Mathematics and Computers in Simulation最新文献

The aim of this work is to conduct a rigorous mathematical analysis for global dynamics and numerical simulation of a recognized viral marketing (VM) model, which is described by a system of ordinary differential equations (ODEs). We first establish positivity and boundedness of solutions and then investigate local and global asymptotic stability properties of possible equilibrium points. As an important consequence, complex dynamics of the VM model is determined fully.

Secondly, we develop the Mickens’ methodology to design a nonstandard finite difference (NSFD) scheme, which is useful in numerical simulation of the VM model. The main advantage of the constructed NSFD scheme is that it has the ability to preserve important mathematical features of the continuous-time model for all finite values of the step size. These features include the positivity and boundedness of solutions, positively invariant sets, equilibrium points and their asymptotic stability properties. Consequently, the NSFD scheme is not only effective to simulate dynamics of the VM model, but also easy to be implemented.

Thirdly, to emphasize implications of the constructed mathematical analysis, an extended version combining the integer-order ODE model under consideration with the Caputo fractional derivative is considered and analyzed. From the mathematical analysis performed for the integer-order VM model, global dynamics of the fractional-order VM model is also investigated rigorously.

Finally, the theoretical insights are supported by a set of illustrative numerical experiments.

The findings of this research not only improve some existing results in the literature, but may also provide several useful real-life applications.

This contribution considers a torque control scheme consisting of model predictive control (MPC) in the inner control loop together with PI reference governor in the outer control loop and a decoupling feedforward control for an isotropic permanent magnet synchronous machine (PMSM). This innovative approach is known in literature as PI-MPC dual loop control. A particular emphasis is given to the control governor strategy which is the outer loop PI reference governor and allows to regulate the machine in the flux weakening region and is therefore only active for field weakening. In this context the analysis of the stability based on Lyapunov’ approach of the control loop in flux weakening region is shown. The desired currents represent the reference currents for the MPC, which forms the inner control loop. The MPC is adapted using an extended Kalman filter (EKF), which estimates inductance of the electrical system in $dq$ coordinates by using a bivariate polynomial. Compared measurements with a hardware-in-the-loop (HIL) system show the effectiveness of the proposed control scheme with respect to a standard PI controller in inner loop (PI-PI scheme) in the presence of saturated inputs and state of a PMSM. The proposed MPC uses just an optimal, proportional control and thus avoids windup effects. Measurement results in the presence of input and state saturations show that MPC is working without overshoot in the currents which leads to less needed power in input.

Analyzing uniform convergence of finite element method for a 2-D singularly perturbed convection–diffusion problem with exponential layers on Bakhvalov-type mesh remains a complex, unsolved problem. Previous attempts to address this issue have encountered significant obstacles, largely due to the constraints imposed by a specific mesh. These difficulties stem from three primary factors: the width of the mesh subdomain adjacent to the transition point, constraints imposed by the Dirichlet boundary condition, and the structural characteristics of exponential layers. In response to these challenges, this paper introduces a novel analysis technique that leverages the properties of interpolation and the relationship between the smooth function and the layer function on the boundary. By combining this technique with a simplified interpolation, we establish the uniform convergence of optimal order $k$ under an energy norm for finite element method of any order $k$. Numerical experiments validate our theoretical findings.

We present new fault jump estimates to guide local refinement in surface approximation schemes with adaptive spline constructions. The proposed approach is based on the idea that, since discontinuities in the data should naturally correspond to sharp variations in the reconstructed surface, the location and size of jumps detected in the input point cloud should drive the mesh refinement algorithm. To exploit the possibility of inserting local meshlines in one or the other coordinate direction, as suggested by the jump estimates, we propose a quasi-interpolation (QI) scheme based on locally refined B-splines (LR B-splines). Particular attention is devoted to the construction of the local operator of the LR B-spline QI scheme, which properly adapts the spline approximation according to the nature and density of the scattered data configuration. A selection of numerical examples outlines the performance of the method on synthetic and real datasets characterized by different geographical features.

Based on the hybrid event-triggered mechanism (HETM), the boundary stabilization issue for fractional-order nonlinear reaction–diffusion systems (FNRDSs) with time-varying delay is studied by using two kinds of measurements. First, when the system state is measurable, a event-triggered feedback controller (ETFC) is designed directly based on the average measured output. Secondly, for the case that the state is unmeasurable, an event-triggered feedback controller based on observer framework is constructed through the boundary point measurement information. Utilizing the Lyapunov method and Wirtinger’s inequality, sufficient conditions for the asymptotic stability of the system are given in the form of linear matrix inequalities (LMIs), respectively, in which the Razumikhin theorem is used to deal with time-varying delay. Meanwhile, it is proved that Zeno behavior can be excluded by the designed HETM. Finally, numerical simulations demonstrate the validity and feasibility of the proposed control scheme.

The study numerically examined a class of nonlinear singular differential problems known as the Lane–Emden differential equation, which emerges in numerous real-world situations. The primary goal of this work is to formulate a computationally efficient iterative technique for solving the nonlinear Lane–Emden initial value problems. The proposed approach is a hybrid of the homotopy perturbation method and the Padé approximation. The nonlinear singular Lane–Emden initial value problem (SLEIVP) is transformed into an equivalent recursive integral employing the Picard’s approach. To resolve the singularity and nonlinearity, the recursive integral equation is transformed into a system of integral equations by using the homotopy notion. Furthermore, to enhance the convergence rate of the technique, Padé approximation is taken into account. The convergence analysis for the proposed approach is also conducted. The present technique is tested on SLEIVPs and numerical findings are compared with the existing techniques, to demonstrate the accuracy, effectiveness and ease of use.

This paper investigates the linear quadratic (LQ) optimal control problem for the stochastic two-dimensional (2-D) systems governed by Roesser models with multiplicative noise. The main contribution is to give the necessary and sufficient optimality condition by proposing a set of novel forward and backward stochastic partial difference equations (FBSPDE), and to further present the explicitly optimal feedback control laws on the finite horizon and on the infinite horizon based on the Riccati-like difference equations and the algebraic equation, respectively. Several numerical simulations are provided to illustrate the performance of the designed controllers.

We construct and analyse nonstandard finite difference (NSFD) schemes for two epidemic optimal control problems. Firstly, we consider the well-known MSEIR system that can be used to model childhood diseases such as the measles, with the vaccination as a control intervention. The second optimal control problem is related to the 2014–2016 West Africa Ebola Virus Disease (EVD) outbreak, that came with the unprecedented challenge of the disease spreading simultaneously in three different countries, namely Guinea, Liberia and Sierra Leone, where it was difficult to control the considerable migrations and travels of people inbound and outbound. We develop an extended SEIRD metapopulation model modified by the addition of compartments of quarantined and isolated individuals. The control parameters are the exit screening of travelers and the vaccination of the susceptible individuals. For the two optimal control problems, we provide the results on: (i) the (global) stability of the disease-free and/or endemic equilibria of the state variable systems; (ii) the positivity and boundedness of solutions of the state variables systems; (iii) the existence, uniqueness and characterization of the optimal control solutions that minimizes the cost functional. On the other hand: (iv) we design Euler-based nonstandard finite difference versions of the Forward-Backward Sweep Method (NSFD-FBSM) that are dynamically consistent with the state variable systems; (v) we provide numerical simulations that support the theory and show the superiority of the nonstandard approach over the classical FBSM. The numerical simulations suggest that significantly increasing the coverage of the vaccine with its implementation for adults as well is essential if the recurrence of measles outbreaks is to be stopped in South Africa. They also show that the optimal control vaccination for the 2014-2016 EVD is more efficient than the exit screening intervention.