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This study investigates the (3 + 1)-dimensional extended Kairat-II model using Lie-Bäcklund symmetry (LBS) and the improved modified extended tanh-function approach (IMETFA). The bifurcation and sensitivity analyses are conducted to understand the stability and chaotic behavior of the model. Through graphical visualizations of phase diagrams, Lyapunov exponents, power spectra, fractal dimension and recurrence plots, the complex dynamics and stability characteristics of the model are elucidated. Additionally, the research focuses on the derivation of dark soliton and various combo soliton solutions of the Kairat-II model. The derived soliton solutions are graphically displayed in 3D and 2D plots. The obtained results are new and have never been reported in the literature for the considered equation. [ABSTRACT FROM AUTHOR]

This paper explores the class C G , consisting of functions g that satisfy a specific subordination relationship with Gregory coefficients in the open unit disk E. By applying certain conditions to related coefficient functionals, we establish sharp estimates for the first five coefficients of these functions. Additionally, we derive bounds for the second and third Hankel determinants of functions in C G , providing further insight into the class's properties. Our study also investigates the logarithmic coefficients of log (g (t) t) and the inverse coefficients of the inverse functions (g − 1) within the same class. [ABSTRACT FROM AUTHOR]

This research deals with a comparative numerical analysis of chaos in two systems with non-integer derivatives. The one-scroll system and circle equilibrium system with different hidden attractors are simulated considering the fractal derivative, Khalil and Atangana conformable derivatives, and the truncated M-derivative considering a constant and variable-order. Phase portraits are shown, as well as bifurcation diagrams, and Lyapunov exponents are obtained. Later, 0–1 test, dynamic death analysis, and sensitivity to initial conditions are considered to choose which derivative produces richer chaotic behaviors. According to those mentioned above, we could observe that the M-derivative not only generalizes Khalil's type conformable derivative but also its two non-integer orders produce interesting dynamic behaviors compared to the remaining derivatives. In the numerical results, we observe that the variable order makes the system more sensitive to the change in the initial conditions. The new chaotic behaviors with constant and variable order are used to develop a synchronization scheme of two identical one-scroll systems via the backstepping technique with the truncated M-derivative involved. [ABSTRACT FROM AUTHOR]

In this work, we studied the dynamics of a chaotic neural network using the fractional-conformable operator and non-integer order derivatives. We first systematically show the dynamics behavior of the network using the conformable operator. Then, we show the dynamics of the network using fractional-order derivatives. We compared the different effects and behaviors when using conformable operators and fractional-order derivatives. Next, we studied the dynamics when applying both operators to the network simultaneously, and we graphically show the effects of each operator on the chaotic neural network. Following the dynamical analysis, we used a fractional-variable order in the derivative operator to compare the system behavior when we used a fractional constant order in the derivative operator. We used a smooth function in all our simulations in the interval (0, 1]. Our results show that either the conformable operator or the fractional derivative, both of them, affect the dynamics of the network showing interesting behaviors. We show that when the order in the conformable operator decreases an attractor is generated which takes the system into a stable state, and when the order in the fractional derivative decreases the system loses its dynamics. Contrary, when we used the fractional-variable order in the derivative operator, the system does not lose its dynamics; instead, it gains more chaotic behavior, which means that the system is susceptible to the initial conditions fractional-variable order function and its initial point. Finally, after depicting the different system behavior under both operators, we proposed a full master-slave synchronization using the fractional-conformable approach. We designed a fractional order controller to reach synchronization using optimal gains means Cuckoo search algorithm. Our results show the different behaviors when using the fractional-conformable approach in the master and slave systems. As we expected, the gains in a combination of the non-integer order allow achieving the synchronization between the master and the slave system even when the dynamics in the slave system was regular. [ABSTRACT FROM AUTHOR]

In the present work, a microchannel that benefits from the simultaneous effect of slip condition and semicircular ribs was studied to boost heat transfer. A numerical method was utilized to examine the thermal and hydraulic behavior. The results reveal that the velocity is not zero since the slip condition exists in the microchannel. Furthermore, the velocity near the wall has a dramatic value when the slip length increases. Although the heat transfer is not remarkable by semicircular ribs, the magnetic field plays a vital role in boosting the heat transfer as a result of the declining thermal boundary layer. The effect of magnetic field on the heat transfer on the low Re number is not like the higher one which means as the Reynolds number (Re) varies from 10 to 90, the heat transfer goes up from 1.12 to 2.63. Furthermore, at Re = 90, a 255% enhancement is seen in the microchannel by affecting magnetic field at Hartmann number = 15. The results of slip condition claim that slip condition is introduced as the third most effective factor in rising and improving the efficiency of the microchannel. There is a 16.23% improvement in heat transfer by using slip condition in the microchannel. More importantly, the figure for heat transfer is enhanced by increasing the radius of ribs. [ABSTRACT FROM AUTHOR]

Applying the electric field to a fluid flowing on an infinite vertical plate is the most recent technique used for studying fluid movement. This technique is known as electro-osmotic flow (EOF). Therefore, the core aim of the present research work is to examine the time-dependent electro-osmotic flow of viscoelastic fluid on a vertical flat plate together with the effects of heat generation and chemical reaction. The classical system of governing equations has been fractionalised by means of Caputo–Fabrizio’s time-fractional derivative. Governing equations have been non-dimensionalised by using relative dimensionless quantities. The exact solutions for the momentum, temperature and concentration equations have been developed by implementing the Laplace transform technique. For graphical analysis, the solutions have been plotted against the inserted parameters using the computational software Mathematica. It is interesting to mention that the time-fractional model provides more than one fluid layer for the analysis of the fluid motion, heat distribution and mass concentration, which is not possible by assuming the classical mathematical model. It is also very important to mention that the velocity profile shows declination for greater values of electro-osmotic parameter Es. [ABSTRACT FROM AUTHOR]

The Klein Kramer equation (KKE) stands for the probability distribution function (PDF) that describes the diffusion of particles subjected an external force in the presence of friction. It is applicable in statistical and stochastic treatments of chemical dynamics, in particular the diffusive description of chemical reactions. Here, we are concerned with finding the exact solutions of the conformable fractional derivative (CFD) KKE. An approach is presented to transform linear partial differential equations (PDEs) to a set of first order PDEs. On the other hand the CFD is shown to be reduced to the classical one's by using similarity transformations. Here, the objective of this work is to find the exact solutions of CFD-KKE. To this issue, the approach presented is applied. The solutions are found by implementing extended unified method. It is found that, the integrability condition is that the external force is constant. The numerical results of the solutions are calculated and the are shown graphically. Calculations are carried by MATHEMATICA 12. [ABSTRACT FROM AUTHOR]

Finite element simulations for the dynamics of Casson fluid flow over time-dependent two-dimensional stretching sheet subjected to magnetic field and variable time and space-dependent temperature are studied numerically through Galerkin finite element method implementation. For this, weak form of the governing boundary value problems is derived through their residuals. Domain is discretized using two nodes per element, and assembly process is performed. The system of algebraic nonlinear equations is linearized through Picard's linearization algorithm. Linear system of algebraic equations is solved iteratively with computational tolerance 10 - 8 . The independent variable is searched through several computational experiments, and code is tested by comparing the results for special case with already published benchmarks. After the validation of code, simulations are performed in order to capture the dynamics of the physical situation against the variation of the pertinent parameters. Behavior of stresses and heat flux for different values of the physical parameters is studied. The temperature decreases when the intensity of radiation in the form of electromagnetic waves is increased. Boundary layer thickness for the Casson fluid is less than the boundary layer thickness of Newtonian fluid. However, opposite trend of thermal boundary layer thickness is noted. The magnetic is responsible for producing a hindrance to flow. Consequently, wall shear stress increases. Heat flux at the surface of stretching sheet increases when the values of unsteadiness parameter are increased, whereas there is a decreasing trend in the rate of heat transfer when the value of Eckert number is increased. Shear stresses are increasing function of the temperature. However, there is an increasing trend in the rate of heat transfer. [ABSTRACT FROM AUTHOR]