Gold nanoparticles are commonly used as a tracer in laboratories. They are biocompatible and can transport heat energy to tumor cells via a variety of clinical techniques. As cancer cells are tiny, properly sized nanoparticles were introduced into the circulation for invasion. As a result, gold nanoparticles are highly effective. Therefore, the current research investigates the magnetohydrodynamic free convection flow of Casson nanofluid in an inclined channel. The blood is considered as a base fluid, and gold nanoparticles are assumed to be uniformly dispersed in it. The above flow regime is formulated in terms of partial differential equations. The system of derived equations with imposed boundary conditions is non-dimensionalized using appropriate dimensionless variables. Fourier's and Fick's laws are used to fractionalize the classical dimensionless model. The Laplace and Fourier sine transformations with a new transformation are used for the closed-form solutions of the considered problem. Finally, the results are expressed in terms of a specific function known as the Mittag-Leffler function. Various figures and tables present the effect of various physical parameters on the achieved results. Graphical results conclude that the fractional Casson fluid model described a more realistic aspect of the fluid velocity profile, temperature, and concentration profile than the classical Casson fluid model. The heat transfer rate and Sherwood number are calculated and presented in tabular form. It is worth noting that increasing the volume percentage of gold nanoparticles from 0 to 0.04 percent resulted in an increase of up to 3.825% in the heat transfer rate.

In this paper, the newly developed Fractal-Fractional derivative with power law kernel is used to analyse the dynamics of chaotic system based on a circuit design. The problem is modelled in terms of classical order nonlinear, coupled ordinary differential equations which is then generalized through Fractal-Fractional derivative with power law kernel. Furthermore, several theoretical analyses such as model equilibria, existence, uniqueness, and Ulam stability of the system have been calculated. The highly non-linear fractal-fractional order system is then analyzed through a numerical technique using the MATLAB software. The graphical solutions are portrayed in two dimensional graphs and three dimensional phase portraits and explained in detail in the discussion section while some concluding remarks have been drawn from the current study. It is worth noting that fractal-fractional differential operators can fastly converge the dynamics of chaotic system to its static equilibrium by adjusting the fractal and fractional parameters.