New Fractional Estimates of Simpson-Mercer Type for Twice Differentiable Mappings Pertaining to Mittag-Leffler Kernel

Abstract

The main motivation of this study is to introduce a novel auxiliary result of Simpson’s formula by employing the Mercer scheme for twice differentiable functions involving the Atangana-Baleanu (AB) fractional integral operator concerned with the Mittag-Leffler as a nonsingular or nonlocal kernel. Thus, by employing Mercer’s convexity on twice differentiable mappings along with Hölder’s and power-mean inequalities, one can develop a variety of new Simpson’s error estimates. Lastly, some applications to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>q</a:mi> </a:math> -digamma function and modified Bessel functions are presented. Furthermore, the graphical illustrations described the efficiency and applicability of the proposed technique with success. We make links between our findings and a number of well-known discoveries in the literature. It is hoped that the proposed methodology will provide a new venue in the numerical techniques for calculating the quadrature formulae.