Adapted Fuzzy Integral: An Application in the Finite Element Method
DOI:
https://doi.org/10.5540/tema.2018.019.01.147Keywords:
Fuzzy Measure, Sugeno Integral, Finite Element Method, Boundary Value Problem.Abstract
In this paper we study and define an adapted fuzzy integral, based on the Sugeno integral. Moreover, we present a numerical integration formula which approximates the value of the adapted fuzzy integral. Thus, we prove that the Riemann integral and the adapted fuzzy integral are equivalent for power functions. Next, we apply the formula proposed in the numerical integration, required in the finite element method, to obtain a numerical solution of a boundary value problem for the one-dimensional Poisson equation. Finally, we observed better results of the approximate solution obtained in the example with the use of our formula when compared with the simple trapezoidal rule.
References
D. Sánchez, L. T. Bassani, L. C. Barros, and E. Esmi, "Ensaios do método de elementos Finitos com integral fuzzy", in Proceedings of IV CBSF. Recentes Avanços em Sistemas Fuzzy, SBMAC, 2016.
M. Sugeno, Theory of fuzzy Integrals and Applications. PhD thesis, Tokyo Institute of Technology, 1974.
L. A. Zadeh, “Information and control”, Fuzzy sets, vol. 8, no. 3, pp. 338-353, 1965.
L. C. Barros, R. C. Bassanezi, and W. A. Lodwick, A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics. New York: Springer, 2017.
L. T. Gomes, L. C. Barros, and B. Bede, Fuzzy Differential Equations in Various Approaches. SBMAC SpringerBriefs, New York: Springer, 2015.
H. Nguyen and E. Walker, A First Course in Fuzzy Logic. CRC Press Taylor & Francis Group, 2006.
G. Arenas-Diáz and E. R. Ramírez-Lamus, "Medidas difusas e integrales difusas", Universitas Scientiarum, vol. 18 (1), no. 1, pp. 7-32, 2013.
M. Asadzadeh, "An Introduction to the Finite Element Method (FEM) for Differential Equations", 2012.
J. Li and Y. Chen, Computational Partial Differential Equations Using MATLAB. CRC Press Taylor & Francis Group, 2008.
W. Pedrycz and F. Gomide, Fuzzy systems engineering toward human-centric Computing. IEEE Press, New Jersey, EUA: John Wiley & Sons, 2007.
T. Murofushi and M. Sugeno, "Fuzzy measures and fuzzy integrals", Fuzzy Measures and Integrals: Theory and Applications, pp. 3-41, 2000.
H. Román-Flores, A. Flores-Franulic, and Y. Chalco-Cano, "The fuzzy integral for monotone functions", Applied Mathematics and Computation, vol. 185, no. 1, pp. 492-498, 2007.
R. J. Leveque, Finite difference methods for ordinary and partial differential equations - steady-state and time-dependent problems. SIAM, 2007.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.