Abstract
In real-life problems, both parameters and data used in mathematical modeling are often vague or uncertain. In fields like system biology, diagnosis, image analysis, fault detection and many others, fuzzy differential equations and stochastic differential equations are an alternative to classical, or in the present context crisp, differential equations. The aim of the paper is to propose a new formulation of fuzzy ordinary differential equations for which the Hukuhara derivative is not needed. After a short review of recent results in the theory of ordered fuzzy numbers, an exemplary application to a dynamical system in mechanics is presented. Departing from the classical framework of dynamics, equations describing the fuzzy representation of the state are introduced and solved numerically for chosen fuzzy parameters and initial data.
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References
Buckley James, J., Eslami, E.: An Introduction to Fuzzy Logic and Fuzzy Sets. Physica-Verlag, Springer, Heidelberg (2005)
Diamond, P., Kloeden, P.: Metric Spaces of Fuzzy Sets. World Scientific, Singapore (1993)
Drewniak, J.: Fuzzy numbers (in Polish). In: Chojcan, J., Łȩski, J. (eds.) Zbiory rozmyte i ich zastosowania, Wydawnictwo Politechniki Śla̧skiej, Gliwice, pp. 103–129 (2001)
Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. System Science 9, 576–578 (1978)
Goetschel Jr., R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets and Systems 18, 31–43 (1986)
Hukuhara, M.: Intégration des applications measurables dont la valeur est un compact convexe (in French). Funkcial. Ekvac. 10, 205–223 (1967)
Kaleva, O.: Fuzzy differential equations. Fuzzy Sets and Systems 24, 301–317 (1987)
Klir, G.J.: Fuzzy arithmetic with requisite constraints. Fuzzy Sets and Systems 91, 165–175 (1997)
Kosiński, W.: On defuzzyfication of ordered fuzzy numbers. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS (LNAI), vol. 3070, pp. 326–331. Springer, Heidelberg (2004)
Kosiński, W.: On fuzzy number calculus. Int. J. Appl. Math. Comput. Sci. 16(1), 51–57 (2006)
Kosiński, W.: On soft computing and modelling. Image Processing Communications 11(1), 71–82 (2006)
Kosiński, W., Frischmuth, K., Piasecki, W.: Fuzzy approach to hyperbolic heat conduction equations. In: Kuczma, M., Wilmański, K., Szajna, W. (eds.) 18th Intern. Confer. on Computational Methods in Mechanics, CMM 2009, Short Papers, Zielona Góra, May 2009, pp. 249–250 (2009)
Kosiński, W., Prokopowicz, P., Kacprzak, D.: Fuzziness - representation of dynamic changes by ordered fuzzy numbers. In: Seising, R. (ed.) Views of Fuzzy Sets and Systems from Different Perspectives. Studies in Fuzziness and Soft Computing, vol. (243), pp. 485–508. Springer, Heidelberg (2009)
Kosiński, W., Prokopowicz, P., Ślȩzak, D.: Fuzzy numbers with algebraic operations: algorithmic approach. In: Kłopotek, M., Wierzchoń, S.T., Michalewicz, M. (eds.) Intelligent Information Systems 2002, Proc. IIS 2002, Sopot, Poland, June 3-6, pp. 311–320. Physica Verlag, Heidelberg (2002)
Kosiński, W., Prokopowicz, P., Ślȩzak, D.: On algebraic operations on fuzzy reals. In: Rutkowski, L., Kacprzyk, J. (eds.) Advances in Soft Computing, Proc. of the Sixth Int. Conference on Neural Network and Soft Computing, Zakopane, Poland, June 11-15 (2002), pp. 54–61. Physica-Verlag, Heidelberg (2003)
Kosiński, W., Prokopowicz, P., Ślȩzak, D.: Ordered fuzzy numbers. Bulletin of the Polish Academy of Sciences, Ser. Sci. Math. 51(3), 327–338 (2003)
Kosiński, W., Prokopowicz, P.: Algebra of fuzzy numbers (in Polish). Matematyka Stosowana 5(46), 37–63 (2004)
Kosiński, W., Prokopowicz, P., Ślȩzak, D.: Calculus with fuzzy numbers. In: Bolc, L., Michalewicz, Z., Nishida, T. (eds.) IMTCI 2004. LNCS (LNAI), vol. 3490, pp. 21–28. Springer, Heidelberg (2005)
Kosiński, W., Słysz, P.: Fuzzy reals and their quotient space with algebraic operations. Bull. Pol. Acad. Sci., Sér. Techn. Scien. 41(30), 285–295 (1993)
Lakshmikantham, V., Mohapatra, R.N.: Theory of Fuzzy Differential Equations and Inclusions. Taylor and Francis, Abington (2003)
Nguyen, H.T.: A note on the extension principle for fuzzy sets. J. Math. Anal. Appl. 64, 369–380 (1978)
Sanchez, E.: Solutions of fuzzy equations with extended operations. Fuzzy Sets and Systems 12, 237–248 (1984)
Wagenknecht, M.: On the approximate treatment of fuzzy arithmetics by inclusion, linear regression and information content estimation. In: Chojcan, J., Łȩski, J. (eds.) Zbiory rozmyte i ich zastosowania, Wydawnictwo Politechniki Śla̧skiej, Gliwice, pp. 291–310 (2001)
Wagenknecht, M., Hampel, R., Schneider, V.: Computational aspects of fuzzy arithmetic based on archimedean t-norms. Fuzzy Sets and Systems 123(1), 49–62 (2001)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning, Part I. Information Sciences 8, 199–249 (1975)
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Kosiński, W., Frischmuth, K., Wilczyńska-Sztyma, D. (2010). A New Fuzzy Approach to Ordinary Differential Equations. In: Rutkowski, L., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds) Artificial Intelligence and Soft Computing. ICAISC 2010. Lecture Notes in Computer Science(), vol 6113. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13208-7_16
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DOI: https://doi.org/10.1007/978-3-642-13208-7_16
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