Error Bounds of Boole's Formula for Diverse Function Classes: Implications for Riemann-Liouville Fractional Integrals and Numerical Analysis

Abstract

In numerical analysis, Boole's formula is a key tool for approximating definite integrals. Accurate approximation of these integrals is vital in numerical methods for solving differential equations, especially in the finite volume method, where high-quality integral approximations lead to improved results. This paper provides rigorous proof of integral inequalities for first-time differentiable convex functions within the context of fractional calculus. We begin by establishing an integral equality that involves fractional integrals, subsequently deriving  Boole's formula-type inequalities for differentiable convex functions. This study examines important functional classes, including convex functions, Lipschitzian functions, bounded functions, and functions of bounded variation. Furthermore, we demonstrate the efficiency of derived inequalities through graphical representations, illustrating their application to specific functions and emphasizing their precision in approximating definite integrals.