International Journal of Nonlinear Analysis and Applications

New representations of the generalized uniform fuzzy partitions: Generalized normal case

Document Type : Research Paper

Author

1 The Hashemite Kingdom of Jordan Ministry of Education, Amman 11118, Jordan

2 Arab Open University – Jordan Branch, Faculty of Computer Studies, Amman, Jordan

Abstract

In this research, new representations of basic functions are proposed based on the new types of fuzzy partition and a subnormal generating function. The generalized uniform fuzzy partitions in subnormal case, i.e. in case a generating function K is not normal (generalized normal case), and simpler form of fuzzy transform (FzT) components based on these new representations of the generalized uniform fuzzy partitions are indicated. The main properties of a new uniform fuzzy partition are suggested. New theorems and lemmas are proved.

Keywords

[1] Z. Alijani, A. Khastan, S. K. Khattri, and S. Tomasiello, Fuzzy transform to approximate solution of boundary value problems via optimal coefficients, 2017 International Conference on High-Performance Computing Simulation (HPCS) (Genoa, Italy), 2017, pp. 466–471.
[2] H.A. Alkasasbeh, I. Perfilieva, M. Zaini Ahmad, and Z. Ridzuan Yahya, New approximation methods based on fuzzy transform for solving SODEs: I, Appl. Syst. Innov. 1 (2018), 29.
[3] H.A. Alkasasbeh, I. Perfilieva, M. Zaini Ahmad, and Z. Ridzuan Yahya, New approximation methods based on fuzzy transform for solving SODEs: II, Appl. Syst. Innov. 1 (2018), 30.
[4] H.A. Alkasasbeh, I. Perfilieva, M. Zaini Ahmad, and Z. Ridzuan Yahya, New fuzzy numerical methods for solving Cauchy problems, Appl. Syst. Innov. 1 (2018), 15.
[5] D. Baleanu, B. Agheli, M. Adabitabar Firozja, and M. Mohamed Al Qurashi, A method for solving nonlinear Volterra’s population growth model of noninteger order, Adv. Differ. Equ. 2017 (2017), no. 1, 368.
[6] B. Bede and I.J. Rudas, Approximation properties of fuzzy transforms, Fuzzy Sets Syst. 180 (2011), no. 1, 20–40.
[7] J.L. Castro and M. Delgado, Fuzzy systems with defuzzification are universal approximators, IEEE Trans. Syst. Man Cybernet. Part B (Cybernetics) 26 (1996), no. 1, 149–152.
[8] W. Chen and Y. Shen, Approximate solution for a class of second-order ordinary differential equations by the fuzzy transform, J. Intell. Fuzzy Syst. 27 (2014), no. 1, 73–82.
[9] R. Ezzati, F. Mokhtari, and M. Maghasedi, Numerical solution of Volterra-Fredholm integral equations with the help of inverse and direct discrete fuzzy transforms and collocation technique, Int. J. Ind. Math. 4 (2012), no. 3, 221–229.
[10] R. Ghosh, S. Chowdhury, G.C. Gorain, and S. Kar, Uniform stabilization of the telegraph equation with a support by fuzzy transform method, QSci. Connect 2014 (2014), no. 1, 19.
[11] P. Hodakova and I. Perfilieva, F1-transform of functions of two variables., EUSFLAT 2013 (Milan, Italy), Atlantis Press, 2013, pp. 547–553.
[12] M. Holcapek, I. Perfilieva, V. Novak, and V. Kreinovich, Necessary and sufficient conditions for generalized uniform fuzzy partitions, Fuzzy Sets Syst. 277 (2015), 97–121.
[13] M. Holcapek and T. Tichy, A smoothing filter based on fuzzy transform, Fuzzy Sets Syst. 180 (2011), no. 1, 69–97.
[14] M. Holcapek and R. Valasek, Numerical solution of partial differential equations with the help of fuzzy transform technique, 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (Naples, Italy), 2017, pp. 1–6.
[15] P. Hurtik and I. Perfilieva, Image compression methodology based on fuzzy transform, Int. Joint Conf. CISIS’12-ICEUTE´12-SOCO´12 Special Sessions (Berlin/Heidelberg, Germany), Springer, 2013, pp. 525–532.
[16] S. Jahedi, M.J. Mehdipour, and R. Rafizadeh, Approximation of integrable function based on o-transform, Soft Comput 18(10) (2013), 2015–2022.
[17] A. Khastan, A new representation for inverse fuzzy transform and its application, Soft Comput. 21 (2017), no. 13, 3503–3512.
[18] A. Khastan, Z. Alijani, and I. Perfilieva, Fuzzy transform to approximate solution of two-point boundary value problems, Math. Meth. Appl. Sci. 40 (2017), no. 17, 6147–6154.
[19] A. Khastan, I. Perfilieva, and Z. Alijani, A new fuzzy approximation method to Cauchy problems by fuzzy transform, Fuzzy Sets Syst. 288 (2016), 75–95.
[20] M. Kokainis and S. Asmuss, Higher degree F-transforms based on B-splines of two variables, Information Processing and Management of Uncertainty in Knowledge-Based Systems (Cham) (Joao Paulo Carvalho, Marie-Jeanne Lesot, Uzay Kaymak, Susana Vieira, Bernadette Bouchon-Meunier, and Ronald R. Yager, eds.), Springer International Publishing, 2016, pp. 648–659.
[21] E. H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. Inst. Elect. Eng. 121 (1974), no. 12, 1585–1588.
[22] G. Patane, Fuzzy transform and least-squares approximation: Analogies, differences, and generalizations, Fuzzy Sets Syst. 180 (2011), no. 1, 41–54.
[23] I. Perfilieva, Fuzzy transform: Application to the reef growth problem, Fuzzy Logic Geo. (Robert V. Demicco and George J. Klir, eds.), Academic Press, Amsterdam, 2003, pp. 275 – 300.
[24] I. Perfilieva, Fuzzy transforms: Theory and applications, Fuzzy Sets Syst. 157 (2006), no. 8, 993–1023.
[25] I. Perfilieva, F-transform versus Takagi-Sugeno models, NAFIPS Ann. Meet. North Amer. Fuzzy Inf. Process. Soc. (Cincinnati, OH, USA), IEEE, 2009, pp. 1–4.
[26] I. Perfilieva, F-transform, ch. 7, pp. 113–130, Springer, Berlin/Heidelberg, Germany, 2015.
[27] I. Perfilieva, M. Dankova, and B. Bede, Towards a higher degree F-transform, Fuzzy Sets Syst. 180 (2011), no. 1, 3–19.
[28] I. Perfilieva and E. Haldeeva, Fuzzy transformation, Proc. Joint 9th IFSA World Cong. 20th NAFIPS Int. Conf. (Vancouver, BC, Canada), vol. 4, IEEE, 2001, pp. 1946–1948.
[29] I. Perfilieva, P. Hodakova, and P. Hurtık, Differentiation by the F-transform and application to edge detection, Fuzzy Sets Syst. 288 (2016), 96–114.
[30] I. Perfilieva, M. Holcapek, and V. Kreinovich, A new reconstruction from the F-transform components, Fuzzy Sets Syst, 288 (2016), 3–25.
[31] E.H. Ruspini, A new approach to clustering, Inf. Control 15 (1969), no. 1, 22–32.
[32] L. Stefanini, F-transform with parametric generalized fuzzy partitions, Fuzzy Sets Syst. 180 (2011), no. 1, 98–120.
[33] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybernet. SMC-15 (1985), no. 1, 116–132.
[34] S. Tomasiello, An alternative use of fuzzy transform with application to a class of delay differential equations, Int. J. Comput. Math. 94 (2017), no. 9, 1719–1726.
[35] S. Tomasiello, A first investigation on the dynamics of two delayed neurons through fuzzy transform approximation, Int. Conf. High Perform. Comput. Simul. (HPCS) (Genoa, Italy), 2017, pp. 460–465.
[36] S. Tomasiello, M. Gaeta, and V. Loia, Quasi–consensus in second–order multi–agent systems with sampled data through fuzzy transform, J. Uncertain Syst. 10 (2016), no. 4, 243–250.
[37] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965), no. 3, 338–353.
[38] M. Zeinali, R. Alikhani, S. Shahmorad, F. Bahrami, and I. Perfilieva, On the structural properties of Fm-transform with applications, Fuzzy Sets and Systems 342 (2018), 32–52.
[39] Sh. Ziari and I. Perfilieva, On the approximation properties of fuzzy transform, J. Intell. Fuzzy Syst. 33 (2017), 171–180.
International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 2
February 2025
Pages 245-253
  • Receive Date: 08 May 2022
  • Accept Date: 15 October 2023