fuzzy approximation

In this paper we provide new several Jackson-type approximations results for continuous fuzzy-number-valued functions which improve several previous ones. We use alternative techniques adapted from Interval Analysis which rely on the gH-difference (which might not exist) and the generalized difference (which might lack the cancellation law ) of fuzzy numbers.

The fuzzy operations on fuzzy numbers of type L-R are much easier than general fuzzy numbers. It would be interesting to approximate a fuzzy number by a fuzzy number of type L-R. In this paper, we state and prove two significant application inequalities in the monotonic functions set. These inequalities show that under a condition, the nearest fuzzy number of type L-R to an arbitrary fuzzy number exists and is unique. After that, the nearest fuzzy number of type L-R can be obtained by solving a linear system. Note that the trapezoidal fuzzy numbers are a particular case of the fuzzy numbers of type L-R. The proposed method can represent the nearest trapezoidal fuzzy number to a given fuzzy number. Finally, to approximate fuzzy solutions of a fuzzy linear system, we apply our idea to construct a framework to find solutions of crisp linear systems instead of the fuzzy linear system. The crisp linear systems give the nearest fuzzy numbers of type L-R to fuzzy solutions of a fuzzy linear system. The proposed method is illustrated with some examples.

Solving fuzzy linear systems has been widely studied during the last decades‎. ‎However‎, ‎there are still many challenges to solving fuzzy linear equations‎, ‎as most of the studies have used the principle of extension‎, ‎which suffers from shortcomings such as the lack of solution‎, ‎achieving solutions under very strong conditions‎, ‎large support of the obtained solutions‎, ‎inaccurate or even incorrect solutions due to not utilizing all the available information‎, complicated process and high computational load‎. ‎These problems motivated us to present a fuzzy Cramer method for solving fuzzy linear equations‎, ‎which uses arithmetic operations based on the Transmission Average (TA)‎. ‎In this study‎, ‎fully fuzzy linear systems in the form of $ \tilde{A}\tilde{X}=\tilde{B} $‎, ‎and dual fuzzy linear systems in the form of $ \tilde{A}\tilde{X}+\tilde{B}=\tilde{C}\tilde{X}+\tilde{D} $ are solved using the proposed fuzzy Cramer method‎, ‎and numerical examples are provided to confirm the effectiveness and applicability of the proposed method.