Numerical solutions of fuzzy differential and integral equations

Fuzzy Sets and Systems

Volumn 106, Issue 1, 1999, Pages 35-48

  Friedman, Menahem ^a,c   Ma, Ming ^a,d   Kandel, Abraham ^b,e  

^a University of South Florida   (United States)

^b TEL AVIV UNIVERSITY   (Israel)

^c NUCLEAR RESEARCH CENTER NEGEV   (Israel)

^d HARBIN INSTITUTE OF TECHNOLOGY   (China)

^e UNIVERSITY OF SOUTH FLORIDA   (United States)

Author keywords

Fuzzy differential equations; Fuzzy dynamical systems; Fuzzy integral equations; Fuzzy numbers; Numerical methods

Indexed keywords

EID: 0009933836     PISSN: 01650114     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0165-0114(98)00355-8     Document Type: Article

References (94)

1. [1] J.P. Aubin, Fuzzy differential inclusions, Problems of Control and Information Theory 19 (1990) 55-57.
2. [2] J.P. Aubin, A survey of viability theory, SIAM J. Control Optim. 28 (1990) 749-788.
3. [3] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12.
4. [4] R. Badard, Fixed point theorems for fuzzy numbers, Fuzzy Sets and Systems 13 (1984) 291-302.
5. [5] R. Badard, Comparison of topological and uniform structures for fuzzy number and the fixed point problem, Fuzzy Sets and Systems 21 (1987) 211-220.
6. [6] V.A. Baidosov, Fuzzy differential inclusions, Prikl. Math. Mekh., USSR 54 (1990) 8-13.
7. [7] J.J. Buckley, Fuzzy eigenvalues and input-output analysis, Fuzzy Sets and Systems 34 (1990) 187-195.
8. [8] J.J. Buckley, Solving fuzzy equations, Fuzzy Sets and Systems 50 (1992) 1-14.
9. [9] D. Butnariu, Fixed points for fuzzy mapping, Fuzzy Sets and Systems 7 (1982) 191-207.
10. [10] D. Butnariu, Measurability concepts for fuzzy mappings, Fuzzy Sets and Systems 31 (1989) 77-82.
11. [11] Byung Soo Moon, A tuning algorithm for the fuzzy logic controllers, EUFIT '95 620-624.
12. [12] S.S.L. Chang, L. Zadeh, On fuzzy mapping and control, IEEE Trans. System Man Cybernet. 2 (1972) 30-34.
13. [13] G. Colombo, V. Krivan, Fuzzy differential inclusions and non-probabilistic likelihood, Dyn-Systems Appl. 1 (1992) 419-439.
14. [14] Z. Ding, A. Kandel, Existence and stability of fuzzy differential equations, J. Fuzzy Math. 5 (1997) 681-697.
15. [15] Z. Ding, M. Ma, A. Kandel, On the observability of fuzzy dynamic control system (I), Fuzzy Sets and Systems, to appear.
16. [16] Z. Ding, A. Kandel, On the observability of fuzzy dynamic control system (II), Fuzzy Sets and Systems, to appear.
17. [17] P. Diamond, Fuzzy least squares, Inform. Sci. 46 (1988) 141-157.
18. [18] P. Diamond, P.E. Kloeden, Characterization of compact sets of fuzzy sets, Fuzzy Sets and Systems 29 (1989) 341-348.
19. [19] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35 (1990) 241-249.
20. [20] P. Diamond, P. Kloeden, Metric spaces and fuzzy sets, Corrigendum, Fuzzy Sets and Systems 45 (1992) 123.
21. [21] J. Dieudonué, Eléments Dánalyse, Academic Press, New York, 1960, p. 136.
22. [22] D. Dubois, H. Prade, Operations on fuzzy numbers, J. Systems Sci. 9 (1978) 613-626.
23. [23] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.
24. [24] D. Dubois, H. Prade, Towards fuzzy differential calculus, Fuzzy Sets and Systems 8 (1982) 1-7, 105-116, 225-233.
25. [25] D. Dubois, H. Prade, Fuzzy number: an overview, in: The Analysis of Fuzzy Information, vol. 1, Mathematics, J.C. Bezdek (Ed.), CRC Press, Boca Raton, FL, 1987, pp. 3-39.
26. [26] M. Ferraro, D. Foster, Differentiation of fuzzy continuous mapping on fuzzy topological vector spaces, J. Math. Anal. Appl. 121 (1987) 589-601.
27. [27] M. Ferraro, D. Foster, Fuzzy manifolds, Fuzzy Sets and Systems 54 (1993) 99-103.
28. [28] M. Friedman, A. Kandel, On the design of a fuzzy intelligent differential equation solver, in: A. Kandel (Ed.), Fuzzy Expert Systems, CRC Press, Boca Raton, 1992, pp. 203-212.
29. [29] M. Friedman, A. Kandel, Fundamentals of Computer Numerical Analysis, CRC Press, Boca Raton, FL, 1994, pp. 307-351.
30. [30] M. Friedman, Ma Ming, A. Kandel, Numerical methods for calculating the fuzzy integral, Fuzzy Sets and Systems 83 (1996) 57-62.
31. [31] M. Friedman, Ma Ming, A. Kandel, On the validity of Peano theorem for fuzzy differential equations, Fuzzy Sets and Systems 86 (1997) 331-334.
32. [32] G. Gerla, A preservation theorem for fuzzy number theory, Inform. Sci. 45 (1988) 51-59.
33. [33] A.N. Godunov, Peano's theorem in Banach spaces, Funct. Anal. Appl. 9 (1975) 53-55.
34. [34] R. Goetschel, W. Voxman, A pseudometric for fuzzy sets and certain related result, J. Math. Anal. Appl. 81 (1981) 507-523.
35. [35] R. Goetschel, W. Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems 10 (1983) 87-99.
36. [36] R. Goetschel, W. Voxman, Eigen fuzzy number sets, Fuzzy Sets and Systems 16 (1985) 75-85.
37. [37] R. Goetschel, W. Voxman, Elementary calculus, Fuzzy Sets and Systems 18 (1986) 31-43.
38. [38] B. Harman, Sum and product of the modified real fuzzy numbers, Kybernetika Suppl. 28 (1992) 34-40.
39. [39] B. Heshmaty, A. Kandel, Fuzzy linear regression and its applications to forecasting in uncertain environments, Fuzzy Sets and Systems 15 (1985) 159-191.
40. [40] H. Hochstadt, Integral Equations, Wiley, New York, 1973, pp. 1-24.
41. [41] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317.
42. [42] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems 35 (1990) 389-396.
43. [43] O. Kaleva, The calculus of fuzzy valued functions, Appl. Math. Lett. 3 (1990) 55-59.
44. [44] A. Kandel, Fuzzy statistics and forecast evaluation, IEEE Trans. Systems Man Cybernet, 8 (1978) 396-401.
45. [45] A. Kandel, W.J. Byatt, Fuzzy differential equations, Proc. Int. Conf. Cybernetics and Society, Tokyo, November 1978, pp. 1213-1216.
46. [46] A. Kandel, W.J. Byatt, Fuzzy sets, fuzzy algebra and fuzzy statistics, Proc. IEEE 66 (1978) 1619-1639.
47. [47] A. Kandel, W.J. Byatt, Fuzzy processes, Fuzzy Sets and Systems 4 (1980) 117-152.
48. [48] A Kandel, Fuzzy dynamical systems and the nature of their solutions, in: P.P. Wang, S.K. Chang (Eds.), Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, pp. 93-122.
49. [49] A. Kandel, Fuzzy Techniques in Pattern Recognition, Wiley, New York, 1982, pp. 64-90.
50. [50] A. Kandel, Fuzzy Mathematical Techniques with Applications, Addison-Wesley, New York, 1986.
51. [51] K. Kirkiakidis, A. Tzez, Fuzzy logic adaptive sliding control design with an application to air-handling systems, EUFIT '95, pp. 954-959.
52. [52] E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables, Proc. Roy. Soc. London Ser. A. 407 (1986) 171-182.
53. [53] P. Kloeden, Remarks on Peano-like theorems for fuzzy differential equations, Fuzzy Sets and Systems 44 (1991) 161-164.
54. [54] R.P. Leland, Fuzzy differential systems and Malliavin calculus, Fuzzy Sets and Systems 70 (1995) 59-73.
55. p) Fuzzy Sets and Systems 56 (1993) 297-301.
56. [56] Ma Ming, On embedding problem of fuzzy number spaces: parts 4, 5, Fuzzy Sets and Systems 55 (1993) 313-318;
57. [56] Ma Ming, On embedding problem of fuzzy number spaces: parts 4, 5, Fuzzy Sets and Systems 55 (1993) 313-318; 58 (1993) 185-193. 58 (1993) 185-193.
58. [57] Ma Ming, M. Friedman, A. Kandel, Numerical solution of fuzzy differential equations, Fuzzy Sets and Systems, to appear.
59. [58] M. Matloka, On fuzzy integrals, Proc. 2nd Polish Symp. on Interval and Fuzzy Mathematics, Politechnika Poznansk, 1987, pp. 167-170.
60. [59] M. Mizumoto, K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems-Comput.-Controls 7 (5) (1976) 73-81.
61. [60] M. Mizumoto, K. Tanaka, Some properties of fuzzy numbers, in: M.M. Gupta, R.K. Ragade, R.R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam, 1979, pp. 156-164.
62. [61] S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems 1 (1978) 97-111.
63. [62] S. Nanda, On integration of fuzzy mappings, Fuzzy Sets and Systems 32 (1989) 95-101.
64. [63] C.V. Negoita, D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975.
65. [64] H.T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl. 64 (1978) 369-380.
66. [65] Ouyang He, Topological properties of the space of regular fuzzy sets, J. Math. Anal. Appl. 129 (1988) 346-361.
67. [66] Ouyang He, Wu Yi, On fuzzy differential equations, Fuzzy Sets and Systems 32 (1989) 321-325.
68. [67] R. Palm, D. Driankov, Fuzzy inputs, Fuzzy Sets and Systems 70 (1995) 315-335.
69. [68] J.Y. Park, Y.C. Kwan, J.V. Jeong, Existence of solutions of fuzzy integral equations in Banach spaces, Fuzzy Sets and Systems 72 (1995) 373-378.
70. [69] M.L. Puri, D. Ralescu, Differential for fuzzy function, J. Math. Anal. Appl. 91 (1983) 552-558.
71. [70] M.L. Puri, D, Ralescu, The concept of normality for fuzzy random variables, Ann. Probab. 13 (1985) 1373-1379.
72. [71] M.L. Puri, D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409-422.
73. [72] H.A. Radstrom, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165-169.
74. [73] D. Ralescu, A survey of the representation of fuzzy concepts and its applications, in: M.M. Gupta, R.K. Ragade, R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam, 1979, p. 77.
75. [74] D. Ralescu, Toward a general theory of fuzzy variables, J. Math. Anal. Appl. 86 (1982) 176-193.
76. [75] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987) 319-330.
77. [76] M. Sugeno, Theory of fuzzy integrals and its application, Ph.D. Dissertation, Tokyo Institute of Technology, 1974.
78. [77] M. Stojakovic, Fuzzy random variables, expectations and martingales, J. Math. Anal. Appl. 184 (1994) 594-606.
79. [78] H. Tanaka et al., Linear regression analysis with fuzzy model, IEEE Trans. Systems Man Cybernet SMC-14 (1984) 325-328.
80. [79] J. Vrba, A note on inverses in arithmetic with fuzzy numbers, Fuzzy Sets and Systems 50 (1992) 267-278.
81. [80] Wu Congxin, Ma Ming, An embedding operator of fuzzy numbers and its application for fuzzy integrals, System Sci. Math. Sci. 8 (1990) 193-199.
82. [81] Wu Congxin, Ma Ming, On the integrals, series and integral equations of fuzzy set-valued functions, J. Harbin Inst. Technol. 21 (1990) 11-19.
83. [82] Wu Congxin, Ma Ming, On embedding problem of fuzzy number spaces, Fuzzy Sets and Systems 44 (1991) 33-38; 46 (1992) 281-286.
84. [82] Wu Congxin, Ma Ming, On embedding problem of fuzzy number spaces, Fuzzy Sets and Systems 44 (1991) 33-38; 45 (1992) 189-202; 45 (1992) 189-202;
85. [82] Wu Congxin, Ma Ming, On embedding problem of fuzzy number spaces, Fuzzy Sets and Systems 44 (1991) 33-38; 45 (1992) 189-202; 46 (1992) 281-286. 46 (1992) 281-286.
86. 1, J. Fuzzy Systems Math. 2 (1988) 1:51-56 (in Chinese).
87. [84] Xue Xiaoping, Ha Minghu, Ma Ming, Random fuzzy number integrals in Banach spaces, Fuzzy Sets and Systems 66 (1994) 97-111.
88. [85] R.R. Yager, Fuzzy prediction based upon regression models, Inform. Sci. 26 (1982) 45-63.
89. [86] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353.
90. [87] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. 8 (1975) 199-249; 301-357;
91. [87] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. 8 (1975) 199-249; 301-357; 9 (1975) 43-80. 9 (1975) 43-80.
92. [88] L.A. Zadeh, Linguistic variables, approximate reasoning and disposition, Med. Inform. 8 (1983) 173-186.
93. [89] Zhao Renhong, R, Govind, Solutions of algebraic equations involving generalized fuzzy number, Inform. Sci. 56 (1991) 199-243.
94. [90] H.J. Zimmerman, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978) 45-55.