Adapted Fuzzy Integral: An Application in the Finite Element Method

Daniel Sánchez, Luana T. Bassani, Laécio C. Barros, Estevão Esmi


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.


Fuzzy Measure, Sugeno Integral, Finite Element Method, Boundary Value Problem.


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.


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