Some novel inequalities of Weddle's formula type for Riemann-Liouville fractional integrals with their applications to numerical integration

In numerical analysis, Weddle's formula is a pivotal tool for approximating definite integrals. The approximation of the definite integrals plays a significant role in numerical methods for differential equations, particularly in the finite volume method. We need to use the best approximation of the integrals to get better results. This paper thoroughly proves integral inequalities for first-time differentiable convex functions in fractional calculus. For this, first, we prove an integral identity involving Riemann-Liouville fractional integrals. Then, with the help of this identity, we prove fractional Weddle's formula-type inequalities for differentiable convex functions. Our approach involves significant functional classes, including convex, Lipschitzian and bounded functions. The primary motivation of this paper is that Weddle's formula should be employed when Simpson's 1/3 formula fails to yield the required precision. Simpson's formula is limited to third-order polynomial approximations, which may only sometimes capture the intricacies of more complex functions. On the other hand, Weddle's formula provides a higher degree of interpolation using sixth-order polynomials, offering a more refined approximation. Additionally, the paper highlights the significance of the Riemann- Liouville fractional operator in addressing problems involving non-integer-order differentiation, showcasing its critical role in enhancing classical inequalities. These new inequalities can help to find the error bounds for different numerical integration formulas in classical calculus. Moreover, we provide some applications to numerical quadrature formulas of these newly established inequalities. These approximations highlight their potential impact on computational mathematics and related fields. Furthermore, we give numerical examples, computational analysis, and graphical representations that show these newly established inequalities are numerically valid.