Sharp Error Bounds for Weddle's Quadrature via a Novel Twice-Differentiable Convex Kernel

This paper introduces novel error bounds for Weddle's quadrature rule—a sixth-degree Newton–Cotes numerical integration method—using a new kernel. Unlike classical approaches requiring six-time differentiability, our results leverage twice-differentiable convex functions to derive tighter error estimates, significantly broadening applicability. We establish a key identity involving the second derivative and use it to prove inequalities for diverse function classes, including convexity, boundedness, and Lipschitz continuity. By using novel identity, we obtained refined bounds based on Hölder's, power-mean, and Young's inequalities, improving upon prior work. Applications to special functions ((Formula presented.) -digamma and Bessel) and composite quadrature rules demonstrate practical utility. Computational analysis and graphical presentations confirm the effectiveness of our results. Furthermore, we quantify the sensitivity of these bounds to integration-interval width, revealing a precise quadratic dependence on ((Formula presented.)) that guides partition-size selection in practice.