Fuzzy Solutions to Partial Differential Equations: Adaptive Approach

• Published in: IEEE transactions on fuzzy systems Vol. 17; no. 1; pp. 116 - 127
• Main Authors: Chen, Yung-Yue, Chang, Yu-Te, Chen, Bor-Sen
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2009; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive fuzzy algorithm; Algorithms; Boundary conditions; Differential equations; Errors; Finite element methods; finite-element method; Fuzzy; fuzzy approximation; Fuzzy logic; Fuzzy set theory; Fuzzy systems; Laws; Mathematical analysis; Nonlinear equations; partial differential equation (PDE); Partial differential equations; Power engineering and energy; Studies; Transforms
• ISSN: 1063-6706; 1941-0034
• DOI: 10.1109/TFUZZ.2008.2005010

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.