This note generalizes several existing results related to Hermite–Hadamard inequality using h-Godunova–Levin and $(h_1,h_2)$-convex functions using a fractional integral operator associated with the Caputo–Fabrizio fractional derivative. This study uses a non-singular kernel and constructs some new theorems associated with fractional order integrals. Furthermore, we demonstrate that the obtained results are a generalization of the existing ones. To demonstrate the correctness of these results, we developed a few interesting non-trivial examples. Finally, we discuss some applications of our findings associated with special means.

This work wishes to model influence of nano-powders with various shapes on acceleration of freezing of water inside the container with rectangular fins. Single phase formulations for features of NEPCM were applied and implicit approach was utilized for transient modeling. Galerkin approach was utilized to find the solution of current work. The governing equations contain two scalars and contours for various level of time and ice front configuration have been demonstrated in outputs. Gravity force role has been ignored and verification test proved the correctness of this assumption. The mesh near the ice front should have higher elements to achieve more accurate results and Galerkin approach offered adaptive grid to reach this purpose. Rate of freezing intensifies around 10.74% with adding nano-powders with highest shape coefficient. By altering shape to platelet leads to decrease in time of freezing about 4.7%.

The evolution of a physical system occurs through a set of variables, and the problems can be treated based on an approach employing multivariable Hermite polynomials. These polynomials possess beneficial properties exhibited in functional and differential equations, recurring and explicit relations as well as symmetric identities, and summation formulae, among other examples. In view of these points, comprehensive schemes have been developed to apply the principle of monomiality from mathematical physics to various mathematical concepts of special functions, the development of which has encompassed generalizations, extensions, and combinations of other functions. Accordingly, this paper presents research on a novel family of multivariable Hermite polynomials associated with Frobenius–Genocchi polynomials, deriving the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified, as well as establishing the series representations, summation formulae, operational and symmetric identities, and recurrence relations satisfied by these polynomials. This proposed scheme aims to provide deeper insights into the behavior of these polynomials and to uncover new connections between these polynomials, to enhance understanding of their properties.