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Issue:Derivative-free Newton's method for solving intuitionistic fuzzy nonlinear equations with an application

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Title of paper: Derivative-free Newton's method for solving intuitionistic fuzzy nonlinear equations with an application
A. O. Umar
Department of Mathematics, Federal University of Agriculture, Zuru, Kebbi, Nigeria
M. Y. Waziri
Department of Mathematics, Bayero University, Kano, Nigeria
A. U. Moyi
Department of Mathematics, Federal University, Gusau, Nigeria
Published in: Notes on Intuitionistic Fuzzy Sets, Volume 28 (2022), Number 2, pages 149–160
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Abstract: In this paper, we present a derivative-free 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 non-membership 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: Derivative-free, Intuitionistic fuzzy nonlinear equation, Parametric form, Zadeh’s fuzzy set.
AMS Classification: 03E72, 94-04.
  1. Abbasbandy, S., & Asady, B. (2004). Newton's method for solving fuzzy nonlinear equations. Applied Mathematics and Computation, 159(2), 349–356.
  2. 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.
  3. Atanassov, K. T. (1983). Intuitionistic fuzzy sets. VII ITKR Session, Sofia, 20-23 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.
  4. Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96.
  5. Atanassov, K. T. (1999). Intuitionistic Fuzzy Sets: Theory and Applications. Physica, Heidelberg.
  6. 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.
  7. Brown, K. M., & Dennis, J. E. (1971). Derivative free analogues of the Levenberg–Marquardt and Gauss algorithms for nonlinear least-squares approximation. Numerische Mathematik, 18(4), 289–297.
  8. Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation, 19(92), 577–593.
  9. Buckley, J. J., & Qu, Y. (1990). Solving linear and quadratic fuzzy equations. Fuzzy Sets and Systems, 38(1), 43–59.160
  10. Buckley, J. J., and Qu, Y. (1991). Solving fuzzy equations: a new solution concept. Fuzzy Sets and Systems, 39(3), 291–301.
  11. 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.
  12. Fang, J. X. (2002). On nonlinear equations for fuzzy mappings in probabilistic normed spaces. Fuzzy Sets and Systems, 131(3), 357–364.
  13. 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).
  14. Keyanpour, M., & Akbarian, T. (2014). Solving intuitionistic fuzzy nonlinear equations. Journal of Fuzzy Set Valued Analysis, 2014, 1–6.
  15. 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.
  16. Omesa, A. U., Sulaiman, I. M., Mamat, M., Waziri, M. Y., Shadi, A., Zaini, M. A., & Sumiati, I. (2021, March). Derivative Free Levenberg-Marquardt Method for Solving Fuzzy Nonlinear Equation. In IOP Conference Series: Materials Science and Engineering, 1115(1), 012002.
  17. Peeva, K. (1992). Fuzzy linear systems. Fuzzy Sets and Systems, 49(3), 339–355.
  18. Rafiq, N., Yaqoob, N., Kausar, N., Shams, M., Mir, N. A., Gaba, Y. U., & Khan, N. (2021). Computer-Based Fuzzy Numerical Method for Solving Engineering and Real-World Applications. Mathematical Problems in Engineering, 2021, Article ID 6916282.
  19. Sulaiman, I. M., Mamat, M., Waziri, M. Y., Mohamed, M. A., & Mohamad, F. S. (2018). Solving Fuzzy Nonlinear Equation via Levenberg-Marquardt Method. Far East Journal of Mathematical Sciences, 103(10), 1547–1558.
  20. 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.
  21. 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.
  22. Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353.
  23. Zadeh, L. A. (1996). Fuzzy sets. In Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh (pp. 394–432).
  24. 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.

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