Hyperbolic Tangent Method Research Articles

The propagation of shape changing soliton in a nonuniform nonlocal media

Magnetization dynamics in uniformly magnetized ferromagnetic media is studied by using Landau—Lifshitz—Gilbert equation. The nonlinear evolution equation is integrable with site-dependent and biquadratic exchange interaction by means of Landau—Lifshitz (LL) equation which is well understood. In the present work, we construct the exact solitary solutions of the nonlinear evolution equation, particularly, we employ the modified extended tangent hyperbolic function method. We show the shape changing property of solitons for the given integrable system in the presence of damping as well as inhomogeneities.

Formation of solitary structures in uniform and nonuniform magnetoplasmas with superthermal electrons: a non-reductive perturbative approach

Electrostatic solitary structures are studied in uniform and nonuniform magnetoplasmas with superthermal electrons. In the linear analysis, the differences in the acoustic frequencies for Maxwellian, Cairns, and Kappa distributed electrons for both homogeneous and inhomogeneous plasmas are highlighted and discussed. It is shown that using the linear dispersion relation, nonlinear Zakharov-Kuznetsov (ZK) equation can be derived both for the homogeneous and inhomogeneous magnetoplasmas. The solution of the ZK equation is presented using the tangent hyperbolic method. It is found that the increasing magnetic field and the angle of propagation enhances the amplitude whereas the increasing number density mitigates the amplitude of the acoustic drift solitary wave. Furthermore, it is observed that the amplitude of the solitary structure is maximum for Cairns, intermediate for Maxwellian, and minimum for the Kappa distributed electrons. The results presented in this paper may be beneficial to understand the formation of electrostatic drift solitary waves in planetary environments where the nonthermal population of electrons are observed by various satellite missions.

Ion acoustic shock waves in plasmas with warm ions and kappa distributed electrons and positrons

The monotonic and oscillatory ion acoustic shock waves are investigated in electron-positron-ion plasmas (e-p-i) with warm ions (adiabatically heated) and nonthermal kappa distributed electrons and positrons. The dissipation effects are included in the model due to kinematic viscosity of the ions. Using reductive perturbation technique, the Kadomtsev-Petviashvili-Burgers (KPB) equation is derived containing dispersion, dissipation, and diffraction effects (due to perturbation in the transverse direction) in e-p-i plasmas. The analytical solution of KPB equation is obtained by employing tangent hyperbolic (Tanh) method. The analytical condition for the propagation of oscillatory and monotonic shock structures are also discussed in detail. The numerical results of two dimensional monotonic shock structures are obtained for graphical representation. The dependence of shock structures on positron equilibrium density, ion temperature, nonthermal spectral index kappa, and the kinematic viscosity of ions are also discussed.

A Note on Exact Travelling Wave Solutions for Nonlinear PHI-Four Equation

The purpose of this work is to reveal the dynamical behavior of the nonlinear PHI-four equation, which is an interesting and very useful model in particle physics and mathematical physics. As a result, travelling wave solutions of nonlinear PHI-four equation are formally derived by employing hyperbolic tangent method in this paper. This paper shows that hyperbolic tangent method can be a powerful tool in obtaining evolution solutions of nonlinear system.

한국 남부해역의 수온약층 추출 알고리즘 개발

A new algorithm was developed, not only to detect the existence of a thermocline, but also to extract the thermocline parameters (such as thermocline thickness, mixed layer thickness, maximum temperature gradient, and temperature difference of thermocline), using the vertical profile of water temperature. According to Kappa analysis, in order to find adequate threshold values of vertical water temperature gradients <TEX>${\Delta}T$</TEX> (<TEX>$^{\circ}C/m$</TEX>), agreement and reliability were 87% and 0.74 respectively, in the conditions of maximum <TEX>${\Delta}T{\geq}0.5$</TEX> and surface and bottom layers <TEX>${\Delta}T</TEX><TEX><</TEX><TEX>{\mid}0.2{\mid}$</TEX>. Also, three different kinds of methods, viz. 1. Gradient method, 2. Hyperbolic tangent method, and 3. Differential hyperbolic tangent method, were tested to extract the key parameters of a thermocline. Comparing the results of three different methods, the differential hyperbolic tangent method was the most appropriate to extract the start and end point of a thermocline curve.

Analytic study of the generalized Burger’s–Huxley equation by hyperbolic tangent method

In this research, exact traveling wave solutions of the generalized Burger’s–Huxley equation are obtained which occurs in the description of many non-linear wave phenomena. The analysis rests mainly on the standard hyperbolic tangent (tanh) method. This method is a powerful technique for the computation of closed form tanh-solutions for nonlinear partial differential equations. The types of exact solutions are analyzed, depending on the values of the equation parameters.

Nonlinear electrostatic shock waves in inhomogeneous plasmas with nonthermal electrons

Density inhomogeneity driven linear and nonlinear ion drift waves are investigated in a plasma consisting of heavy ions and non-thermal electrons. The dissipation is introduced in the system by the ion-neutral collision frequency. The nonlinear Korteweg de Vries Burgers (KdVB) and Burgers like equations are derived in the small amplitude limit, and the solution is obtained using the tangent hyperbolic method. It is found that the system under consideration admits rarefactive shock structures. It is observed that the ion-neutral collision frequency, nonthermal electron population, inverse density inhomogeneity scalelength, and the ambient magnetic field affect the propagation characteristics of the drift shock waves. The present study may be applicable in regions of space where nonthermal electrons and heavy ions have been observed.

Ion-acoustic shock waves in a plasma with weakly relativistic warm ions, thermal positrons and a background electron nonextensivity

A weakly nonlinear analysis is carried out to derive a Korteweg–de Vries-Burgers-like equation for small, but finite amplitude, ion-acoustic waves in a dissipative plasma consisting of weakly relativistic ions, thermal positrons and nonextensive electrons. The travelling wave solution has been acquired by employing the tangent hyperbolic method. Our results show that in a such plasma, ion-acoustic shock waves, the strength and steepness of which are significantly modified by relativistic, nonextensive and dissipative effects, may exist. Interestingly, we found that because of ion kinematic viscosity, an initial solitonic profile develops into a shock wave. This later evolves towards a monotonic profile (dissipation-dominant case) as the electrons deviate from their Maxwellian equilibrium. Our investigation may help to understand the dissipative structures that may occur in high-energy astrophysical plasmas.

Exact solutions for the transformed reduced Ostrovsky equation via the [formula omitted]-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions

Although the F-expansion method is a relatively old method for obtaining exact solutions for nonlinear partial differential equations, its advantage over the other auxiliary equation methods is shown in this work for solving the transformed reduced Ostrovsky equation. In fact, to the authors’ knowledge, this is the first time that a largest number of exact solutions for the auxiliary equation F′(ω)=PF4(ω)+QF2(ω)+R has been presented. It is used in this work to get 52 types of exact solution: six for Weierstrass-elliptic function solutions and the rest for Jacobian-elliptic function solutions. However, in consideration of Parkes (2010) [9], who states that the term ‘new’ has to be used very carefully, it is very important to mention that the solutions obtained are not completely different. Many examples for proving Parkes’s view [9] have been examined. Moreover, soliton-like solutions and trigonometric-function solutions are also obtained as limiting cases. The previous results determined by the hyperbolic tangent method (Yusufoğlu and Bekir, 2007) [5] and exponential function approach (Kangalgil and Ayaz, 2008) [6] are found to be limited and special cases of the current results.

Protonic transport through solitons in hydrogen-bonded systems

We offer an alternative route for investigating soliton solutions in hydrogen-bonded (HB) chains. We invoke the modified extended tangent hyperbolic function method coupled with symbolic computation to solve the governing equation of motion for proton dynamics. We investigate the dynamics of proton transfer in HB chains through bell-shaped soliton excitations, which trigger the bio-energy transport in most biological systems. This solitonic mechanism of proton transfer could play functional roles in muscular contraction, enzymatic activity and oxidative phosphorylation.

Dissipative cylindrical fast magnetoacoustic waves in planetary magnetospheres

Nonlinear cylindrical fast magnetoacoustic waves are investigated in a dissipative magnetoplasma comprising of electrons, positrons, and ions. In this regard, cylindrical Kadomtsev-Petviashvili-Burgers (CKPB) equation is derived using the small amplitude perturbation expansion method. Furthermore, cylindrical Burgers-Kadomtsev-Petviashvili (Cyl Burgers-KP) for a fast magnetoacoustic wave is derived, for the first time, for spatial scales larger than the electron/positron skin depths, c/ω p(e,p). Using the tangent hyperbolic method, the solutions of both planar KPB and Burgers-KP equations are obtained and then subsequently used as an initial profile to solve their respective counterparts in the cylindrical geometry. The effect of positron concentration, kinematic viscosity, and plasma β are explored both for the KPB and the Burgers-KP shock waves and the differences between the two are highlighted. The temporal evolution of the cylindrical fast magnetoacoustic wave is also numerically investigated. The present study may be beneficial to study the propagation characteristics of nonlinear electromagnetic shock waves in planetary magnetospheres.

25 Year Perspective Recent developments in analysis of high temperature creep and creep fracture behaviour

Over many decades, the creep and creep fracture properties of metals and alloys have commonly been described using power law relationships. However, with this approach, controversies still surround the mechanisms proposed to account for the stress and temperature dependencies of the minimum creep rate and rupture life. Similarly, power law and related expressions do not allow extrapolation of short term data sets to predict the long term properties required for engineering design purposes. For these reasons, over the last 25 years, three UK approaches have evolved, termed the θ projection concept, the hyperbolic tangent method and the Wilshire equations. These methodologies are now applied to results for the aluminium alloy Al 2124, recommending reappraisal of current reliance on power law expressions for theoretical and practical purposes.

Nonlinear electrostatic shock waves in inhomogeneous dense dusty magnetoplasmas

The Linear and nonlinear propagation characteristics of drift dust acoustic waves were studied in an inhomogeneous electron–ion-dust (e–i–d) quantum magnetoplasma with neutrals in the background using the well-known quantum hydrodynamic (QHD) model. The Korteweg–de Vries–Burgers (KdVB) and Kadomtsev–Petviashvili–Burgers (KPB) equations are obtained. Furthermore, the solutions of the KdVB and KPB equations are presented using the tangent hyperbolic (tanh) method. The variation of the shock profile with the quantum Bohm potential, collision frequency, ratio of drift to shock velocity in the co-moving frame and effect of magnetic field is also investigated. The relevance of the present investigation to dense astrophysical environments is also highlighted.

An alternate approach to study electrostatic solitary waves in homogeneous and inhomogeneous quantum magnetoplasmas

Ion acoustic drift solitary wave is studied in both linear and nonlinear regimes in an electron-ion quantum magnetoplasma. It is shown that using the linear dispersion relation, nonlinear Zakharov–Kuznetsov (ZK) equation can be derived in an inhomogeneous quantum magnetoplasma for ion acoustic drift solitary wave in a certain parametric range. The solution of the ZK equation is also presented using the tangent hyperbolic method. It is found that the quantum Bohm potential (via increasing number density), angle of propagation, and the magnetic field affect the propagation characteristics of the nonlinear quantum ion acoustic drift wave. The results presented in this paper may be beneficial to understand the formation of electrostatic drift solitary waves in dense magnetized plasmas.

Forecasting Financial Stocks Using Data Mining

This study presents a Business Intelligence (BI) approach to forecast daily changes in seven financial stocks’ prices from September 1, 1998 to April 30, 2008 with 267 independent variables. The purpose of our paper is to compare the performance of Ordinary Least Squares model and Neural Network model to see which model better predicts the changes in the stock prices and to identify critical predictors to forecast stock prices to increase forecasting accuracy for the professionals in the market. We used SPSS to perform stepwise regression to create a unique regression model for each company. Then, we ran the neural network with Alyuda NueroIntelligence to create a NN model by performing data analysis, data preprocessing, network design with hyperbolic tangent method, training with batch back propagation, testing, and query. We did data manipulation by using the first derivative and adding 0.1 to the absolute value of the minimum value in each variable to avoid minus sign from the rounding. Finally, we tested the model with the paired t-test in 152 randomly selected data points. Our result showed that the neural network model (batch back propagation algorithm) outperformed OLS model. The %error for NN and OLS mean ranges from 2.13%-3.27% and 4%-32% and standard deviation ranges from 1.78%-3.39% and 2.46%-8%. The OLS model is easy to use, validate, and works fast with lower forecasting accuracy because it is a linear model. NN has a better forecasting accuracy with no explanation of the relationship between interacting variables with dynamic results due to the learning setup. Some critical success factors to train NN are the network architecture, network design algorithm, training algorithm, and stop training conditions. Data normalization can make a huge difference to the result. We recommend more forecasting method and independent variables (e.g. expert opinion) to be included for future studies.

Nonlinear Korteweg–de Vries–Burger equation for ion acoustic shock waves in a weakly relativistic electron-positron-ion plasma with thermal ions

The nonlinear propagation of ion acoustic waves in electron-positron-ion plasma comprising of Boltzmannian electrons, positrons, and relativistic thermal ions has been examined. The Korteweg–de Vries–Burger equation has been derived by reductive perturbation technique, and its shock like solution is determined analytically through tangent hyperbolic method. The effect of various plasma parameters on strength and structure of shock wave is investigated. The pert graphical view of the results has been presented for illustration. It is observed that strength and steepness of the shock wave enervate with an increase in the ion temperature, relativistic streaming factor, positron concentrations, electron temperature and they accrue with an increase in coefficient of kinematic viscosity. The convective, dispersive, and dissipative properties of the plasma are also discussed. It is determined that the electron temperature has remarkable influence on the propagation and structure of nonlinear wave in such relativ...

A new equation in two dimensional fast magnetoacoustic shock waves in electron-positron-ion plasmas

Nonlinear properties of the two dimensional fast magnetoacoustic waves are studied in a three-component plasma comprising of electrons, positrons, and ions. In this regard, Kadomtsev–Petviashvili–Burger (KPB) equation is derived using the small amplitude perturbation expansion method. Under the condition that the electron and positron inertia are ignored, Burger–Kadomtsev–Petviashvili (Burger-KP) for a fast magnetoacoustic wave is derived for the first time, to the best of author’s knowledge. The solutions of both KPB and Burger-KP equations are obtained using the tangent hyperbolic method. The effects of positron concentration, kinematic viscosity, and plasma β are explored both for the KPB and the Burger-KP shock waves and the differences between the two are highlighted. The present investigation may have relevance in the study of nonlinear electromagnetic shock waves both in laboratory and astrophysical plasmas.

Nonplanar electrostatic shock waves in dense plasmas

Two-dimensional quantum ion acoustic shock waves (QIASWs) are studied in an unmagnetized plasma consisting of electrons and ions. In this regard, a nonplanar quantum Kadomtsev–Petviashvili–Burgers (QKPB) equation is derived using the small amplitude perturbation expansion method. Using the tangent hyperbolic method, an analytical solution of the planar QKPB equation is obtained and subsequently used as the initial profile to numerically solve the nonplanar QKPB equation. It is observed that the increasing number density (and correspondingly the quantum Bohm potential) and kinematic viscosity affect the propagation characteristics of the QIASW. The temporal evolution of the nonplanar QIASW is investigated both in Cartesian and polar planes and the results are discussed from the numerical stand point. The results of the present study may be applicable in the study of propagation of small amplitude localized electrostatic shock structures in dense astrophysical environments.

Coupled nonlinear drift and ion acoustic waves in dense dissipative electron-positron-ion magnetoplasmas

Linear and nonlinear propagation characteristics of drift ion acoustic waves are investigated in an inhomogeneous electron-positron-ion (e-p-i) quantum magnetoplasma with neutrals in the background using the well known quantum hydrodynamic model. In this regard, Korteweg–de Vries–Burgers (KdVB) and Kadomtsev–Petviashvili–Burgers (KPB) equations are obtained. Furthermore, the solutions of KdVB and KPB equations are presented by using the tangent hyperbolic (tanh) method. The variation in the shock profile with the quantum Bohm potential, collision frequency, and the ratio of drift to shock velocity in the comoving frame, v*∕u, is also investigated. It is found that increasing the positron concentration and collision frequency decreases the strength of the shock. It is also shown that when the localized structure propagates with velocity greater than the diamagnetic drift velocity (i.e., u>v*), the shock strength decreases. However, the shock strength is observed to increase when the localized structure pro...

Ion acoustic shock waves in a relativistic electron–positron–ion plasmas

The Korteweg–de Vries–Burger (KdVB) equation is derived for ion acoustic shock waves in a weakly relativistic electron–positron–ion plasma. Electrons, positrons are considered isothermal and ions are relativistic. The travelling wave solution has been acquired by employing the tangent hyperbolic method. The vivid display of the graphical results is presented and analyzed. It is observed that amplitude and steepness of the shock wave decrease with increase of the relativistic streaming factor, the positron concentration and they increase with the increase of the coefficient of kinematic viscosity and vice versa. It is determined that at low temperature the shock wave propagates, whereas at very high temperature the solitary wave propagates in the system. The results may have relevance in astrophysical plasmas as well as in inertial confinement fusion plasmas.