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In our article, we are primarily concentrating on approximate controllability results for fractional Sobolev type Volterra‐Fredholm integro‐differential inclusions of order 1 < r < 2. By applying the results and ideas belongs to the cosine function of operators, fractional calculus and fixed point approach, the main results are established. Initially, we establish the approximate controllability of the considered fractional system, then continue to examine the system with the concept of nonlocal conditions. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order 1 < r < 2. By applying the results and facts belongs to the cosine function of operators, fractional calculus, the measure of noncompactness and fixed point approach, the main results are established. Initially, we focus the existence of mild solution, and later we establish the controllability of the considered fractional system. Finally, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In our paper, we primarily concentrate on the existence and controllability results for fractional evolution inclusions of order 1 < r < 2 with Clarke's subdifferential type. By applying the facts related to the measure of noncompactness, fractional calculus, and fixed‐point technique, the principal results are proved. Initially, we are concentrating the existence and continue to prove the controllability of the fractional evolution system. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In our article, we primarily concentrate on the approximate controllability results for Sobolev type fractional stochastic evolution hemivariational inequalities. By applying the facts related to fractional calculus, and fixed‐point technique, the principal results are proved. Initially, we are concentrating the existence and continue to prove the controllability of the fractional evolution system. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

The fluid flow between the rotating disk has wide range of applications and are used in medical equipment's, car brake system, gas turbines, manufacturing of glass and plastic films. Inspired from these applications, we scrutinized the behavior of an axisymmetric expansion–contraction rotating disk on three‐dimensional flow of hybrid nanoparticles in the presence of microorganisms. Here, we considered copper and silica nanoparticles suspended in base fluid water. It is known that, the inequality of concentration in chemically reactive species affecting mass transfer mechanism and activation energy plays an important role in increasing the production speed of chemical reactions. Hence, the Arrhenius activation energy and chemical reaction effects are added in the mass equation. Autorelated transformations were adopted to reduce PDE's to ODE's and then, we utilized the RKF‐45 by adopting shooting skill to solve reduced ODE's. The analysis of rate of heat and mass transfer is done through graphs. Furthermore, change in velocity, thermal and concentration profiles for various nondimensional parameters are deliberated briefly by using suitable plots. Result reveals that rise in wall expansion/contraction parameter supports up the velocity gradients, higher the values of bioconvection Peclet number and concentration difference parameter declines the microorganism profile and Nusselt number decreases for rise in values of thermophoresis parameter. [ABSTRACT FROM AUTHOR]

In the Nidovirales order of the Coronaviridae family, where the coronavirus (crown‐like spikes on the surface of the virus) causing severe infections like acute lung injury and acute respiratory distress syndrome. The contagion of this virus categorized as severed, which even causes severe damages to human life to harmless such as a common cold. In this manuscript, we discussed the SARS‐CoV‐2 virus into a system of equations to examine the existence and uniqueness results with the Atangana–Baleanu derivative by using a fixed‐point method. Later, we designed a system where we generate numerical results to predict the outcome of virus spreadings all over India. [ABSTRACT FROM AUTHOR]