Band 22, Heft 7-8

To study the effect of environmental noise on the spread of the disease, a stochastic Susceptible, Infective, Removed and Susceptible (SIRS) model with two viruses is introduced in this paper. Sufficient conditions for global existence of positive solution and stochastically asymptotic stability of disease-free equilibrium in the model are given. Then, it is shown that the positive solution is stochastically ultimately bounded and the moment average in time of the positive solution is bounded. Our results mean that the environmental noise suppresses the growth rate of the individuals and drives the disease to extinction under certain conditions. Finally, numerical simulations are given to illustrate our main results.

This paper deals with the Constrained Regulation Problem (CRP) for linear continuous-times fractional-order systems. The aim is to find the existence conditions of linear feedback control law for CRP of fractional-order systems and to provide numerical solving method by means of positively invariant sets. Under two different types of the initial state constraints, the algebraic condition guaranteeing the existence of linear feedback control law for CRP is obtained. Necessary and sufficient conditions for the polyhedral set to be a positive invariant set of linear fractional-order systems are presented, an optimization model and corresponding algorithm for solving linear state feedback control law are proposed based on the positive invariance of polyhedral sets. The proposed model and algorithm transform the fractional-order CRP problem into a linear programming problem which can readily solved from the computational point of view. Numerical examples illustrate the proposed results and show the effectiveness of our approach.

A general numerical framework is designed for the two-dimensional convection–diffusion–reaction (CDR) system. The compatibility of differential quadrature and finite difference methods (FDM) are utilized for the formulation. The idea is to switch one numerical scheme to another numerical scheme without changing the formulation. The only requirement is to input the weighting coefficients associated with the derivative discretizations to the general algorithm. Three numerical schemes comprising combinations of differential quadrature and FDMs are studied using the general algorithm. Properties of numerical schemes and the algorithm are analyzed by using the simulations of two-dimensional linear CDR system, Burgers’ equation, and Brusselator model.

In this paper the problems of the existence and stability of positive periodic solutions of inertial neural networks with time-varying delays are discussed by the use of Mawhin’s continuation theorem and Lyapunov functional method. Some sufficient conditions are obtained for guaranteeing the existence and stability of positive periodic solutions of the considered system. Finally, a numerical example is given to illustrate the effectiveness of the obtained results.

This paper aims to discuss a class of discontinuous bidirectional associative memory (BAM) neural networks with discrete and distributed delays. By using the set-valued map, differential inclusions theory and fundamental solution matrix, the existence of almost-periodic solutions for the addressed neural network model is firstly discussed under some new conditions. Subsequently, based on the non-smooth analysis theory with Lyapunov-like strategy, the global exponential stability result of the almost-periodic solution for the proposed neural network system is also established without using any additional conditions. The results achieved in the paper extend some previous works on BAM neural networks to the discontinuous case and it is worth mentioning that it is the first time to investigate the almost-periodic dynamic behavior for the BAM neural networks like the form in this paper. Finally, in order to demonstrate the effectiveness of the theoretical schemes, simulation results of two topical numerical examples are delineated.

This article is devoted to establish the existence of solution of ( α , β ) $\left(\alpha ,\beta \right)$ -order coupled implicit fractional differential equation with initial conditions, using Laplace transform method. The topological degree theory is used to obtain sufficient conditions for uniqueness and at least one solution of the considered system. Beside this, Ulam’s type stabilities are discussed for the proposed system. To support our main results, we present an example.

In this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

In this paper, we introduce the mild solution for a new class of noninstantaneous and nonlocal impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps. The existence of the mild solution is derived for the considered system by using fractional calculus, stochastic analysis and Sadovskii’s fixed point theorem. Finally, an example is also given to show the applicability of our obtained theory.

The fissures are ubiquitous in deep rock masses, and they are prone to instability and failure under dynamic loads. In order to study the propagation attenuation of dynamic stress waves in rock mass with different number of fractures under confining pressure, nonlinear theoretical analysis, laboratory model test and numerical simulation are used respectively. The theoretical derivation is based on displacement discontinuity method and nonlinear fissure mechanics model named BB model. Using ABAQUS software to establish a numerical model to verify theoretical accuracy, and laboratory model tests were carried out too. The research shows that the stress attenuation coefficient decreases with the increase of the number of fissures. The numerical simulation results and experimental results are basically consistent with the theoretical values, which verifies the rationality of the propagation equation.

For the entry guidance problem of hypersonic gliding vehicles (HGVs), an analytical predictor–corrector guidance method based on feedback control of bank angle is proposed. First, the relative functions between the velocity, bank angle and range-to-go are deduced, and then, the analytical relation is introduced into the predictor–corrector algorithm, which is used to replace the traditional method to predict the range-to-go via numerical integration. To eliminate the phugoid trajectory oscillation, a method for adding the aerodynamic load feedback into the control loop of the bank angle is proposed. According to the quasi-equilibrium gliding condition, the function of the quasi-equilibrium glide load along with the velocity variation is derived. For each guidance period, the deviation between the real-time load and the quasi-equilibrium gliding load is revised to obtain a smooth reentry trajectory. The simulation results indicate that the guidance algorithm can adapt to the mission requirements of different downranges, and it also has the ability to guide the vehicle to carry out a large range of lateral maneuvers. The feedback control law of the bank angle effectively eliminates the phugoid trajectory oscillation and guides the vehicle to complete a smooth reentry flight. The Monte Carlo test indicated that the guidance precision and robustness are good.

In recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

In this contribution, a non-linear arctic sea-ice model of fractional Duffing oscillator is given. The solution of the model was obtained using a new proposed analytical method, which is an elegant combination of asymptotic and Laplace methods. The result obtained showed that this method is a very powerful and efficient technique for finding the analytical solution of nonlinear fractional differential equation. From the analysis of the result, we observed that the impact of anthropogenic climate change on arctic sea-ice could lead to flooding in many coastal areas and low-lying island nations.

In this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

This paper considers two categories of fractional-order population growth models, where a time component is defined by Riemann–Liouville derivatives. These models are studied under the Lie symmetry approach, and we reduce the fractional partial differential equations to nonlinear ordinary differential equations. Subsequently, solutions of the latter are determined numerically or with the aid of Laplace transforms. Graphical representations for integral and trigonometric solutions are presented. A key feature of these models is the connection between spatial patterning of organisms versus competitive coexistence.