Volume 21, Issue 3-4 -

In this paper, we present a numerical simulation of a round impacting jet using coupled Smoothed Particle Hydrodynamics (SPH) and Finite Element (FE) methods. Numerical results are compared with the results of another simulation carried out by the CEL (Coupled Eulerian-Lagrangian) method. A water jet with a spherical head was used at an initial speed of 570 m/s to impact a flat plate made of Polymethyl-Methacrylate (PMMA). To model the entire process, the SPH method was used to model the water jet and the FE method for the PMMA structure. The distribution of the pressure on the impact surface and the resulting deformation of the structure were discussed. A Numerical model was developed using ABAQUS/Explicit version 6.14. Results of the coupled SPH-FE simulation were further validated. It is demonstrated that the CEL method presents smoother curves compared to the SPH method. These comparisons serve not only to validate the numerical simulation but also to give guidance in formulating the SPH-FEM numerical model.

The kill signals are alert about possible viruses that infect computer network to decrease the danger of virus propagation. In this work, we focus on a fractional-order SEIR-KS model in the sense of Caputo derivative to analyze the effects of kill signal nodes on the virus propagation. For this purpose, we first prove the existence and uniqueness of the model and give qualitative analysis. Then, we obtain the numerical solution of the model by using the Adams–Bashforth–Moulton algorithm. Finally, the effects of model parameters are demonstrated with graphics drawn by MATLAB program.

In this work, we established new travelling wave solutions for some nonlinear evolution equations with dual-power-law nonlinearity namely the Zakharov–Kuznetsov equation, the Benjamin–Bona–Mahony equation and the Korteweg–de Vries equation. The functional variable method was used to construct travelling wave solutions of nonlinear evolution equations with dual-power-law nonlinearity. The travelling wave solutions are expressed by generalized hyperbolic functions and the rational functions. This method presents a wider applicability for handling nonlinear wave equations.

This paper deals with the mathematical analysis of a general class of epidemiological models with multiple infectious stages for the transmission dynamics of a communicable disease. We provide a theoretical study of the model. We derive the basic reproduction number R0$\mathcal R_0$ that determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever R0≤1$\mathcal R_0 \leq 1$, while when R0>1$\mathcal R_0 \gt 1$, the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is globally asymptotically stable. A case study for tuberculosis (TB) is considered to numerically support the analytical results.

In this paper, we have investigated the Langevin and Brownian equations on fractal time sets using F α -calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the classical method of deriving the Fokker–Planck equation in order to obtain the Fokker–Planck equation on fractal time sets.

In this work, the (4+1)-dimensional Fokas equation, which is an important physics model, is under investigation. Based on the obtained soliton solutions, the new rational solutions are successfully constructed. Moreover, based on its bilinear formalism, a concise method is employed to explicitly construct its rogue-wave solution and interaction solution with an ansätz function. Finally, the main characteristics of these solutions are graphically discussed. Our results can be helpful for explaining some related nonlinear phenomena.

We propose a non-integer order model for the dynamics of the coinfection of HIV and HSV-2. We calculate the reproduction number of the model and study the local stability of the disease-free equilibrium. Simulations of the model for the variation of epidemiologically relevant parameters and the order of the non-integer order derivative, α , reveal interesting dynamics. These results are discussed from an epidemiologically point of view.

A nonlocal elasticity theory is a popular growing technique for mechanical analysis of the micro- and nanoscale structures which captures the small-size effects. In this paper, a comprehensive study was carried out to investigate the influence of the nonlocal parameter on the bifurcation behavior of a capacitive clamped-clamped nano-beam in the presence of the electrostatic and centrifugal forces. By using Eringen’s nonlocal elasticity theory, the nonlocal equation of the dynamic motion for a nano-beam has been derived using Euler–Bernoulli beam assumptions. The governing static equation of motion has been linearized using step by step linearization method; then, a Galerkin based reduced order model have been used to solve the linearized equation. In order to study the bifurcation behavior of the nano-beam, the static non-linear equation is changed to a one degree of freedom model using a one term Galerkin weighted residual method. So, by using a direct method, the equilibrium points of the system, including stable center points, unstable saddle points and singular points have been obtained. The stability of the fixed points has been investigated drawing motion trajectories in phase portraits and basins of attraction set and repulsion have been illustrated. The obtained results have been verified using the results of the prior studies for some cases and a good agreement has been observed. Moreover, the effects of the different values of the nonlocal parameter, angular velocity and van der Waals force on the fixed points have been studied using the phase portraits of the system for different initial conditions. Also, the influence of the nonlocal beam theory and centrifugal forces on the dynamic pull-in behavior have been investigated using time histories and phase portraits for different values of the nonlocal parameter.

The unsteady self-similar flow due to a permeable shrinking sheet is analyzed in this investigation. The current theoretical study is enacted in the light of correct self-similar formulation proposed by (Mehmood. A. Viscous flows: Stretching and shrinking of surfaces, Springer , 2017). For the existence of a meaningful solution the retarded boundary-layer developed due to retarded shrinking wall velocity uwx=±xt\left( {{u_w}\left( x \right) = {{ \pm x} \over t}} \right) is supported by the sufficient mass suction introduced at the shrinking surface. Main theme of this paper is to utilize the correct mathematical formulation for the shrinking sheet flow in order to investigate the existence of dual solutions of the considered problem. By using suitable self-similar transformations, the nonlinear Navier–Stokes equations are converted to corresponding non-linear ordinary differential equations. A reliable numerical technique (shooting method) is applied to solve the resulting ODEs for the involved physical parameters. Particular to this problem, it is observed that a sufficient amount of wall suction is mandatory S<Sc;Sc≤0\left( {S \lt {S_c}; {S_c} \le 0} \right) for the existence of meaningful solutions; and also the solutions are non-unique. As the magnitude of wall suction is reduced (i. e. S→ScS \to {S_c}) the solution seizes to exist by having non-zero coefficient of wall skin-friction, reflecting the absence of any flow separation. On the other hand, a stronger flow augmentation is noted for the increasing values of the wall suction parameter SS. The results are presented through graphs as well as in tabulated form. It is noticed that dual solutions exist for sufficiently large magnitude of the suction parameterS>Sc\left( { \left| S \right| \gt \left| {{S_c}} \right| } \right).

In this article, we study the existence and uniqueness of solutions of a switched coupled implicit ψ -Hilfer fractional differential system. The existence and uniqueness results are obtained by using fixed point techniques. Further, we investigate different kinds of stability such as Hyers–Ulam stability and Hyers–Ulam–Rassias stability. Finally, an example is provided to illustrate the obtained results.

Cournot’s game is one of the most distinguished and influential economic models. However, the classical integer order derivatives utilized in Cournot’s game lack the efficiency to simulate the significant memory characteristics observed in many economic systems. This work aims at introducing a dynamical study of a more realistic proposed competition Cournot-like duopoly game having fractional order derivatives. Sufficient conditions for existence and uniqueness of the new model’s solution are obtained. The existence and local stability analysis of Nash equilibrium points along with other equilibrium points are examined. Some aspects of global stability analysis are treated. More significantly, the effects of seasonal periodic perturbations of parameters values are also explored. The multiscale fuzzy entropy measurements for complexity are employed for this case. Numerical simulations are presented in order to verify the analytical results. It is observed that the time-varying parameters induce very complicated dynamics in perturbed Cournot duopoly game compared with the unperturbed game.

The Sylvester–Kac matrix, also known as Clement matrix, has many extensions and applications. The evaluation of determinant and spectra of many of its generalizations sometimes are hard to compute. Recently, E. Kılıç and T. Arikan proposed an extension the Sylvester–Kac matrix, where the main diagonal is a 2-periodic sequence. They found its determinant using a spectral technique. In this short note, we provide a simple proof for that result by calculating directly the determinant.

Air pollution is caused by contamination of air due to various natural and anthropogenic activities. The growing air pollution has diverse adverse effects on human health and other living species. However, a significant reduction in the concentration of air pollutants has been observed during the rainy season. Recently, a number of studies have been performed to understand the mechanism of removal of air pollutants due to the rain. These studies have found that rain is helpful in removing many air pollutants from the environment. In this paper, we proposed a mathematical model to investigate the role of rain in the removal of air pollutants and its subsequent impacts on the human population.

In this paper, the extended Boiti–Leon–Manna–Pempinelli equation (eBLMP) is first proposed, and by Ma’s [1] method, a class of lump and lump–kink soliton solutions is explicitly generated by symbolic computations. The propagation orbit, velocity and extremum of the lump solutions on ( x , y ) plane are studied in detail. Interaction solutions composed of lump and kink soliton are derived by means of choosing appropriate real values on obtained parameter solutions. Furthermore, 3-dimensional plots, 2-dimensional curves, density plots and contour plots with particular choices of the involved parameters are depicted to demonstrate the dynamic characteristics of the presented lump and lump–kink solutions for the potential function v  = 2ln(  f ( x )) x .

In this article, the differential transform method (DTM) is used to solve the nonlinear boundary value problems describing heat transfer in continuously moving fins undergoing convective-radiative heat dissipation. The thermal conductivity is variable and temperature dependent. The surface of the moving fin is assumed to be gray with a constant emissivity ɛ . The flow in the surrounding medium provides a constant heat transfer coefficient h over the entire surface of the moving fins. The effects of some physical parameters such as the Peclet number, Pe , thermal conductivity parameter, β , convection-conduction parameter, N c , radiation-conduction parameter, N r , and dimensionless convection-radiation sink temperature, θ a , on the temperature distribution are illustrated and explained.

This work presents the constrained optimal control of a fractionally damped elastic beam in which the damping characteristic is described with the Caputo fractional derivative of order 1/2. To achieve the optimal control that involves energy optimal control index with fixed endpoints, the fractionally damped elastic beam problem is first converted to a state space form of order 1/2 by using a change of coordinates. Then, the state and the costate equations are set in terms of Hamiltonian formalism and the constrained control law is acquired from Pontryagin Principle. The numerical solution of the problem is obtained with Grünwald-Letnikov approach by utilizing the link between the Riemann-Liouville and the Caputo fractional derivatives. Application of the formulations is demonstrated with an example and the illustrations are figured by MATLAB. Also, the effectiveness of the Grünwald-Letnikov approach is exhibited by comparing it with an iterative method which is one-step Adams-Bashforth-Moulton method.