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In the current study, a well-known fractional nonlinear differential equation called Zakharov–Kuznetsov equation which has many applications in engineering and physics is investigated. An effective computational techniques namely Daftardar–Jafari Method (DJM) and He’s fractional derivative are employed to model the problem for obtaining a high accuracy of the analytical solution. The governing equations are solved and comparison of the results with exact solutions is presented to evaluate the precision of the presented method. The outcomes reveal that the DJM is efficient and easy to utilize. Further, three-dimensional contour plots are depicted according to the suitable parameters values.

A general concept of the long-range elastic interactions in continuous medium is proposed. The interaction is determined as a consequence of symmetry breaking of the elastic field distribution produced by the topological defect as isolated inclusions. It is proposed to treat topological defects as the source of elastic field that can be described in terms of this field. The source is considered as a nonlinear object which determines the effective charge of the field at large distances in the linear theory. The models of the nonlinear source are proposed.

We treat here the process of simulation of ion micro- and nanoprobes in detail using the matrix formalism for Lie algebraic tools. Similar approach allows realizing necessary analytical and numerical modeling procedures.

Nowadays ion micro- and nanoprobes are extensively applied in different branches of science and industry. It is known that similar facilities are very sensitive to certain of steering parameters of the systems. In other words, similar beam lines are high precision systems, requiring preliminary modeling for thorough analysis of possible optimal working regimes. In this paper we consider analytical and numerical models, which allow one to study effect of various aberrations on basic beam characteristics. Research process performs from linear to nonlinear model with step by step including nonlinear effects of different nature.

Previous papers of the authors consider some aspects of nonlinear models. The present paper deals with full conception of modeling process, generalizing most essential aberrations and providing adequate solution methods.

In this study, through the $(m+\frac{1}{G^{\prime }})$-expansion method, we extract soliton solutions to the coupled-Higgs equation. The studied nonlinear model is known to describe Higgs mechanism. The Higgs mechanism is essential to explain the generation mechanism of the property “mass” for gauge bosons. The proposed method is one of the most powerful methods for constructing soliton solutions for nonlinear partial differential equations. The obtained wave solutions include exponential, hyperbolic, and distinct structures of complex function solutions. The presented results may be helpful in explaining the physical features of various nonlinear physical phenomena. In order to analyze the dynamic behavior of all obtained solutions, we plot three-dimensional and two-dimensional graphs for the obtained solutions.

This paper studies the Lax pair (LP) of the $(1+1)$-dimensional Benjamin–Bona–Mahony equation (BBBE). Based on the LP, initial solution and Darboux transformation (DT), the analytic one-soliton solution will also be obtained for BBBE. This paper contains a bunch of soliton solutions together with bright, dark, periodic, kink, rational, Weierstrass elliptic and Jacobi elliptic solutions for governing model through the newly developed sub-ODE method. The BBBE has a wide range of applications in modeling long surface gravity waves of small amplitude. In addition, we will evaluate generalized breathers, Akhmediev breathers and standard rogue wave solutions for stated model via appropriate ansatz schemes.

This paper shows how the appearance and disappearance of high-frequency oscillations in a batch reactor can be predicted by means of a minimal model describing the fast dynamics of the reaction. The analysis is based on the bifurcations of the model with respect to the slowly varying species, namely product and reactants. The method of analysis can, in principle, be applied to any slow-fast system provided the evolution of the slow variables is monotonic.

A model is proposed for the population dynamics of an annual plant with a seed bank (i.e. in which a proportion of seeds remain dormant for at least one year). In this model, demographic parameters (dormancy and germination rate) of the seeds of the year are different from those of the seeds of the seed bank. First, a simple linear matrix model is deduced from the life cycle graph and a more complicated model is built by introducing density dependence effect. The obtained system, nonlinear with delay, can be simplified by a change of variables. A non-trivial fixed point of this system is obtained and the conditions of stability are studied. Under certain conditions (choice of exponential law for functional response of density dependence and absence of seed mortality before germination) we show that conditions of stability depend only on 3 parameters, the dormancy rate of the seeds of the year, dormancy rate of the seeds of the seed bank and the maximum potential fecundity of adults. Study of the behaviour of this model in the parameter space shows that the domain of demographic stability can be reduced if the dormancy rate of seeds of the year is low, even if the dormancy rate of seeds of the seed bank is high.

Electronic sensor devices in geophysical processes are required to measure and automate different tasks. Throughout history, people have created multiple type devices, but acoustics have an important application such as the content form description in deep wells, watersheds, lakes, caves, among others. The acoustic signal is capable of reflecting where other types of signals cannot operate, either by drawbacks or where fluid is displaced. A mathematical model is presented in this paper described in state space as a basic acoustic sensor description. The objective is to adjust the parameters allowing the acoustic device to describe a signal in its trajectory, representing in geophysical manner the cavity form. Therefore, the control is performed on the response of the acoustic sensor model, adjusted with a parameter estimation process. The simulation results counts convergence between the reference and identified signals.

This study aims to develop a realistic mathematical model of fabric. In contrast to other studies on fabric modeling as a deformable surface, the model described in this article takes into account the geometry of the object. Moreover, it integrates the nonlinear phenomena of the dynamic behavior of material. As input parameters, the weaving data that define the 3D structure of the object and the mechanical properties of the yarn that express its dynamics are used. Thus, the fabric model is composed of a geometrical model of fabric (structure) on which a model of yarn (material characterization) is added. This hypothesis may be reasonable since a fabric shows the result of a three-dimensional assembly of yarns judiciously disposed. Since these yarns interact dynamically: the main difficulty consists of defining the yarn model. In our case, it is composed of various nonlinear functions representing the dynamic behavior of yarn. In order to characterize the flexibility of material, the weight, the elasticity and any other mechanical characteristics defining the relation between the strain and the stretching out of the shape should be taken into account. Firstly, several works dealing with realistic mathematical models of fabric are described. A taxonomic classification is achieved in order to position our study (in comparison to the scientific community). Secondly, the model of the fabric is described. A geometrical model of the object is presented. It allows one to dimension the object in a 3D space and then to position it at its initial state. Subsequently, a nodal model of yarns is described, step by step, in order to demonstrate the separability of the various dynamic behaviors. These nodal links make it simple to integrate the proposed model in the global geometrical model. Thus, the methods of numerical resolution used to simulate the complete model of the fabric are exposed. One method is selected and used in order to improve the performances of the fabric simulator and to obtain better stability. Several simulations illustrate the quality of the results obtained.

Following the linear theory of thermoelastic materials with voids, a generalized Hamilton variational principle is extended to the thermoelastic plates with voids under the case of large deflection, and a 3D nonlinear mathematical model is presented. In this process, the balance equation of the entropy is converted to an equivalent form without the first-order time-derivative by integral, and the concept of the moments for the volume fraction of voids and temperature field is introduced. As application, the nonlinear dynamic and aerodynamic characteristics of simply-supported rectangular thermoelastic plates with voids for four different materials are studied and compared by using a Galerkin approach. The effects of the initial deflections and material parameters are considered in detail. In addition to providing a generalized Hamilton variational principle and a 3D nonlinear mathematical model, it is also provided a valuable numerical method to solve the dynamic problem directly in the paper. The theory and the method can be applied to solving various problems of the thermoelastic plates with voids easily.

Tensile cracking at the position where geometry changes is a typical failure mode of gravity dams under strong earthquakes. Steel reinforcement has been proposed to reduce the degree of dam cracking. In this paper, a nonlinear model is presented to consider the interaction effect between the steel reinforcement and the dam concrete in a combined concrete damage model and distributed-steel model. In the model, a composite constitutive model of a steel reinforcement-concrete element is proposed. The approach can model the process of gradual degradation of the concrete loading capacity and load transfer to the steel reinforcement by establishing the concrete and steel models separately. Taking a typical gravity dam with positions where geometry changes upstream and downstream as a case study, the influence of steel reinforcement on seismic damage of the gravity dam is investigated. The analytical results show that the steel reinforcement strengthening prevents cracks thoroughly around the elevation of the downstream slope change. However, the cracking around the elevation of the upstream slope change extends to the downstream direction. This reflects the transfer of fracture energy release during the cracking process.

A nonlinear dynamic model of microbial growth is established based on the theories of the diffusion response of thermodynamics and the chemotactic response of biology. Except for the two traditional variables, i.e. the density of bacteria and the concentration of attractant, the pH value, a crucial influencing factor to the microbial growth, is also considered in this model. The pH effect on the microbial growth is taken as a Gaussian function G₀e^{-(φ- φ_c)2/G₁}, where G₀, G₁ and φ_c are constants, φ represents the pH value and φ_c represents the critical pH value that best fits for microbial growth. To study the effects of the reproduction rate of the bacteria and the pH value on the stability of the system, three parameters a, G₀ and G₁ are studied in detail, where a denotes the reproduction rate of the bacteria, G₀ denotes the impacting intensity of the pH value to microbial growth and G₁ denotes the bacterial adaptability to the pH value.

When the effect of the pH value of the solution which microorganisms live in is ignored in the governing equations of the model, the microbial system is more stable with larger a. When the effect of the bacterial chemotaxis is ignored, the microbial system is more stable with the larger G₁ and more unstable with the larger G₀ for φ₀ > φ_c. However, the stability of the microbial system is almost unaffected by the variation G₀ and G₁ and it is always stable for φ₀ < φ_c under the assumed conditions in this paper. In the whole system model, it is more unstable with larger G₁ and more stable with larger G₀ for φ₀ < φ_c. The system is more stable with larger G₁ and more unstable with larger G₀ for φ₀ > φ_c. However, the system is more unstable with larger a for φ₀ < φ_c and the stability of the system is almost unaffected by a for φ₀ > φ_c.

The results obtained in this study provide a biophysical insight into the understanding of the growth and stability behavior of microorganisms.

This paper presents a guideline to systematically design and construct a micro quadrotor unmanned aerial vehicle (UAV), capable of autonomous flight. The designed micro UAV has a gross weight of less than 40 g including power supply sufficient for an 8-min flight. The design is divided into three parts. First, investigation is made on the structural design of a conventional quadrotor. The quadrotor frame is then carefully designed to avoid any potential structural natural frequencies within the range of rotors operating speeds, based on simulation results obtained from MSC Nastran. Second, avionic system of the aircraft will be discussed in detail, mainly focusing on the design of printed circuit boards which include sensors, microprocessors and four electronic speed controllers, specially catered for micro quadrotor design. Last, a mathematical model for the micro quadrotor is derived based on Newton–Euler formalism, followed by methods of identifying the parameters. The flight test results are later described, analyzed and illustrated in this paper.

Force production involves the coordination of multiple muscles, and the produced force levels can be attributed to the electrophysiology activities of those related muscles. This study is designed to explore the activity modes of extensor carpi radialis longus (ECRL) using surface electromyography (sEMG) at the presence of different handgrip force levels. We attempt to compare the performance of both the linear and nonlinear models for estimating handgrip forces. To achieve this goal, a pseudo-random sequence of handgrip tasks with well controlled force ranges is defined for calibration. Eight subjects (all university students, five males, and three females) have been recruited to conduct both calibration and voluntary trials. In each trial, sEMG signals have been acquired and preprocessed with Root–Mean–Square (RMS) method. The preprocessed signals are then normalized with amplitude value of Maximum Voluntary Contraction (MVC)-related sEMG. With the sEMG data from calibration trials, three models, Linear, Power, and Logarithmic, are developed to correlate the handgrip force output with the sEMG activities of ECRL. These three models are subsequently employed to estimate the handgrip force production of voluntary trials. For different models, the Root–Mean–Square–Errors (RMSEs) of the estimated force output for all the voluntary trials are statistically compared in different force ranges. The results show that the three models have different performance in different force ranges. Linear model is suitable for moderate force level (30%–50% MVC), whereas a nonlinear model is more accurate in the weak force level (Power model, 10%–30% MVC) or the strong force level (Logarithmic model, 50%–80% MVC).

This chapter aims to summarize the theories, models, methods and tools of modern econometrics which we have covered in the previous chapters. We first review the classical assumptions of the linear regression model and discuss the historical development of modern econometrics by various relaxations of the classical assumptions. We also discuss the challenges and opportunities for econometrics in the Big data era and point out some important directions for the future development of econometrics.

Because of its significant traits, magnetorheological (MR) damper becomes to be one of the most promising devices for vibration reduction. Many investigations have been done in the fields as automobiles, civil engineering and medical treatment. However, the applications of vibration-reduction under impulsive loads, which are essential for practical uses such as rocket launcher, weapon recoil system and many other applications are not well explored. A lot of dynamic models have been developed to describe the dynamic characteristics of MR damper for its employment when the load is random and smooth. While, when the loads are impulsive, little dynamic model can be used to describe the dynamic behaviour of the MR damper. In this paper, a novel MR impact damper for impulsive load with two damping passages in series, four long-thin flow passages in the piston head was exhibited and the model of this impact damper was developed for its use under impulsive load. Inertia damping force, which is caused by abrupt acceleration of the damper, was introduced. It is indicated that damping force of this novel MR damper generated is quite large, while the dynamic range of this impact damper is relative small.