In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. The problems of single-well, double-well and double-hump Van der Pol-Dufing equations are studied in this paper. The Chebyshev Neural Network (ChNN) model will be applied to obtain the numerical solutions of these types of equations for the first time. The hidden layer is eliminated by expanding the input pattern by Chebyshev polynomials which employs a single layer neural network. In order to modify the network parameters and to minimize the computed error function, a feed forward neural network model with error back propagation principle is used. The obtained numerical results form the ChNN model will be compared with the analytical solutions, namely Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), Differential Transform Method (DTM) and exact. Comparisons of the solutions obtained with existing numerical results show that this method is a capable tool for solving this kind of nonlinear problems.

The aim of this article is to use the Homotopy Analysis Method (HAM) to pinpoint the optimal location of leakage in an inclined pipeline containing hydrogen-natural gas mixture by obtaining quick and accurate analytical solutions for nonlinear transportation equations. The homotopy analysis method utilizes a simple and powerful technique to adjust and control the convergence region of the infinite series solution using auxiliary parameters. The auxiliary parameters provide a convenient way of controlling the convergent region of series solutions. Numerical solutions obtained by HAM indicate that the approach is highly accurate, computationally very attractive and easy to implement. The solutions obtained with HAM have been shown to be in good agreement with those obtained using the method of characteristics (MOC) and the reduced order modelling (ROM) technique.

Poverty and crime are two major problem areas. The economic theory of crime shows a direct correlation between poverty and crime. In this study, we propose a model that shows the correlation between poverty and crime. Then we obtain the dynamic system of the proposed model. In the next step, we will compute the reproductive number by finding the maximum eigenvalue of Jacobian matrix to study of stability and un-stability of the presented model. Poverty and crime are two major problem areas. The economic theory of crime shows a direct correlation between poverty and crime. In this study, we propose a model that shows the correlation between poverty and crime. Then we obtain the dynamic system of the proposed model. In the next step, we will compute the reproductive number by finding the maximum eigenvalue of Jacobian matrix to study of stability and un-stability of the presented model.

In this article, fractional calculus has been applied to study the motion of ions in a three-dimensional radio frequency quadrupole ion trap; we have called this arrangement a fractional quadrupole ion trap. The main purpose of the article is to show that by controlling the fractional parameter of a trapped ion, one can gain a more efficient mass separation. In what follows, we will see that with decreasing the fractional parameter, we can achieve a smaller first stability region. Note that a small stability diagram will result in a good and acceptable mass separation. Various methods can be proposed to obtain a desired ion acceleration with a sufficient accuracy for good mass separation, which is similar to the one obtained by a fractional ion trap. Some of these methods are using the effects of a damping force, a magnetic field or both on the confinement of particles in the quadrupole ion trap. The first stability regions are plotted for all of the aforementioned methods, and simulation results are provided to compare them with those for the fractional case.

We investigate the dynamics of an ion confined in a Paul–trap supplied by a fractional periodic impulsional potential. The Cantor–type cylindrical coordinate method is a powerful tool to convert differential equations on Cantor sets from cantorian–coordinate systems to Cantor–type cylindrical coordinate systems. By applying this method to the classical Laplace equation, a fractional Laplace equation in the Cantor–type cylindrical coordinate is obtained. The fractional Laplace equation is solved in the Cantor–type cylindrical coordinate, then the ions is modelled and studied for confined ions inside a Paul–trap characterized by a fractional potential. In addition, the effect of the fractional parameter on the stability regions, ion trajectories, phase space, maximum trapping voltage, spacing between two signals and fractional resolution is investigated and discussed.

We study importance of influential nodes in spreading of epidemic COVID-19 in a complex network. We will show that quarantine of important and influential nodes or consider of health protocols by efficient nodes is very helpful and effective in the controlling of spreading epidemic COVID-19 in a complex network. Therefore, identifying influential nodes in complex networks is the very significant part of dependability analysis, which has been a clue matter in analyzing the structural organization of a network. The important nodes can be considered as a person or as an organization. To find the influential nodes we use the technique for order preference by similarity to ideal solution (TOPSIS) method with new proposed formula to obtain the efficient weights. We use various centrality measures as the multi-attribute of complex network in the TOPSIS method. We define a formula for spreading probability of epidemic disease in a complex network to study the power of infection spreading with quarantine of important nodes. In the following, we use the Susceptible–Infected (SI) model to figure out the performance and efficiency of the proposed methods. The proposed method has been examined for efficiency and practicality using numerical examples.