In this study, the Fredholm hypersingular integral equation of the first kind with a singular right-hand function on the interval $\left[-1,1\right]$ is solved. The discontinuous solution on the domain $\left[-1,1\right]$ is approximated by a piecewise polynomial, and a collocation method is introduced to evaluate the unknown coefficients. This method, which can be applied to both linear and nonlinear integral equations, is very simple and straightforward. The presented illustrations relate that the results are very accurate compared to the other methods in the literature.

The complex Ginzburg-Landau model appears in the mathematical description of wave propagation in nonlinear optics. In this paper, the fractional complex Ginzburg-Landau model is investigated using the generalized exponential rational function method. The Kerr law and parabolic law are considered to discuss the nonlinearity of the proposed model. The fractional effects are also included using a novel local fractional derivative of order $\alpha$ . Many novel solutions containing trigonometric functions, hyperbolic functions, and exponential functions are acquired using the generalized exponential rational function method. The 3D-surface graphs, 2D-contour graphs, density graphs, and 2D-line graphs of some retrieved solutions are plotted using Maple software. A variety of exact traveling wave solutions are reported including dark, bright, and kink soliton solutions. The nature of the optical solitons is demonstrated through the graphical representations of the acquired solutions for variation in the fractional order of derivative. It is hoped that the acquired solutions will aid in understanding the dynamics of the various physical phenomena and dynamical processes governed by the considered model.

In this note, we establish an integral inequality for compact and orientable real hypersurfaces in complex two-plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ , involving the shape operator $A$ and the Reeb vector field $\xi$ . Moreover, this integral inequality is optimal in the sense that the real hypersurfaces attaining the equality are completely determined. As direct consequences, some new characterizations of the real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ with isometric Reeb flow can be presented.

Due to high rate of electrical device usage of all types and high consumption of electricity, it is necessary to search for alternative way. There are so many models of nature describing many phenomena in energy, magnetic field, heat transfer, and so on. These models are reflected in the delicate nonlinear partial differential equations. The system of coupled Konno-Oono equations is one of those models. We extract some vital solutions for this model, which describe some vital phenomena in applied science. Namely, we applied unified technique in order to perform this mechanism in a completely unified way. Finally, some simulations are performed by utilizing Matlab 18 to exhibit the behaviour of these solutions.

In this investigation, the exact solutions of variable coefficients of generalized Zakharov-Kuznetsov (ZK) equation and the Gardner equation are studied with the help of an extended generalized $\left(G^{\prime }/G\right)$ expansion method. The main objective of this study is to establish the closed-form solutions and dynamics of analytical solutions to the generalized ZK equation and the Gardner equation. The generalized ZK equation and the Gardner equation govern the behavior of nonlinear wave phenomena in the presence of magnetic field in plasma dynamics, turbulence, bottom topography, and quantum field theory. We construct innovative solutions to the models under consideration using various computing tools and a recently developed extended generalized $\left(G^{\prime }/G\right)$ expansion technique. The extended generalized $\left(G^{\prime }/G\right)$ expansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial of $\left(G^{\prime }/G\right)$ . The derived solutions for both the equations are the hyperbolic, trigonometric, and rational functions. The obtained solutions have shock/kink waves and multisoliton, which depict the dynamical representations of the acquired solutions through the three-dimensional surface plots and the contour plots.

The present analysis is aimed at examining MHD micropolar nanofluid flow past a radially stretchable rotating disk with the Cattaneo-Christov non-Fourier heat and non-Fick mass flux model. To begin with, the model is developed in the form of nonlinear partial differential equations (PDEs) for momentum, microrotation, thermal, and concentration with their boundary conditions. Employing suitable similarity transformation, the boundary layer micropolar nanofluid flows governing these PDEs are transformed into large systems of dimensionless coupled nonlinear ordinary differential equations (ODEs). These dimensionless ODEs are solved numerically by means of the spectral local linearization method (SLLM). The consequences of more noticeable involved parameters on different flow fields and engineering quantities of interest are thoroughly inspected, and the results are presented via graph plots and tables. The obtained results confirm that SLLM is a stable, accurate, convergent, and computationally very efficient method to solve a large coupled system of equations. The radial velocity grows while the tangential velocity, temperature, and concentration distributions turn down as the value of the radial stretching parameter improves, and hence, in practical applications, radial stretching of the disk is helpful to advance the cooling process of the rotating disk. The occurrence of microrotation viscosity in microrotation parameters ( $A_1-A_6$ ) declines the radial velocity profile, and the kinetic energy of the fluid is reduced to some extent far away from the surface of the disk. The novelty of the study is the consideration of microscopic effects occurring from the micropolar fluid elements such as micromotion and couple stress, the effects of non-Fourier’s heat and non-Fick’s mass flux, and the effect of radial stretching disk on micropolar nanofluid flow, heat, and mass transfer.

The current study examined the effects of magnetohydrodynamics (MHD) on time-dependent mixed convection flow of an exothermic fluid in a vertical channel. Convective heating and Navier’s slip conditions are considered. The dimensional nonlinear flow equations are transformed into dimensionless form with suitable transformation. For steady-state flow formations, we apply homotopy perturbation approach. However, for the unsteady-state governing equation, we use numerical technique known as the implicit finite difference approach. Flow is influenced by several factors, including the Hartmann number, Newtonian heating, Navier slip parameter, Frank-Kamenetskii parameter, and mixed convection parameter. Shear stress and heat transfer rates were also investigated and reported. The steady-state and unsteady-state solutions are visually expressed in terms of velocity and temperature profiles. Due to the presence of opposing force factors such as the Lorentz force, the research found that the Hartmann number reduces the momentum profile. Fluid temperature and velocity increase as the thermal Biot number and Frank-Kamenetskii parameter increase. There are several scientific and infrastructure capabilities that use this type of flow, such flow including solar communication systems exposed to airflow, electronic devices cooled at room temperature by airflow, nuclear units maintained during unscheduled shutoffs, and cooling systems occurring in low circumstances. The current findings and the literature are very consistent, which recommend the application of the current study.

This study examines the flow of hyperbolic nanofluid over a stretching sheet in three dimensions. The influence of velocity slip on the flow and heat transfer properties of a hyperbolic nanofluid has been investigated. The partial differential equations for nanoparticle solid concentration, energy, and motion were turned into ordinary differential equations. Nanoparticle mass fluxes at boundaries are assumed to be zero, unlike surface concentrations. The influence of the main parameters on flow characteristics, surface friction coefficients, and the Nusselt number has been visualized. The results suggest that Brownian motion has a negligible impact on the heat transfer rate. The ratio of the elastic force to the viscosity force was found to decrease the fluid velocity. The resulting thermophysical properties of nanofluids are in agreement with previous research. The present findings can be used to expand the potential for using nanofluids as a coolant in critical thermophysical and industrial installations.

The combined quasineutral and zero-viscosity limits of the bipolar Navier-Stokes-Poisson system with boundary are rigorously proved by establishing the nonlinear stability of the approximate solutions. Based on the conormal energy estimates, we showed that the solutions for the original system converge strongly in $H^3$ space towards the solutions of the one-fluid compressible Euler system as long as the amplitude of the boundary layers is small enough.

In this paper, we aim to investigate the ( $2+1$ )-dimensional Heisenberg ferromagnetic spin chain equation that is used to describe the nonlinear dynamics of magnets. Two recent effective technologies, namely, the variational method and subequation method, are employed to construct the abundant soliton solutions. By these two methods, diverse solutions such as the bright soliton, dark soliton, bright-dark soliton, perfect periodic soliton, and singular periodic soliton are successfully extracted. The numerical results are illustrated in the form of 3-D plots and 2-D curves by choosing proper parametric values to interpret the dynamics of wave profiles. Finally, the physical interpretation of the acquired results is elaborated in detail. The results obtained in this study are helpful to explain some physical meanings of some nonlinear physical models in electromagnetic waves.