Some New Results and Application of Caputo-Fractional-Order Ostrowski’s-Type Inequality

Abstract

This article is about a study of a new concept of fractional order Ostrowski's type inequality for interval-valued functions using Caputo generalized Hukuhara (gH) differentiability. As fractional calculus gains traction in mathematical modeling, particularly in contexts involving uncertainty, our focus lies on extending Ostrowski's type inequality—a fundamental result in numerical analysis—within this framework. We begin by defining the Caputo gH-derivative and exploring its properties as applied to interval-valued functions. Through rigorous analysis, we derive a general form of Ostrowski's type inequality that incorporates fractional order derivatives, demonstrating how this approach accommodates the inherent variability of interval-valued data. We also give a comparison of fractional order Ostrowski's type inequality with the classical order Ostrowski's type inequality. Existence and uniqueness of caputo integral inequity is verified with Lipschitz property and also stability analysis is derived with Lyapunov function for Ostrowski's type Inequality.  The results not only enhance the theoretical landscape of fractional calculus but also have significant implications for practical applications across various scientific fields, including engineering and economics, where interval data frequently arises.  To validate our findings, we provide illustrative examples that showcase the applicability of the generalized Ostrowski's type inequality, highlighting its potential to manage complex phenomena in interval-valued functions.Our results are new, novel and generalization of the classical Ostrowski's type inequality. Ultimately, this work aims to bridge the gap between fractional calculus and interval analysis, paving the way for future research in this promising area and expanding the toolkit available for mathematicians and practitioners alike.