We investigate some solitary wave results of time fractional evolution equations. By employing the extended rational $\exp ( (-\frac{{\psi }^{\prime }}{\psi }) ( \eta ) )$exp((−ψ′ψ)(η)) -expansion method, a few different results including kink, singular-kink, multiple soliton, and periodic wave solutions are formally generated. It is worth mentioning that the solutions obtained are more general with more parameters. The exact solutions are constructed in the form of exponential, trigonometric, rational, and hyperbolic functions. With the choice of proper values of parameters, graphs to some of the obtained solutions are drawn. On comparing some special cases, our solutions are in good agreement with the results published previously and the remaining are new.

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.

Nanoparticle-mediated drug delivery offers a promising approach to targeted cancer therapy, leveraging the ability of nanoparticles to deliver therapeutic agents directly to cancerous tissues with minimal impact on surrounding healthy cells. The presence of these nanoparticles is governed by a concentration equation, which accounts for the diffusion, convection, and reaction of the nanoparticles with the blood components. It is well-known that whenever a disease or infection occurs in a human, in 80% of cases a rise in the concentration of hydrogen peroxide in the blood occurs. This is the reason why blood is assumed to contain hydrogen peroxide (in the present study), which is a biomarker of oxidative stress and inflammation. This study explores the integration of machine learning (ML) techniques into the optimization of drug delivery processes within the human cardiovascular system, focusing on the enhancement of these processes through the application of magnetic fields. By employing ML algorithms, we analyze and predict the behavior of nanoparticles as they navigate the complex fluid dynamics of the cardiovascular system, particularly under the influence of an external magnetic field. The predictive power of ML models enables the precise control of nanoparticle trajectories, optimizing their accumulation in cancerous tissues and improving the efficacy of the drug delivery system. The findings of this study demonstrate that ML-enhanced magnetic targeting can significantly enhance the precision and effectiveness of nanoparticle-mediated drug delivery, offering a new paradigm in cancer treatment strategies. This approach has the potential to revolutionize the field by providing personalized and highly efficient therapeutic solutions for cancer patients.