Title of paper:

Derivativefree Newton's method for solving intuitionistic fuzzy nonlinear equations with an application

Author(s):

A. O. Umar

Department of Mathematics, Federal University of Agriculture, Zuru, Kebbi, Nigeria

umarabdul64@gmail.com

M. Y. Waziri

Department of Mathematics, Bayero University, Kano, Nigeria

mywaziri@gmail.com

A. U. Moyi

Department of Mathematics, Federal University, Gusau, Nigeria

aliyumoyik@gmail.com


Published in:

Notes on Intuitionistic Fuzzy Sets, Volume 28 (2022), Number 2, pages 149–160

DOI:

https://doi.org/10.7546/nifs.2022.28.2.149160

Download:

PDF (550 Kb, Info)

Abstract:

In this paper, we present a derivativefree Newton’s method that avoids computing the derivative by generating an approximation of the derivative for the intuitionistic fuzzy nonlinear equation. We first consider transforming the intuitionistic fuzzy quantities into their equivalent membership and nonmembership parametric forms and insert the approximation from the forward difference method applied to [math]\displaystyle{ F'(x_k) = 0 }[/math] in Newton’s method to avoid computing the Jacobian matrix. Numerical experiments were carried out, which shows that the approach is a good option for computing Jacobian and is an efficient one.

Keywords:

Derivativefree, Intuitionistic fuzzy nonlinear equation, Parametric form, Zadeh’s fuzzy set.

AMS Classification:

03E72, 9404.

References:

 Abbasbandy, S., & Asady, B. (2004). Newton's method for solving fuzzy nonlinear equations. Applied Mathematics and Computation, 159(2), 349–356.
 Amma, B. B., Melliani, S., & Chadli, L. S. (2016). Numerical solution of intuitionistic fuzzy differential equations by Euler and Taylor methods. Notes on Intuitionistic Fuzzy Sets, 22(2), 71–86.
 Atanassov, K. T. (1983). Intuitionistic fuzzy sets. VII ITKR Session, Sofia, 2023 June 1983 (Deposed in Centr. Sci.Techn. Library of the Bulg. Acad. of Sci., 1697/84) (in Bulgarian). Reprinted: International Journal of Bioautomation, 2016, 20(S1), S1–S6.
 Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96.
 Atanassov, K. T. (1999). Intuitionistic Fuzzy Sets: Theory and Applications. Physica, Heidelberg.
 Biswas, S., Banerjee, S., & Roy, T. K. (2016). Solving intuitionistic fuzzy differential equations with linear differential operator by Adomian decomposition method. Notes on Intuitionistic Fuzzy Sets, 22(4), 25–41.
 Brown, K. M., & Dennis, J. E. (1971). Derivative free analogues of the Levenberg–Marquardt and Gauss algorithms for nonlinear leastsquares approximation. Numerische Mathematik, 18(4), 289–297.
 Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation, 19(92), 577–593.
 Buckley, J. J., & Qu, Y. (1990). Solving linear and quadratic fuzzy equations. Fuzzy Sets and Systems, 38(1), 43–59.160
 Buckley, J. J., and Qu, Y. (1991). Solving fuzzy equations: a new solution concept. Fuzzy Sets and Systems, 39(3), 291–301.
 Ettoussi, R., Melliani, S., Elomari, M., & Chadli, L. S. (2015). Solution of intuitionistic fuzzy differential equations by successive approximations method. Notes on Intuitionistic Fuzzy Sets, 21(2), 51–62.
 Fang, J. X. (2002). On nonlinear equations for fuzzy mappings in probabilistic normed spaces. Fuzzy Sets and Systems, 131(3), 357–364.
 Nehi, H. M., & Maleki, H. R. (2005, July). Intuitionistic fuzzy numbers and it’s applications in fuzzy optimization problem. In Proceedings of the 9th WSEAS International Conference on Systems (pp. 1–5). Athens, Greece: World Scientific and Engineering Academy and Society (WSEAS).
 Keyanpour, M., & Akbarian, T. (2014). Solving intuitionistic fuzzy nonlinear equations. Journal of Fuzzy Set Valued Analysis, 2014, 1–6.
 Omesa, U. A., Mamat, M., Sulaiman, I. M., & Sukono, S. (2020). On Quasi Newton method for solving fuzzy nonlinear equations. International Journal of Quantitative Research and Modeling, 1(1), 1–10.
 Omesa, A. U., Sulaiman, I. M., Mamat, M., Waziri, M. Y., Shadi, A., Zaini, M. A., & Sumiati, I. (2021, March). Derivative Free LevenbergMarquardt Method for Solving Fuzzy Nonlinear Equation. In IOP Conference Series: Materials Science and Engineering, 1115(1), 012002.
 Peeva, K. (1992). Fuzzy linear systems. Fuzzy Sets and Systems, 49(3), 339–355.
 Rafiq, N., Yaqoob, N., Kausar, N., Shams, M., Mir, N. A., Gaba, Y. U., & Khan, N. (2021). ComputerBased Fuzzy Numerical Method for Solving Engineering and RealWorld Applications. Mathematical Problems in Engineering, 2021, Article ID 6916282.
 Sulaiman, I. M., Mamat, M., Waziri, M. Y., Mohamed, M. A., & Mohamad, F. S. (2018). Solving Fuzzy Nonlinear Equation via LevenbergMarquardt Method. Far East Journal of Mathematical Sciences, 103(10), 1547–1558.
 Umar, A. O., Waziri, M. Y., & Sulaiman, I. M. (2018). Solving Dual Fuzzy Nonlinear Equations via a Modification of Shamanskii Steps. Malaysian Journal of Computing and Applied Mathematics, 1(2), 1–9.
 Waziri, M. Y., & Moyi, A. U. (2016). An alternative approach for solving dual fuzzy nonlinear equations. International Journal of Fuzzy Systems, 18(1), 103–107.
 Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353.
 Zadeh, L. A. (1996). Fuzzy sets. In Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh (pp. 394–432).
 Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning, Parts 1–3. Information Sciences, 8: 199–249, 301–357, and 9: 43–80.

Citations:

The list of publications, citing this article may be empty or incomplete. If you can provide relevant data, please, write on the talk page.

