A new methodology via using an adaptive fuzzy algorithm to obtain solutions of Two-dimensional Navier-Stokes equations (2-D NSE) is presented in this investigation. The design objective is to find two fuzzy solutions to satisfy precisely the 2-D NSE frequently encountered in practical applications. In this study, a rough fuzzy solution is formulated with adjustable parameters firstly, and then, a set of adaptive laws for optimally tuning the free parameters in the consequent parts of the proposed fuzzy solutions are derived from minimizing an error cost function which is the square summation of approximation errors of boundary conditions, continuum equation and Navier-Stokes equations. In addition, elegant approximated error bounds between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors have also been proven. Furthermore, the error equations in mesh points can be proven to converge to zero for the 2-D NSE with two sufficient conditions.
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