4244-2752-9. S2CID 5618541.
- ^ Min, Chao; Zhong, Yihua; Yang, Yan; Liu, Zhibin (May 2012). "On the implicit fuzzy differential inclusions". 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery. pp. 117–119. doi:10.1109/FSKD.2012.6234283. ISBN 978-1-4673-0024-7. S2CID 1952893.
- ^ Antonelli, Peter L.; Křivan, Vlastimil (1992). "Fuzzy differential inclusions as substitutes for stochastic differential equations in population biology". Open Systems & Information Dynamics. 1 (2): 217–232. doi:10.1007/BF02228945. JSTOR 24898205. S2CID 123026730.