On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order.

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz's equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered-it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.

Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense.

Fuzzy differential equations provide a crucial tool for modeling numerous phenomena and uncertainties that potentially arise in various applications across physics, applied sciences and engineering. Reliable and effective analytical methods are necessary to obtain the required solutions, as it is very difficult to obtain accurate solutions for certain fuzzy differential equations. In this paper, certain fuzzy approximate solutions are constructed and analyzed by means of a residual power series (RPS) technique involving some class of fuzzy fractional differential equations. The considered methodology for finding the fuzzy solutions relies on converting the target equations into two fractional crisp systems in terms of ρ-cut representations. The residual power series therefore gives solutions for the converted systems by combining fractional residual functions and fractional Taylor expansions to obtain values of the coefficients of the fractional power series. To validate the efficiency and the applicability of our proposed approach we derive solutions of the fuzzy fractional initial value problem by testing two attractive applications. The compatibility of the behavior of the solutions is determined via some graphical and numerical analysis of the proposed results. Moreover, the comparative results point out that the proposed method is more accurate compared to the other existing methods. Finally, the results attained in this article emphasize that the residual power series technique is easy, efficient, and fast for predicting solutions of the uncertain models arising in real physical phenomena.

Influence of flexoelectricity on coupled electromechanical response of 2D MXene/graphene-based hybrid piezocomposites.

This study investigates the influence of flexoelectricity on the coupled electromechanical behavior of MXene/graphene-based hybrid piezocomposite (MGHPC) plates. We developed an analytical model based on Navier's solution and Kirchhoff's plate theory, as well as an approximate model based on the Ritz method for validation purposes. A three-phase micromechanical modeling is developed for determining the effective properties of MGHPC composed of 2D MXene and graphene nano-reinforcements embedded in an epoxy matrix. These micromechanical models were implemented to predict the static and dynamic electromechanical response of MGHPC plates subject to various edge support and loading conditions. Both the analytical and approximate solutions provided unequivocal evidence of the profound impact of the flexoelectric effect on the bending and modal analysis of MGHPC nanoplates. The flexoelectric effect enhanced the stiffness of the nanoplate, irrespective of the support condition. This implies that MGHPC plates can be tailored for precise resonance frequencies and static deflection within nanoelectromechanical systems. This can be achieved by manipulating parameters such as boundary conditions and geometric attributes, including plate thickness/aspect ratio and graphene/MXene nano-reinforcements volume fractions.

Mathematical analysis of isothermal study of reverse roll coating using Micropolar fluid.

This article demonstrates a mathematical model and theoretical analysis of the Micropolar fluid in the reverse roll coating process. It is important because micropolar fluids account for the microstructure and microrotation of particles within the fluid. These characteristics are significant for accurately describing the behavior of complex fluids such as polymer solutions, biological fluids, and colloidal suspensions. First, we modeled the flow equations using basic laws of fluid dynamics. The flow equations are made modified using low Reynolds number theory. The simplified equations are solved analytically. The exact expression for velocity and pressure gradient are obtained, while pressure is calculated numerically using Simpson Rule. Graphical depictions are carried out to comprehend the impact of the newly emerged physical constraints. The influence of micropolar and microrotation parameters on the velocity, pressure and pressure gradient are elaborated with the help of different graphs.

Controlling wind turbine tower vibration under external force by applying control systems combination.

The global focus has recently shifted away from fuel-based power sources, and one of the most important projects for energy production is wind energy. To maintain low costs, the current research examines the problem of vibrations affecting wind turbine towers' performance (WTTs). In particular, the tower, resulting from excessive vibrations, can negatively affect a structure's power output and service life, as it can cause fatigue. Therefore, we conducted numerical tests on various types of controlled systems. Our tests revealed that combining a new technique cubic negative velocity control (CNVC) and linear negative acceleration control (LNAC) was the most effective and cost-efficient option for vibration damping. This solution was derived by using an approximation method for the averaging technique. The external force is an important component of a nonlinear dynamic system and can be characterized by two-degree-of-freedom (2-DOF) differential coupled equations. After implementing the control measures, we conducted a numerical analysis of the vibration values before and after the operation. Stability is studied numerically. The numerical and approximate solutions were confirmed through the frequency response equation and time history with fourth-order Runge-Kutta (RK-4). Finally, we investigated the effect of parameters and compared our results with those of previously published studies.

Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative.

This paper presents the Elzaki homotopy perturbation transform scheme (EHPTS) to analyze the approximate solution of the multi-dimensional fractional diffusion equation. The Atangana-Baleanu derivative is considered in the Caputo sense. First, we apply Elzaki transform (ET) to obtain a recurrence relation without any assumption or restrictive variable. Then, this relation becomes very easy to handle for the implementation of the homotopy perturbation scheme (HPS). We observe that HPS produces the iterations in the form of convergence series that approaches the precise solution. We provide the graphical representation in 2D plot distribution and 3D surface solution. The error analysis shows that the solution derived by EHPTS is very close to the exact solution. The obtained series shows that EHPTS is a very simple, straightforward, and efficient tool for other problems of fractional derivatives.

On nonlinear dynamics of COVID-19 disease model corresponding to nonsingular fractional order derivative.

This manuscript is devoted to investigate the mathematical model of fractional-order dynamical system of the recent disease caused by Corona virus. The said disease is known as Corona virus infectious disease (COVID-19). Here we analyze the modified SEIR pandemic fractional order model under nonsingular kernel type derivative introduced by Atangana, Baleanu and Caputo ([Formula: see text]) to investigate the transmission dynamics. For the validity of the proposed model, we establish some qualitative results about existence and uniqueness of solution by using fixed point approach. Further for numerical interpretation and simulations, we utilize Adams-Bashforth method. For numerical investigations, we use some available clinical data of the Wuhan city of China, where the infection initially had been identified. The disease free and pandemic equilibrium points are computed to verify the stability analysis. Also we testify the proposed model through the available data of Pakistan. We also compare the simulated data with the reported real data to demonstrate validity of the numerical scheme and our analysis.

Mathematical analysis of SIRD model of COVID-19 with Caputo fractional derivative based on real data.

We discuss a fractional-order SIRD mathematical model of the COVID-19 disease in the sense of Caputo in this article. We compute the basic reproduction number through the next-generation matrix. We derive the stability results based on the basic reproduction number. We prove the results of the solution existence and uniqueness via fixed point theory. We utilize the fractional Adams-Bashforth method for obtaining the approximate solution of the proposed model. We illustrate the obtained numerical results in plots to show the COVID-19 transmission dynamics. Further, we compare our results with some reported real data against confirmed infected and death cases per day for the initial 67 days in Wuhan city.

Mathematical Analysis of Pseudoplastic Polymers during Reverse Roll-Coating.

This article presents a mathematical model and theoretical analysis of coating of a thin film of non-Newtonian polymers as they travel through a small space between two reverse-rotating rolls. The dimensionless forms of the governing equations are simplified with the help of the lubrication approximation theory (LAT). By using the perturbation technique, the analytical solutions for velocity, flow rate and pressure gradient were obtained. From an engineering point of view, some significant results such as thickness of the coated web, pressure distribution, separation points, separation force and power input were computed numerically. The effect of velocities ratio k and Weissenberg number We on these physical quantities is presented graphically; others are shown in tabular form. It is noted that the involved material parameters provide a mechanism to control the flow rate, pressure distribution, the thickness of coating, separation force and power input. Moreover, the separation point is shifted toward the nip region by increasing velocities ratio k.

Approximating distance between sets by multivalued coupling with application to uniformly convex Banach spaces.

In this paper, our aim is to ascertain the distance between two sets iteratively in two simultaneous ways with the help of a multivalued coupling define for this purpose. We define the best proximity points of such couplings that realize the distance between two sets. Our main theorem is deduced in metric spaces. As an application, we obtain the corresponding results in uniformly convex Banach spaces using the geometry of the space. We discuss two examples.