On the structural properties for the cross product of fuzzy numbers with applications

Subject Areas : Statistics

Robab Alikhani ^{1 *}

1 - Applied Mathematics-Faculty of Mathematics Sciences-University of Tabriz-Tabriz

Received: 2018-11-26 Accepted : 2019-10-25 Published : 2020-04-01

Keywords: ضرایب متغییر فازی, ضرب خارجی, اعداد فازی مثلثی, مشتق‌پذیری تعمیم یافته, معادلات دیفرانسیل فازی خطی,

Abstract :

In the fuzzy arithmetic, the definitions of addition and multiplication of fuzzy numbers are based on Zadeh’s extension principle. From theoretical and practical points of view, this multiplication of fuzzy numbers owns several unnatural properties. Recently, to avoid this shortcoming, a new multiplicative operation of product type is introduced, the so-called cross-product of fuzzy numbers. The main advantage is that this product preserves the shape of triangular or trapezoidal fuzzy numbers under multiplication and from computational point of view the cross product is more applicable than the usual product. The above mentioned properties motivate us to use the cross product in applications as a possible alternative of the product obtained by Zadeh's extension Principle. The aim of the present paper is to give an explicit formula for the cross product of triangular fuzzy numbers based on the scalar product of fuzzy numbers and then, explicit formulas for the length of cross product of triangular fuzzy numbers and fuzzy derivative of cross product of triangular fuzzy functions. As an application, we apply the cross product concept for the first order linear fuzzy differential equations with fuzzy variable coefficients and obtain its triangular solutions under generalized differentiability. Finally, some examples are given to illustrate the theoretical results.

References:

[1] L. A. Zadeh. Fuzzy sets. Information and Control 8:338-353 (1965)

[2] B. Bede, J. Fodor. Product Type Operations between Fuzzy Numbers and their Applications in Geology. Acta Polytechnica Hungarica 3:123–139 (2006)                                                                        

[3] B. Bede. Mathematics of fuzzy sets and fuzzy logic. Springer, London (2013)

 

[4] R. Alikhani.  Interval fractional integrodifferential equations without singular kernel by fixed point in partially ordered sets. Computational Methods for Differential Equations 5: 12-29 (2017)

 

[5] B. Bede, S. G. Gal. Generalizations of the differetiability of fuzzy-number-valued functions with applications to fuzzy differential equation. Fuzzy Sets and Systems 151:581- 599 (2005)

 

[6] O. Kaleva. Fuzzy differential equations. Fuzzy Sets and Systems 24: 301-317(1987)

 

[7] Y. Chalco-Cano, H. Romn-Flores, On new solutions of fuzzy differential equations. Chaos, Solitons and Fractals 38: 112-119 (2008)

 

[8] R. Alikhani, F. Bahrami.  Fuzzy partial differential equations under the cross product of fuzzy Numbers. Information Sciences 494:80-99 (2019)

 

[9] T. Allahviranloo. A method for solving nth order fuzzy linear differential equations. International Journal of Computer Mathematics 89: 730-742 (2009)

 

[10] R. Alikhani, F. Bahrami, S. Parvizi. Differential calculus of fuzzy multi-variable functions and its applications to fuzzy partial differential equations. Fuzzy Sets and Systems 375:100-120 (2019)

 

[11] J. J. Buckley, T. Feuring. Fuzzy differential equations. Fuzzy sets and Systems 110:43-54 (2000)

 

[12] M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Román-Flores, R. C. Bassanezi. Fuzzy differential equations and the extension principle. Information Sciences 177:3627-3635 (2007)

 

[13] F. Bahrami, R. Alikhani, A. Khastan. Transport equation with fuzzy data. Iranian Journal of Fuzzy Systems 15: 67-78 (2018)

 

[14] N. Gasilov, A. G. Fatullayev, S. E. Amrahov,  A. Khastan, A new approach to fuzzy initial value problem. Soft Computing 18:217– 225 (2014)

 

[15] N. Gasilov, S. E. Amrahov, A. G. Fatullayev. Solution of linear differential equations with fuzzy boundary values. Fuzzy Sets and Systems 257:169–183 (2014)

 

[16] B. Bede, I. J. Rudas, A. L. Bencsik. First order linear fuzzy differential equations under generalized differentiability. Information Sciences 177:1648–1662 (2007)

 

[17] L. Jamshidi, T. Allahviranloo. Solution of the first order fuzzy differential equations with generalized differentiability. Journal of Linear and Topological Algebra. 3:159-171 (2014)

 

[18] A. Khastan, J. J. Nieto and R. R. Lopez. Variation of constant formula for first order fuzzy differential equations. Fuzzy Sets and Systems. 177:20-33 (2011)

 

[19] D. Vivek, K. Kanagarajan, S. Harikrishnan. Numerical solution of first-order fully fuzzy differential equations by Runge-Kutta Fehlberg method under strongly generalized H-differentiability. Journal of Soft Computing and Applications 2017:1-23 (2017)

 

[20] P. Darabi, S. Moloudzadeh, H. Khandani. A numerical method for solving first-order fully fuzzy differential equation under strongly generalized H-differentiability. Soft Computing. 20:4085-4098 (2016)

 

[21] M. Chehlabi, T. Allahviranloo.  Positive or negative solutions to first-order fully fuzzy linear differential equations under generalized differentiability. Applied Soft Computing 70:359-370 (2018)