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This paper seeks to present an optimization method to estimate the solutions of nonlinear oscillation equations of fractional order. The mentioned method is based on Bernstein polynomials (Bps). In the presented numerical approach, the operational matrices of the ordinary and fractional derivatives of Bernstein polynomials are utilized to estimate the solution of the model under the study. In this technique, the unknown function is expanded in terms of Bps. By using the residual function and its 2-norm, the problem under consideration is converted into a constrained nonlinear optimization one. So that, the constraint equations are obtained from the given initial conditions and the object function is obtained from the residual function. Finally, we obtain the unknown coefficients optimally by a set of unknown Lagrange multipliers. The main advantage of this approach is that it reduces such problems to those optimization problems, which greatly simplifies them and also leads to obtain a good approximate solution for them. The accuracy and efficiency of the presented method are supported by some examples. At the end, we compare the numerical results with other results.

This analysis deals the steady and the incompressible MHD fluid flow by sinusoidal walls of a hexagonal cavity with a cylindrical obstacle at the center. Finite element method (FEM), a numerical modeling method is used to examine the heat transfer and fluid flow controlled by the energy equation, the continuity equation and Navier-Stokes equations, which are converted to dimensionless form through suitable parameterization, which are tackled by finite element method. The temperature distribution and velocity fields are displayed for various parameters which are Richardson number, Hartmann number, Reynolds number, velocities’ amplitudes ratio, temperatures’ amplitudes ratio and phase deviation. This graphical study shows good convergent results of temperature and velocity for variation of involved parameters. At the end, the significant effects of the heat transfer rate are discussed in terms of the Nusselt number.

A mathematical model of discrete fractional equations with initial condition is constructed to study the tumor-immune interactions in this research. The model is a system of two nonlinear difference equations in the sense of Caputo fractional operator. The applications of Banach’s and Leray–Schauder’s fixed point theorems are used to analyze the existence results for the proposed model. Additionally, we developed several kinds of Ulam’s stability results for the suggested model. The tumor-immune fractional map’s dynamic behavior is numerical analyzed for some special cases. Further, adaptive control law is proposed to stabilize the fractional map and a control scheme is introduced to enhance the synchronization of the fractional model.

Due to realistic implementations of heat transfer flows, this study communicates with the two cases of the prescribed thermal process. A comparative analysis of the time-dependent flow of a micropolar fluid between two problems of linear stretching sheet and exponential stretching sheet is demonstrated. In the context of the linear stretching sheet, the investigation covers situations involving prescribed surface temperature (PST) and prescribed heat flux (PHF). Likewise, for the exponential stretching sheet, the study examines cases of prescribed exponential order temperature (PEST) and prescribed exponential order heat flux (PEHF). For both problems, the two cases of the thermal process are examined for heat transfer analysis considering joule heating and nonlinear thermal radiation. The two-dimensional flow in the presence of a variable magnetic field is explored for both the linear and exponential stretching sheet problems. Graphical representations are employed to provide a comparative illustration of flow properties concerning relevant parameters for both cases. From the graphical and tabular analysis, we deduce that an exponential stretching sheet provides more consequential results as compared to a linear stretching sheet. Furthermore, the material parameter demonstrates an increase in the velocity field for both of the considered sheets.

In this paper, a general system of quadratically perturbed system of modified fractional differential equations (FDEs) is considered for the solution existence, solution uniqueness, stability results, numerical scheme and computational applications. The presumed perturbed system is more general and several preexisting problems become its special cases. Fixed point results are applied for the theoretical results. The Lagrange’s polynomial is used to approximate the nonlinear system and a useful numerical scheme is established. For an application, a complex system of chaotic attractor is given and is computational studied via some graphical presentation.

This article investigates the modeling and numerical simulation of Magnetohydrodynamic (MHD) buoyancy-driven convection flow in a differentially heated, square enclosure. Left vertical side is given a high temperature and the right vertical side is sustained at a low temperature. Horizontal sides of the enclosure are insulated. A constant magnetic field is presumed horizontally. Findings of the governing differential equations are explored numerically considering the impact of Magneto-hydrodynamic (MHD). Problem is deciphered by Galerkin finite element approach in COMSOL Multiphysics. Numerical solutions are computed for different values of Rayleigh number ranging 103≤Ra≤107 and Hartmann number ranging 0≤Ha≤40. Rate of heat that passes from the heated side is affected by increasing Rayleigh and Hartmann numbers. Comportment of MHD free convection heat flow from transient to steady state is numerically examined for a period of 0 to 1 s. The numerical solutions are discussed in respect of streamlines, iso-contours, and isotherms. In addition, physical quantities such as velocity and Nusselt number are studied. It is seen that with increasing values of Rayleigh number there is increase in local Nusselt number distribution on heated side of the cavity. Velocity distribution in the flow domain decreases in variations with increasing Hartmann number.

Taking two efficient analytical techniques, including the generalized exponential rational function method and the extended sinh–Gordon equation expansion method, into account several optical solutions to a variety of nonlinear Schrödinger’s equation with group velocity dispersion are constructed. We are investigated several families of localized structures (bright solitons) for attractive nonlinearities including localized structures in both angular directions in addition to localized structures in one-directional and homogeneous in the other. To provide a better understanding of the obtained results, some numerical simulations for the obtained solutions are carried out. Further, the obtained solutions are represented graphically in order to clarify the vision and deep understanding of the mechanism of all nonlinear phenomena described by these solutions. The acquired results in this survey are new and have not been obtained in previous literature. Moreover, both utilized techniques can be assumed as an important tool for the explanation of some nonlinear physical phenomena such as the modulated envelope localized structures in plasma physics and fluid mechanics. Also, the obtained solutions can help researchers interested in studying the acoustic nonlinear modulated structures in different plasma models especially bright envelope solitons.

In this study, a new attempt has been made using mathematical modelling to study dynamic behaviour and estimate the final size of spread of the psychological stress arising due to sudden outbreak of COVID-19 in India. The proposed mathematical model examines and includes different behaviours of transition from one process to another in current situation and study their propagation mode. We propose a mathematical model, where two different type of psychological stresses occur due to COVID-19 situation and its impact on people’s life such as their mental well being and happiness. We present some sufficient conditions for the vanishing or spreading of the psychological stress through qualitative and quantitative analysis. The basic reproduction number (R0) of the model is computed and the local and the global stabilities of different equilibria are studied. Moreover, to better understand the level of psychological stress and decreasing mental well-being during the COVID-19 outbreak in India, we also conducted an online survey. Our findings establish several factors associated with level of psychological impact and mental health status. Based on the empirical analysis, we found that psychological stress has a significant negative influence on mental well being. Further, this study confirms that coping strategies with stress have significantly contributed towards the betterment in the mental well-being of the people. Numerical simulations are also given to illustrate the theoretical results. The results of the present study can be generalized to the society, Government, and others that they can adopt different strategies to avoid stressful situations during COVID-19 outbreak. The findings suggest that policy-makers, Government officials should focus on coping strategies to combat with pandemic disease.