Analytical results for positivity of discrete fractional operators with approximation of the domain of solutions.

We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where $ \left({}_{{c_0}}^{C{F_R}}\nabla^{\theta} \mathtt{F}\right)(t) > -\epsilon\, \Lambda(\theta-1)\, \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1) $ such that $ \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1)\geq 0 $ and $ \epsilon > 0 $. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of $ \epsilon $ and $ \theta $.

Soret and Dufour effects on MHD squeezing flow of Jeffrey fluid in horizontal channel with thermal radiation.

The fluid flow with chemical reaction is one of well-known research areas in the field of computational fluid dynamic. It is potentially useful in the modelling of flow on a nuclear reactor. Motivated by the implementation of the flow in the industrial application, the aim of this study is to explore the time-dependent squeeze flow of magnetohydrodynamic Jeffrey fluid over permeable medium in the influences of Soret and Dufour, heat source/sink and chemical reaction. The presence of joule heating, joule dissipation and radiative heat transfer are analyzed. The flow is induced due to compress of two surfaces. Conversion of partial differential equations (PDEs) into ordinary differential equations (ODEs) is accomplished by imposing similarity variables. Then, the governing equations are resolved using Keller-box approach. The present outcomes are compared with previously outcomes in the literature to validate the precision of present outcomes. Both outcomes are shown in close agreement. The tabular and graphical results demonstrate that wall shear stress and velocity profile accelerate with the surfaces moving towards one another. Moreover, the concentration, temperature and velocity profiles decreasing for the increment of Hartmann numbers and Jeffrey fluid parameters. The impacts of heat generation/absorption, joule dissipation and Dufour numbers enhance the heat transfer rate and temperature profile. In contrast, the temperature profile drops and the heat transfer rate boosts when thermal radiation increases. The concentration profile decelerates, and the mass transfer rate elevates with raise in Soret number. Also, the mass transfer rate rises for destructive chemical reaction and contrary result is noted for convective chemical reaction.

Positivity and monotonicity results for discrete fractional operators involving the exponential kernel.

This work deals with the construction and analysis of convexity and nabla positivity for discrete fractional models that includes singular (exponential) kernel. The discrete fractional differences are considered in the sense of Riemann and Liouville, and the υ1-monotonicity formula is employed as our initial result to obtain the mixed order and composite results. The nabla positivity is discussed in detail for increasing discrete operators. Moreover, two examples with the specific values of the orders and starting points are considered to demonstrate the applicability and accuracy of our main results.