Hybrid error estimates of Weddle's formula and their applications for various classes of functions

This study presents new bounds on Weddle's type formula for differentiable functions using proportional Caputo-Hybrid operators ($P_{cap}$). These operators provide hybrid estimates; for $\alpha =1$, they yield first-derivative estimates, while for $\alpha =0$, they approximate the second derivative. By applying $P_{cap}$, we generalize Weddle's formula for optimal approximations, particularly for polynomials of degree six. This enhances its applicability when Simpson's 1/3 rule fails to achieve the required precision. Our findings extend Weddle's formula to a broader class of functions, improving error bounds in inequality theory and calculus. We demonstrate applications in special means and functions, validating our approach through rigorous computational analysis. Additionally, we propose future research directions, including extensions to q-calculus, symmetrized q-calculus, and multiplicative calculus, as well as exploring multidimensional spaces using alternative fractional integral operators.