TY - JOUR
T1 - Fuzzy solutions to partial differential equations
T2 - Adaptive approach
AU - Chen, Yung Yue
AU - Chang, Yu Te
AU - Chen, Bor Sen
N1 - Funding Information:
Manuscript received January 10, 2008; revised May 2, 2008; accepted May 26, 2008. First published August 26, 2008; current version published February 4, 2009. This work was supported by the National Science Council under Contract NSC 95-2221-E-007-196.
PY - 2009
Y1 - 2009
N2 - A new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations (PDEs) is presented. The design objective is to find a fuzzy solution to satisfy precisely the PDEs with boundary conditions. According to the adaptive scheme of fuzzy logic systems, a fuzzy solution with adjustable parameters for the PDE is first described. Then, a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function. In addition, an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived. Furthermore, the convergence of error equations in mesh points is also discussed from the energy perspective. In this paper, we show that the proposed method can solve a variety of PDEs encountered in engineering. Comparisons are also made with solutions obtained by the finite-element method.
AB - A new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations (PDEs) is presented. The design objective is to find a fuzzy solution to satisfy precisely the PDEs with boundary conditions. According to the adaptive scheme of fuzzy logic systems, a fuzzy solution with adjustable parameters for the PDE is first described. Then, a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function. In addition, an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived. Furthermore, the convergence of error equations in mesh points is also discussed from the energy perspective. In this paper, we show that the proposed method can solve a variety of PDEs encountered in engineering. Comparisons are also made with solutions obtained by the finite-element method.
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U2 - 10.1109/TFUZZ.2008.2005010
DO - 10.1109/TFUZZ.2008.2005010
M3 - Article
AN - SCOPUS:60549112223
SN - 1063-6706
VL - 17
SP - 116
EP - 127
JO - IEEE Transactions on Fuzzy Systems
JF - IEEE Transactions on Fuzzy Systems
IS - 1
ER -