A study of improved error bounds for Simpson type inequality via fractional integral operator

Abstract

Fractional integral operators have been studied extensively in the last few decades, and many different types of fractional integral operators have been introduced by various mathematicians. In 1967 Michele Caputo introduced Caputo fractional derivatives, which defined one of these fractional operators, the Caputo Fabrizio fractional integral operator. The main aim of this article is to established the new integral equalities related to Caputo-Fabrizio fractional integral operator. By incorporating this identity and convexity theory to obtained a novel class  of Simpson type inequality. In this paper, we present a novel generalization of Simpson type inequality via s-convex and quasi-convex functions. Then, utilizing this identity the bounds of classical Simpson type inequality is improved. Finally, we discussed some applications to Simpson^{,}s quadrature rule.