Fuzzy product rule with applications

Document Type : Research Paper


Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran


In this paper, we establish the GH-derivative of multiplication of fuzzy functions, the so-called product rule. For the first time, a product rule is constructed while both of the multiplied functions are assumed to be fuzzy without any restriction on the signs of multiplied functions. The rule is extracted based on the MCE-product and its property of distributivity. Then, we propose two important applications of the fuzzy product rule: An integration by parts formula for fuzzy functions and solving a nonlinear fuzzy differential equation. Some illustrative examples are given to verify the theoretical results.


[1] R. Alikhani, F. Bahrami, Fuzzy partial differential equations under the cross product of fuzzy numbers, Information Sciences, 494 (2019), 80-99.
[2] A. Armand, T. Allahviranloo, Z. Gouyandeh, Some fundamental results on fuzzy calculus, Iranian Journal of Fuzzy Systems, 15(3) (2018), 27-46.
[3] A. Ban, B. Bede, Properties of the cross product of fuzzy numbers, Journal of Fuzzy Mathematics, 14(3) (2005), 513.
[4] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, 2013.
[5] B. Bede, J. Fodor, Product type operations between fuzzy numbers and their applications in geology, Acta Polytechnica Hungarica, 3(1) (2006), 123-139.
[6] B. Bede, S. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151(3) (2005), 581-599.
[7] B. Bede, I. Rudas, A. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences, 177(7) (2007), 1648-1662.
[8] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230(1) (2013), 119-141.
[9] A. M. Bica, D. Fechete, I. Fechete, Towards the properties of fuzzy multiplication for fuzzy numbers, Kybernetika, 55(1) (2019), 44-62.
[10] C. Carlsson, R. Full´er, On additions of interactive fuzzy numbers, Proceedings of the Fifth International Symposium of Hungarian Researchers on Computational Intelligence, Budapest, (2004), 227-238.
[11] Y. Chalco-Cano, R. Rodríguez-López, M. D. Jiménez-Gamero, Characterizations of generalized differentiable fuzzy functions, Fuzzy Sets and Systems, 295 (2016), 37-56.
[12] Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores, M. D. Jiménez-Gamero, Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and Systems, 219 (2013), 49-67.
[13] S. S. Chang, L. A. Zadeh, On fuzzy mapping and control. In fuzzy sets, fuzzy logic, and fuzzy systems: Selected papers by Lotfi A Zadeh, World Scientific, (1996), 180-184.
[14] D. Dubois, H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9(6) (1978), 613-626.
[15] E. Esmi, L. C. Barros, V. F. Wasques, Some notes on the addition of interactive fuzzy numbers, International Fuzzy Systems Association World Congress, Springer, Cham, Jun 2019.
[16] R. Fullér, T. Keresztfalvi, t-norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 51(2) (1992), 155-159.
[17] J. R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1) (1986), 31-43.
[18] D. H. Hong, On shape-preserving additions of fuzzy intervals, Journal of Mathematical Analysis and Applications, 267(1) (2002), 369-376.
[19] D. H. Hong, S. Y. Hwang, The convergence of T-product of fuzzy numbers, Fuzzy Sets and Systems, 85(3) (1997), 373-378.
[20] M. Ma, M. Friedman, A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems, 108(1) (1999), 83-90.
[21] M. Mizumoto, K. Tanaka, The four operations of arithmetic on fuzzy numbers, Computer Control Systems, 7(5) (1976), 73-81.
[22] H. T. Nguyen, A note on the extension principle for fuzzy sets, Journal of Mathematical Analysis and Applications, 64(2) (1978), 369-380.
[23] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems, 161(11) (2010), 1564-1584.
[24] V. F. Wasques, E. Esmi, L. C. Barros, P. Sussner, The generalized fuzzy derivative is interactive, Information Sciences, 519 (2020), 93-109.
[25] L. A. Zadeh. Fuzzy sets, Information and Control, 8(3) (1965), 338-353.
[26] M. Zeinali, The existence result of a fuzzy implicit integro-differential equation in semilinear Banach space, Computational Methods for Differential Equations, 5(3) (2017), 232-245.
[27] M. Zeinali, G. Eslami, Uncertainty analysis of temperature distribution in a thermal fin using the concept of fuzzy derivative, Journal of Mechanical Engineering, 51(4) (2022), 527-536.
[28] M. Zeinali, S. Shahmorad, An equivalence lemma for a class of fuzzy implicit integro-differential equations, Journal of Computational and Applied Mathematics, 327 (2018), 388-399.
[29] M. Zeinali, S. Shahmorad, K. Mirnia, Fuzzy integro-differential equations: Discrete solution and error estimation, Iranian Journal of Fuzzy Systems, 10(1) (2013), 107-122.
[30] D. Zhang, W. Feng, Y. Zhao, J. Qiu, Global existence of solutions for fuzzy second-order differential equations under generalized H-differentiability, Computers and Mathematics with Applications, 60(6) (2010), 1548-1556.