Fuzzy differential inclusion is the extension of differential inclusion to fuzzy sets introduced by Lotfi A. Zadeh.[1][2]
with
![{\displaystyle x'(t)\in [f(t,x(t))]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a01062827cbe73779ab92ee5ef1dff4cfbc7340)
![{\displaystyle x(0)\in [x_{0}]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae55b8b969824cb3b921a247a0224f1cd91c1da6)
Suppose is a fuzzy valued continuous function on Euclidean space. Then it is the collection of all normal, upper semi-continuous, convex, compactly supported fuzzy subsets of .
Second order differential
The second order differential is
where , is trapezoidal fuzzy number , and is a trianglular fuzzy number (-1,0,1).
![{\displaystyle x''(t)\in [kx]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2872d4ef491f0c5f1a123b179c0d383f53cbb5)
![{\displaystyle k\in [K]^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7aac88efa972bfb4a3ecbfa72e60d62b2284fd1)
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]
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.
This page was last edited on 25 May 2024, at 05:31