Document Type : Research Paper

**Authors**

Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.

**Abstract**

In this paper, the numerical solution of an algebraic complex fuzzy equation of degree n, based on the parametric fuzzy numbers, is discussed. The unknown variable and right-hand side of the equation are considered as fuzzy complex numbers, whereas, the coefficients of the equation, are considered to be real crisp numbers. The given method is a numerical method and proposed based on the separation of the real and imaginary parts of the equation and using the parametric forms of the fuzzy numbers in the form of polynomials of degree at most m. In this case, a system of nonlinear equations is achieved. To get the solutions of the system, we used the Gauss-Newton iterative method. We also very briefly explain the conjugate of the solution of such equations. Finally, the efficiency and quality of the given method are tested by applying it to some numerical examples.

**Keywords**

- [1] S. Abbasband and M. Amirfakhrian, The nearest approximation of a fuzzy quantity in parametric form, App. Math. Comput., 172 (2006), 624–632.
- [2] S. Abbasband and M. Amirfakhrian, Numerical approximation of fuzzy functions by fuzzy polynomials, App. Math. Comput., 174 (2006), 669–675.
- [3] S. Abbasbandy and B. Asady, Newton,s method for solving fuzzy nonlinear equations, App. Math. Comput., 159 (2004), 349–356.
- [4] S. Abbasbandy and M. Otadi, Numerical solution of fuzzy polynomials by fuzzy neural network, App. Math. Comput., 181 (2006), 1084–1089.
- [5] T. Allahviranloo, M. Otadi, and M. Moslem, Iterative method for fuzzy equations, Soft Comput., 12 (2007) 935–939.
- [6] T. Allahviranloo and S. Asari, Numerical solution of fuzzy polynomials by Newton-Rephson method, J. App. Math., 7 (2011), 17–24.
- [7] T. Allahviranloo and L. G. Moazam, The solution of fuzzy quadratic equation based on optimization theory, Sci. World J., 2014 (2014) 1–6.
- [8] M. Amirfahkrian, Numerical solution of algebraic fuzzy equations with crisp variable by Gauss- Newton method, Appl. Math. Model., 32 (2008), 1859–1868.
- [9] M. Amirfahkrian, An iterative Gauss- Newton method to solve an algebraic fuzzy equation with crisp coeﬃcients, J. Intell Fuzzy Syst., 22 (2011) 207– 216.
- [10] M. Amirfahkrian, Some approximation methods in fuzzy logic, Lambert Academic Publishing, (2012).
- [11] J. J. Buckley, The fuzzy mathematics of finance, Fuzzy Sets Syst., 21 (1987), 257–273.
- [12] J. J. Buckley, Solving fuzzy equations in economics and finance, Fuzzy Sets Syst., 48 (1992), 289–296.
- [13] J. J. Buckley and Y. Qu, Solving systems of linear fuzzy equations, Fuzzy Sets Syst., 43 (1991), 33–43.
- [14] J. J. Buckley and Y. Qu, On using α-cuts to evaluate fuzzy equations, Fuzzy Sets Syst., 38 (1990) 309–312.
- [15] J. J. Buckley and Y. Qu, Solving fuzzy equations: A new Solution Concept, Fuzzy Sets Syst., 39 (1991) 291–301.
- [16] J. J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets Syst., 38 (1990), 43–59.
- [17] J. Chen and W. Li, Local convergence results of Gauss-Newton,s like method in weak conditions, J. Math. Anal. Appl., 2 (2006), 1381–1394.
- [18] J. Chen and W. Li, Convergence of Gauss-Newton,s method and uniqueness of the solution, Appl. Math. Comput., 1 (2005), 686–705.
- [19] S. H. Chen and X. W. Yang, Interval finite element method for beam structhures, Finite Elem. Anal. Des., 34 (2000), 75–88.
- [20] P. Deuflhard, Newton methods for nonlinear problems, Springer Seris in Computational Mathematics, Springer- Verlag Berlin Heidelberg, Berline, 2004. .
- [21] P. Deuflhard and A. Hohmann, Numerical analysis in modern scientific computing, Texts in Applied Mathematics, 2nd edition, Springer-Verlag New york, New York, 2003. .
- [22] X. Fu and Q. Shen, Fuzzy complex numbers and their application for classifires performance evaluation, Pattern Recogni., 7 (2011), 1403–1417.
- [23] M. Friedman, M. Ming, and A. Kndel, Fuzzy linear systems, Fuzzy Set. Syst., 2 (1998), 201–209.
- [24] R. Goestschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Set. Syst., 1 (1989), 31–43.
- [25] G. J. Klir and B. Yuan, Fuzzy sets and fuzzy logic theory and applications, Prentice Hall of India Pvt. Ltd., Delhi, 2002.
- [26] C. Li, W. H. Zhang, and X. Q. Jin, Convergence and uniqueness properties of Gauss-Newton,s method, Comput. Math. Appl., 47 (2004), 1057–1067.
- [27] M. Li Calzi, Towards a general setting for the fuzzy mathematics of finance, Fuzzy Set. Syst, 35 (1990), 265–280..
- [28] Q. X. Li and S. F. Liu, The foundation of the grey matrix and the grey input-output analysis, Appl. Math. Model., 32 (2008), 267– 291.
- [29] F. A. Mazarbhuiya, A. K. Mahanta, and H. K. Baruah, Solution of fuzzy equation A + X = B using the method of superimposition, Appl. Math., 2 (2011), 1039–1045.
- [30] M. Otadi and M. Mosleh, Solution of fuzzy polynomial equations by modified Adomian Decomposition method, Soft Comput., 15 (2011), 187–192. .
- [31] L. G. Moazam, The solution of fuzzy quadratic equations based on restricted varation, Int. J. Ind. Math., 8 (2016), 395–400.
- [32] M. Ma, M. Friedman, and A. Kandel, A new fuzzy arithmetic, fuzzy set. Syst., 108 (1999), 83–90.
- [33] H. Rouhparvar,Solving fuzzyp polynomial equation by ranking method, First Joint Congress On Fuzzy and Intel- ligent Systems Ferdowsi University of Mashhad, (2007), 1–5.
- [34] L. Stefanini, L. Sorini, and M. L. Guerra, Parametric representation of fuzzy numbers and applications to calculus, Fuzzy set. syst., 157 (2006), 2423–2455.
- [35] C. C. Wu and N. B. Chang, Grey input-output analysis and its application for environmental cost allocation, Eur. J. of Oper. Res., 145 (2003), 175–201.

January 2022

Pages 77-92

**Receive Date:**12 November 2019**Revise Date:**15 January 2021**Accept Date:**18 January 2021