Search Results (16)

This study establishes a fractional-order model (FOM) to describe the rumor spreading process. Members of society in this FOM are classified into three categories that change with time—the population that is ignorant of the rumors and does not know them, the population that is aware of the truth of the rumors but does not believe them, and the spreaders of rumors—taking into consideration awareness programs (APs) through media reports as a subcategory that changes over time where paying attention to these APs makes ignorant individuals avoid believing rumors and become better-informed individuals. We prove the positivity and boundedness of the FOM solutions. The feasible equilibrium points (EPs) and their local asymptotical stability (LAS) are analyzed based on the control reproduction number (CRN). Then, we examine the influence of model parameters that emerge with the CRN through a sensitivity analysis.A fractional optimal control problem (FOCP) is formulated by considering three time-dependent control measures in the suggested FOM to capture the spread of rumors; $u_1$, $u_2$, and $u_3$ represent the contact control between rumor spreaders and ignorant people, control media reports, and control rumor spreaders, respectively. We derive the necessary optimality conditions (NOCs) by applying Pontryagin’s maximum principle (PMP). Different optimal control strategies are proposed to reduce the negative effects of rumor spreading and achieve the maximum social benefit. Numerical simulation is implemented using a forward–backward sweep (FBS) approach based on the predictor–corrector method (PCM) to clarify the efficiency of the proposed strategies in order to decrease the number of rumor spreaders and increase the number of aware populations. Full article

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid’s fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme. Full article

In this article, we consider a delayed system of first-order hyperbolic differential equations. The presence of the delay term in first-order hyperbolic delay differential equations poses significant challenges in both analysis and numerical solutions. The delay term also makes it more difficult to use standard numerical methods for solving differential equations, as these methods often require that the differential equation be evaluated at the current time step. To overcome these challenges, specialized numerical methods and analytical techniques have been developed for solving first-order hyperbolic delay differential equations. We investigated and presented analytical results, such as the maximum principle and stability results. The propagation of discontinuities in the solution was also discussed, providing a framework for understanding its behavior. We presented a fractional-step method using a backward finite difference scheme and showed that the scheme is almost first-order convergent in space and time through the derivation of the error estimate. Additionally, we demonstrated an application of the proposed method to the problem of variable delay differential equations. We demonstrated the practical application of the proposed method to solving variable delay differential equations. The proposed algorithm is based on a numerical approximation method that utilizes a finite difference scheme to discretize the differential equation. We validated our theoretical results through numerical experiments. Full article

Purpose: This paper studies a simple SVIR (susceptible, vaccinated, infected, recovered) type of model to investigate the coronavirus’s dynamics in Saudi Arabia with the recent cases of the coronavirus. Our purpose is to investigate coronavirus cases in Saudi Arabia and to predict the early eliminations as well as future case predictions. The impact of vaccinations on COVID-19 is also analyzed. Methods: We consider the recently introduced fractional derivative known as the generalized Hattaf fractional derivative to extend our COVID-19 model. To obtain the fitted and estimated values of the parameters, we consider the nonlinear least square fitting method. We present the numerical scheme using the newly introduced fractional operator for the graphical solution of the generalized fractional differential equation in the sense of the Hattaf fractional derivative. Mathematical as well as numerical aspects of the model are investigated. Results: The local stability of the model at disease-free equilibrium is shown. Further, we consider real cases from Saudi Arabia since 1 May–4 August 2022, to parameterize the model and obtain the basic reproduction number ${\mathcal{R}}_0^v\approx 2.92$. Further, we find the equilibrium point of the endemic state and observe the possibility of the backward bifurcation for the model and present their results. We present the global stability of the model at the endemic case, which we found to be globally asymptotically stable when ${\mathcal{R}}_0^v>1$. Conclusion: The simulation results using the recently introduced scheme are obtained and discussed in detail. We present graphical results with different fractional orders and found that when the order is decreased, the number of cases decreases. The sensitive parameters indicate that future infected cases decrease faster if face masks, social distancing, vaccination, etc., are effective. Full article

The coupling effects of sandstorm and dust from coal bases themselves can have a major impact on the atmospheric environment as well as on human health. The typical coal resource city of Wuhai in Inner Mongolia was selected in order to study these impacts during a severe sandstorm event in March 2021. Particulate matter (PM₁, PM_2.5 and PM₁₀) and total suspended particulate matter (TSP) samples were collected during the sandstorm event of 15–19 March 2021 and non-sandstorm weather (11–13 March 2021) and analyzed for their chemical composition. The concentrations of PM₁, PM_2.5, PM₁₀ and TSP in Wuhai city during the sandstorm were 2.2, 2.6, 4.8 and 6.0 times higher than during non-sandstorm days, respectively. Trace metals concentrations in particles of different sizes generally increased during the sandstorm, while water-soluble ions decreased. Positive matrix fraction (PMF) results showed that the main sources of particles during both sandstorm and non-sandstorm days were industrial emissions, traffic emissions, combustion sources and dust. The proportion of industrial emissions and combustion sources increased compared with non-sandstorm days, while traffic emissions and dust decreased. The backward trajectory analysis results showed that airflows were mainly transported over short distances during non-sandstorm days, and high concentration contribution source areas were from southern Ningxia, southeast Gansu and western Shaanxi. The airflow was mainly transported over long distances during the sandstorm event, and high concentration contribution source areas were from northwestern Inner Mongolia, southern Russia, northern and southwestern Mongolia, and northern Xinjiang. A health risk analysis showed that the risk to human health during sandstorm days related to the chemical composition of particles was generally 1.2–13.1 times higher than during non-sandstorm days. Children were more susceptible to health risks, about 2–6.3 times more vulnerable than adults to the risks from heavy metals in the particles under both weather conditions. Full article

In this paper, we introduce new three-point fractional formulas which represent three generalizations for the well-known classical three-point formulas; central, forward and backward formulas. This has enabled us to study the function’s behavior according to different fractional-order values of $\alpha$ numerically. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability. Full article

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have $\min \{2-\alpha ,2-\beta \}$-order convergence in time and spectral accuracy in space for smooth solutions, where $\alpha ,\beta$ are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow. Full article

An exponential-type function was discovered to transform known difference formulas by involving a shifted parameter $\theta$ to approximate fractional calculus operators. In contrast to the known $\theta$ methods obtained by polynomial-type transformations, our exponential-type $\theta$ methods take the advantage of the fact that they have no restrictions in theory on the range of $\theta$ such that the resultant scheme is asymptotically stable. As an application to investigate the subdiffusion problem, the second-order fractional backward difference formula is transformed, and correction terms are designed to maintain the optimal second-order accuracy in time. The obtained exponential-type scheme is robust in that it is accurate even for very small $\alpha$ and can naturally resolve the initial singularity provided $\theta =-\frac{1}{2}$, both of which are demonstrated rigorously. All theoretical results are confirmed by extensive numerical tests. Full article

In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results. Full article

In this paper, we present a systematic study concerning the developments of the oscillation results for the fractional difference equations. Essential preliminaries on discrete fractional calculus are stated prior to giving the main results. Oscillation results are presented in a subsequent order and for different types of equations. The investigation was carried out within the delta and nabla operators. Full article

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from $\mathcal{O}\left(N_s^2\right)$ to $\mathcal{O}\left(N_s\right)$ and the computational complexity from $\mathcal{O}\left(N_s^3\right)$ to $\mathcal{O}(N_{s}log{N}_s)$ in each iterative step, where $N_s$ is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method. Full article

The paper presents time, frequency, and real-time properties of a fractional-order PID controller (FOPID) implemented at a STM 32 platform. The implementation uses CFE approximation and discrete version of a Grünwald–Letnikov operator (FOBD). For these implementations, experimental step responses and Bode frequency responses were measured. Real-time properties of the approximations are also examined and analyzed. Results of tests show that the use of CFE approximation allows to better keep the soft real-time requirements with an accuracy level a bit worse than when using the FOBD. The presented results can be employed in construction-embedded fractional control systems implemented at platforms with limited resources. Full article

The flow in a Francis turbine at full load is characterised by the development of an axial vortex rope in the draft tube. The vortex rope often promotes cavitation if the turbine is operated at a sufficiently low Thoma number. Furthermore, the vortex rope can evolve from a stable to an unstable behaviour. For CFD, such a flow is a challenge since it requires solving an unsteady cavitating flow including rotor/stator interfaces. Usually, the numerical investigations focus on the cavitation model or the turbulence model. In the present works, attention is paid to the strategy used for the time integration. The vortex rope considered is an unstable cavitating one that develops downstream the runner. The vortex rope shows a periodic behaviour characterized by the development of the vortex rope followed by a strong collapse leading to the shedding of bubbles from the runner area. Three unsteady RANS simulations are performed using the ANSYS CFX 17.2 software. The turbulence and cavitation models are, respectively, the SST and Zwart models. Regarding the time integration, a second order backward scheme is used excepted for the transport equation for the liquid volume fraction, for which a first order backward scheme is used. The simulations differ by the time step and the number of internal loops per time step. One simulation is carried out with a time step equal to one degree of revolution per time step and five internal loops. A second simulation used the same time step but 15 internal loops. The third simulations used three internal loops and an adaptive time step computed based on a maximum CFL lower than 2. The results show an influence of the time integration strategy on the cavitation volume time history both in the runner and in the draft tube with a risk of divergence of the solution if a standard set up is used. Full article

In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald–Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in embedded systems, due to their more convenient form compared to Caputo and Riemann–Liouville definitions or Laplace transforms, based on the discrete convolution operation. A well-known difficulty relates to the non-locality of the operator, implying continually increasing numbers of processed samples, which may reach the limits of available memory or lead to exceeding the desired computation time. In the study presented here, several promising software optimization techniques were analyzed and tested in the evaluation of the variable fractional-order backward difference and derivative on two different Arm^® Cortex^®-M architectures. Reductions in computation times of up to 75% and 87% were achieved compared to the initial implementation, depending on the type of Arm^® core. Full article

This paper presents integer and linear time-invariant fractional order (FO) models of a closed-loop electric individual-wheel drive implemented on an autonomous platform. Two discrete-time FO models are tested: non-commensurate and commensurate. A classical model described by the second-order linear difference equation is used as the reference. According to the sum of the squared error criterion (SSE), we compare a two-parameter integer order model with four-parameter non-commensurate and three-parameter commensurate FO descriptions. The computer simulation results are compared with the measured velocity of a real autonomous platform powered by a closed-loop electric individual-wheel drive. Full article