Using t- moduli of smoothness with some concepts of fuzzy set theory to estimate the error for Runge-Kuttas and Adams numerical methods

The estimation of the error which appears in applying Runge-Kuttas and Adams numerical methods to find the solution of Cauchy-problem was obtained in [1] and [2] by using the new moduli of smoothness, so called tmoduli of smoothness. In the present paper, the emergence of some concepts of fuzzy set theory, which gives a powerful way to reach our goal, is implemented to improve the approximate solution and to minimize the difference (the error) between the exact solution and the approximate one. Through this paper we mentioned the main definitions and notations related to both of the averaged moduli of smoothness and of the fuzzy set theory separately.