Perceptions about function changes are represented by rules like “If X is SMALL then Y is QUICKLY INCREASING.” The consequent part of a rule describes a granule of directions of the function change when X is increasing on the fuzzy interval given in the antecedent part of the rule. Each rule defines a granular differential and a rule base defines a granular derivative. A reconstruction of a fuzzy function given by the granular derivative and the initial value given by the rule is similar to Euler’s piecewise linear solution of an initial value problem. The solution method is based on a granulation of the directions of the function change, on an extension of the initial value in directions and on a propagation of fuzzy constraints given in antecedent parts of rules on possible function values. The proposed method is illustrated with an example.
@article{bwmeta1.element.bwnjournal-article-amcv12i3p403bwm,
author = {Batyrshin, Ildar},
title = {On granular derivatives and the solution of a granular initial value problem},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {12},
year = {2002},
pages = {403-410},
zbl = {1081.34501},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv12i3p403bwm}
}
Batyrshin, Ildar. On granular derivatives and the solution of a granular initial value problem. International Journal of Applied Mathematics and Computer Science, Tome 12 (2002) pp. 403-410. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv12i3p403bwm/
[000] Batyrshin I. and Fatkullina R. (1995): Context-dependent fuzzy scales and context-free rules for dependent variables. - Proc. 6-th Int. Fuzzy Systems Association World Congress, IFSA'95, Sao Paulo, Brasil, Vol. 1, pp. 89-91.
[001] Batyrshin I. and Panova A. (2001): On granular description of dependencies. -Proc. 9-th Zittau Fuzzy Colloquium, Zittau, Germany, pp. 1-8.
[002] Batyrshin I., Zakuanov R. and Bikushev G. (1994): Expert system based on algebra of uncertainties with memory in process optimization. - Proc. 1-st Int. Workshop Fuzzy Logic and Intelligent Technologies in Nuclear Science, Mol, Belgium, World Scientific, pp. 156-159.
[003] De Kleer J. and Brawn J. (1984): A qualitative physics based on confluences. -Artif. Intell., Vol. 24, pp. 7-83.
[004] Dubois D. and Prade H. (1982): Towards fuzzy differential calculus: Part 3: Differentiation. - Fuzzy Sets Syst., Vol. 8, pp. 225-233. | Zbl 0499.28009
[005] Forbus K.D. (1984): Qualitative process theory. - Artif. Intell., Vol. 24, pp. 85-168.
[006] Jang J.S.R., Sun C.T. and Mizutani E. (1997): Neuro-Fuzzy and Soft Computing. A Computational Approach to Learning and Machine Intelligence. - New York: Prentice Hall.
[007] Kuipers B. (1984): Commonsense reasoning about causality: deriving behavior from structure. - Artif. Intell., Vol. 24, pp. 169-203.
[008] Ma M., Friedman M. and Kandel A. (1999): Numerical solutions of fuzzy differential equations. - Fuzzy Sets Syst., Vol. 105, pp. 133-138. | Zbl 0939.65086
[009] Nieto J.J. (1999): The Cauchy problem for continuous fuzzy differential equations. - Fuzzy Sets and Syst., Vol. 102, pp. 259-262. | Zbl 0929.34005
[010] Park J.K. and Han H.K. (2000): Fuzzy differential equations. - Fuzzy Sets Syst., Vol. 110, pp. 69-77.
[011] Song S. and Wu C. (2000): Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations. - Fuzzy Sets Syst., Vol. 110, pp. 55-67. | Zbl 0946.34054
[012] Vorobiev D. and Seikkala S. (2002): Towards the theory of fuzzy differential equations. - Fuzzy Sets Syst., Vol. 125, pp. 231-237. | Zbl 1003.34046
[013] Zadeh L.A. (1966): The shadows of fuzzy sets. - Problems of Information Transmission, Vol. 2, pp. 37-44 (in Russian). | Zbl 0263.02028
[014] Zadeh L.A. (1975): The concept of a linguistic variable and its application to approximate reasoning, Part I. - Inf. Sci., Vol. 8, pp. 199-249; Part II. - Inf. Sci., Vol. 8, pp. 301-357; Part III - Inf. Sci., Vol. 9, pp. 43-80. | Zbl 0397.68071
[015] Zadeh L.A. (1997): Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. - Fuzzy Sets Syst., Vol. 90, pp. 111-127. | Zbl 0988.03040
[016] Zadeh L.A. (1999): From computing with numbers to computing with words - from manipulation of measurements to manipulation of perceptions. - IEEE Trans. Circ. Syst. - 1: Fund. Th. Applic., Vol. 45, No. 1, pp. 105-119. | Zbl 0954.68513