On granular derivatives and the solution of a granular initial value problem

Ildar Batyrshin

International Journal of Applied Mathematics and Computer Science (2002)

  • Volume: 12, Issue: 3, page 403-410
  • ISSN: 1641-876X

Abstract

top
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.

How to cite

top

Batyrshin, Ildar. "On granular derivatives and the solution of a granular initial value problem." International Journal of Applied Mathematics and Computer Science 12.3 (2002): 403-410. <http://eudml.org/doc/207597>.

@article{Batyrshin2002,
abstract = {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.},
author = {Batyrshin, Ildar},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {initial value problem; fuzzy differential; fuzzy granule; cylindrical extension},
language = {eng},
number = {3},
pages = {403-410},
title = {On granular derivatives and the solution of a granular initial value problem},
url = {http://eudml.org/doc/207597},
volume = {12},
year = {2002},
}

TY - JOUR
AU - Batyrshin, Ildar
TI - On granular derivatives and the solution of a granular initial value problem
JO - International Journal of Applied Mathematics and Computer Science
PY - 2002
VL - 12
IS - 3
SP - 403
EP - 410
AB - 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.
LA - eng
KW - initial value problem; fuzzy differential; fuzzy granule; cylindrical extension
UR - http://eudml.org/doc/207597
ER -

References

top
  1. 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. 
  2. Batyrshin I. and Panova A. (2001): On granular description of dependencies. -Proc. 9-th Zittau Fuzzy Colloquium, Zittau, Germany, pp. 1-8. 
  3. 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. 
  4. De Kleer J. and Brawn J. (1984): A qualitative physics based on confluences. -Artif. Intell., Vol. 24, pp. 7-83. 
  5. Dubois D. and Prade H. (1982): Towards fuzzy differential calculus: Part 3: Differentiation. - Fuzzy Sets Syst., Vol. 8, pp. 225-233. Zbl0499.28009
  6. Forbus K.D. (1984): Qualitative process theory. - Artif. Intell., Vol. 24, pp. 85-168. 
  7. 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. 
  8. Kuipers B. (1984): Commonsense reasoning about causality: deriving behavior from structure. - Artif. Intell., Vol. 24, pp. 169-203. 
  9. Ma M., Friedman M. and Kandel A. (1999): Numerical solutions of fuzzy differential equations. - Fuzzy Sets Syst., Vol. 105, pp. 133-138. Zbl0939.65086
  10. Nieto J.J. (1999): The Cauchy problem for continuous fuzzy differential equations. - Fuzzy Sets and Syst., Vol. 102, pp. 259-262. Zbl0929.34005
  11. Park J.K. and Han H.K. (2000): Fuzzy differential equations. - Fuzzy Sets Syst., Vol. 110, pp. 69-77. 
  12. 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. Zbl0946.34054
  13. Vorobiev D. and Seikkala S. (2002): Towards the theory of fuzzy differential equations. - Fuzzy Sets Syst., Vol. 125, pp. 231-237. Zbl1003.34046
  14. Zadeh L.A. (1966): The shadows of fuzzy sets. - Problems of Information Transmission, Vol. 2, pp. 37-44 (in Russian). Zbl0263.02028
  15. 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. Zbl0397.68071
  16. 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. Zbl0988.03040
  17. 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. Zbl0954.68513

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.