In the present paper, we aim to prove a new lemma and quantum Simpson’s type inequalities for functions of two variables having convexity on co-ordinates over $[\alpha ,\beta ]\times [\psi ,\varphi ]$ . Moreover, our deduction introduce new direction as well as validate the previous results.

The objective of this paper is to derive Hermite-Hadamard type inequalities for several higher order strongly $\mathfrak{h}$ -preinvex functions via Riemann-Liouville fractional integrals. These results are the generalizations of the several known classes of preinvex functions. An identity associated with k-times differentiable function has been established involving Riemann-Liouville fractional integral operator. A number of new results can be deduced as consequences for the suitable choices of the parameters $\mathfrak{h}$ and $\sigma$ . Our outcomes with these new generalizations have the abilities to be implemented for the evaluation of many mathematical problems related to real world applications.

In this paper, we present the preliminaries of $(p,q)$ -calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over $[a,b]\times [c,d]$ by using the $(p,q)$ -calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the $(p,q)$ -calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the $(p,q)$ -calculus of the functions of two variables.

Highlighting the importance of artificial intelligence and machine learning approaches in engineering and fluid mechanics problems, especially in heat transfer applications is main goal of the presented article. With the advancement in Artificial Intelligence (AI) and Machine Learning (ML) techniques, the computational efficiency and accuracy of numerical results are enhanced. The theme of the study is to use machine learning techniques to examine the thermal analysis of MHD boundary layer flow of Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder embedded in a porous medium with heat source/sink and viscous dissipation effects. The considered base fluid is water ($H_{2}O$) and hybrid nanoparticles titanium oxide (${TiO}_2$) and Copper oxide ($CuO$). The governing flow equations are nonlinear PDEs. Non-similar system of PDEs are obtained with efficient conversion variables. The dimensionless PDEs are truncated using a local non-similarity approach up to third level and numerical solution is evaluated using MATLAB built-in-function bvp4c. Artificial Neural Networks (ANNs) simulation approach is used to trained the networks to predict the solution behavior. Thermal boundary layer improves with the enhancement in the value of $Rd$. The accuracy and reliability of ANNs predicted solution is addressed with computation of correlation index and residual analysis. The RMSE is evaluated [0.04892, 0.0007597, 0.0007596, 0.01546, 0.008871, 0.01686] for various scenarios. It is observed that when concentration of hybrid nanoparticles increases then thermal characteristics of the Eyring-Powell Hybrid Nanofluid (EPHNFs) passing a horizontal cylinder.

In this article, firstly, we establish a novel definition of weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ using an another function $\vartheta \left(\dot{\zeta }\right)$. As an additional observation, it is noted that the new class of weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ by employing an additional function $\vartheta \left(\dot{\zeta }\right)$ characterizes a variety of new classes as special cases, which is a generalization of the previous class. Secondly, we prove a new version of the Hermite-Hadamard-Fejér type inequality for h-convex interval-valued functions using weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ according to another function $\vartheta \left(\dot{\zeta }\right).$ Finally, by using weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ according to another function $\vartheta \left(\dot{\zeta }\right)$, we are establishing a new Hermite-Hadamard-Fejér type inequality for harmonically h-convex interval-valued functions that is not previously known. Moreover, some examples are provided to demonstrate our results.

In this study, the assumption of being differentiable for the convex function f in the (p, q)-Hermite-Hadamard inequality is removed. A new identity for the right-hand part of (p, q)-Hermite-Hadamard inequality is proved. By using established identity, some (p, q)-trapezoid integral inequalities for convex and quasi-convex functions are obtained. The presented results in this work extend some results from the earlier research.