Heat and mass transfer study of hybrid nanomaterial Casson fluid with time-dependent flow over a vertical Riga sheet was deliberated under the stagnation region. In the presence of the Riga sheet in fluid flow models, this formulation was utilized to introduce Lorentz forces into the system. We considered the three models of hybrid nanomaterial fluid flow: namely, Yamada Ota, Tiwari Das, and Xue models. Two different nanoparticles, namely, SWCNT and MWCNT under base fluid (water) were studied. Under the flow suppositions, a mathematical model was settled using boundary layer approximations in terms of PDEs (partial differential equations). The system of PDEs (partial differential equations) was reduced into ODEs (ordinary differential equations) after applying suitable transformations. The reduced system, in terms of ODEs (ordinary differential equations), was solved by a numerical scheme, namely, the bvp4c method. The inspiration of the physical parameters is presented through graphs and tables. The curves of the velocity function deteriorated due to higher values of $M$. The Hartmann number is a ratio of electric force to viscous force. The electric forces increased due to higher values of the modified Hartmann number, ultimately declining the velocity function. The skin friction was reduced due to an incremental in $\varpi$, while the Nusselt number raised with higher values of $\varpi$. Physically, the Eckert number increased, which improved kinetic energy and, as a result, skin friction declined. The heat transfer rate increased as kinetic energy increased, and the Eckert number increased. The skin friction reduced due to physical enhancement of ${\beta }_1$, the shear thinning was enhanced which reduced the skin friction.

In this analysis, the generalized Fourier and Fick's law for Second-grade fluid flow at a slendering vertical Riga sheet is examined along with thermophoresis and Brownian motion effects. Boundary layer approximations in terms of PDE's (Partial Differential Equations) are used to build the mathematical model. An appropriate transformation has been developed by using the Lie symmetry method. PDE's (Partial Differential Equations) are transformed into ODE's (Ordinary Differential Equations) by implementing the suitable transformation. A numerical method called bvp4c is used to explain the dimensionless system (ODE's). Graphs and tables are used to interpret the impact of the significant physical parameters. The curves of temperature function declined due to enchanting the values of the thermophoresis Parameter. The temperature is produced at a low level due to enchanting the values of thermophoresis because this force transports burn at a low 10 μm diameter so the temperature becomes lessor. Increments of thermophoresis parameter which enhanced the values of concentration Function. As the concentration boundary layer increased which declined the mass transfer due increment in thermophoresis. The curves of temperature function are increasing due to enhancing the values of the Brownian parameter because addition in the Brownian motion, improved the movement of particles ultimately increasing the kinematic energy of fluid which improved the heat transfer phenomena. Increments of Brownian parameter which declined the values of concentration function. Physically, the kinematic energy improved which declined the mass transfer rate near the surface.

The construction of contraction conditions plays an important role in science for formulating new findings in fixed point theories of mappings under a set of specific conditions. The aim of this work is to take advantage of the idea of extended $ b $-metric spaces in the sense introduced by Kamran et al. [A generalization of $ b $-metric space and some fixed point theorems, Mathematics, 5 (2017), 1–7] to construct new contraction conditions to obtain new results related to fixed points. Our results enrich and extend some known results from $ b $-metric spaces to extended b-metric spaces. We construct some examples to show the usefulness of our results. Also, we provide some applications to support our results.

In this study, we have proposed a mathematical model to describe the dynamics of the spread of Middle East Respiratory Syndrome disease. The model consists of six-coupled ordinary differential equations. The existence of the corona-free equilibrium and endemic equilibrium points has been proved. The threshold condition for which the disease will die out or becomes permanent has been computed. That is the corona-free equilibrium point is locally asymptotically stable whenever the reproduction number is less than unity, and it is globally asymptotically stable (GAS) whenever the reproduction number is greater than unity. Moreover, we have proved that the endemic equilibrium point is GAS whenever the reproduction number is greater than unity. The results of the model analysis have been illustrated by numerical simulations.