Hyperbolic Function Method Research Articles

Optical solutions of cold bosonic atoms in a zig-zag optical lattice

In this paper, using the continuum approximation and from a discrete equation governing a system of cold bosonic atoms in zig-zag optical lattices derived an equation in a continuum domain. With two different methods, namely the $$exp_a$$ function method and the Hyperbolic function method, found many solutions of the equation considered. These solutions are Hyperbolic solutions and trigonometric solutions. Discrete representations have been done to observe their behaviors. These results will help to understand the dynamics of cold bosonic atoms in zig-zag optical lattices and several other systems. More importantly, in this article derived the Lagrangian and the stability condition of the model studied. The methods employed here are very simple and concise.

Nonlinear Mass Transfer at the Electrodes with Reversible Homogeneous; Reactions:Taylor'sSeries and Hyperbolic Function Method

Nonlinear Mass Transfer at the Electrodes with Reversible Homogeneous; Reactions:Taylor'sSeries and Hyperbolic Function Method

New soliton solutions of the 2D‐chiral nonlinear Schrodinger equation using two integration schemes

In this paper, the new exact solutions for 2D‐chiral nonlinear Schrodinger equation (CNLSE) are acquired using two proficient integration tool, namely, the new extended direct algebraic method (EDAM) and extended hyperbolic function method (EHFM). We develop soliton and some other solutions by utilizing the particular values for the parameters involved in these methods. These methods are devoted to secure different kinds of new soliton solutions as well as trigonometric and hyperbolic function solutions. Furthermore, for the physical representation of the obtained solutions of the CNLSE, various 2D and 3D graphs are presented.

Exact Soliton Solutions for Nonlinear Perturbed Schrödinger Equations with Nonlinear Optical Media

The nonlinear perturbed Schrödinger equations (NPSEs) with nonlinear terms as Kerr law, power law, quadratic-cubic law, and dual-power law nonlinearity media play an important role in optical fibers. In this article we implement the rational solitary wave method to study the NPSEs when nonlinear terms take some different forms. Additionally, we use the q-deformed hyperbolic function and q-deformed trigonometric function methods to study the exact solutions to NPSEs. Different kind of soliton solutions are obtained such as bright, dark, and singular periodic solutions to the NPSEs.

Soil liquefaction potential evaluation – An update of the HBF method focusing on research and practice in Taiwan

This paper presents an update of the Hyperbolic Function (HBF) method to evaluate the liquefaction potential of sandy soils. The HBF method was created after the 1999 Chi-Chi, Taiwan Earthquake (Mw = 7.6) using the database of case histories developed based on field observations and in situ and laboratory tests in areas affected by the 1999 event. The updated HBF method reported herein reflects the 20-year efforts of research and practice in Taiwan after the 1999 event. Here, the original database created by Hwang and his co-workers was first re-examined and updated. The updated Chi-Chi database was supplemented with the cases derived from the 2016 Meinong, Taiwan Earthquake, and some selected data from other parts of the world. This new database was then used to update the parameters of the HBF model through rigorous statistical methods. The HBF model is unbiased (as it corresponds to a probability of 50%). The updated HBF method, a deterministic model, is further extended into a probabilistic version through rigorous statistical calibration. This probabilistic version is expressed as a mapping function that relates the factor of safety (FS) against liquefaction to the probability of liquefaction (PL). The design recommendation for engineering practice in Taiwan is provided based on the calibrated PL-FS mapping function. The updated HBF method is demonstrated through examples to be an easy-to-use procedure for liquefaction potential evaluation. Use of Bayes' theorem for updating the liquefaction probability and the new concept of liquefaction likelihood ratio are also discussed.

The conformable space–time fractional Fokas–Lenells equation and its optical soliton solutions based on three analytical schemes

This paper is about the study of space–time fractional Fokas–Lenells equation that describes nonlinear wave propagation in optical fibers. Three prominent schemes are employed for extracting different types of exact soliton solutions. In particular, the [Formula: see text] function method, the hyperbolic function method and the simplest Riccati equation scheme are investigated for the said model. As a sequel, a series of soliton solutions are obtained and verified through MATHEMATICA. The obtained solutions are significant additions in some specific fields of physics and engineering. Furthermore, the 3D graphical descriptions are left to analyze the pulse propagation for the reader.

Exact solutions of (2 + 1)‐dimensional Schrödinger's hyperbolic equation using different techniques

AbstractIn this paper, we derive new optical soliton solutions to (2 + 1)‐dimensional Schrödinger's hyperbolic equation using extended direct algebraic method and new extended hyperbolic function method. New acquired solutions have the form of bright, dark, combined dark‐bright, singular, and combined bright‐singular solitons solutions. These solutions reveal that our techniques are straightforward and dynamic. The solutions are also demonstrated through 3‐d and 2‐d plots to make clear the physical structures for such kind of model. The obtained results illustrate the power of the present method to determine soliton solution of nonlinear evolution equations.

New Mathematical Analysis for Nonlinear Simultaneous Differential Equation in Micro-Disk Biosensor Using Hyperbolic Function Method

In this study, a mathematical model of immobilized enzyme system, which follows the Michaelis-Menten kinetics for micro-disk biosensor is discussed. It is based on substrate and hydrogen peroxide profile with enzyme reaction within the biosensor under steady state condition. Accordingly, simple and compact analytical expression for substrate as well as hydrogen peroxide concentrations and electrode current for micro-disk biosensor are obtained. In addition, it acquired as a function of reaction diffusion parameter and saturated parameter using hyperbolic function method. Therefore, this hyperbolic function analysis of the proposed model is an efficient tool to predict the Michaelis-Menten constant. Furthermore, to confirm the validity and accuracy of the proposed method, the results are compared with those obtained by using well-established Homotopy analysis method and Modified Adomian decomposition method. The numerical results obtained are provided by the highly reputed program pdex2 and pdex4 in MATLAB. Tabular compilations of concentrations and current are also explored for typical values of the governing parameters.

Solving Nonlinear Partial Differential Transport-Density Equation with Diffusion Traffic Flow Model using the Generalized Hyperbolic Functions Method (GHFM) and the First Integral Method (FIM)

Solving Nonlinear Partial Differential Transport-Density Equation with Diffusion Traffic Flow Model using the Generalized Hyperbolic Functions Method (GHFM) and the First Integral Method (FIM)

On the relations between some well-known methods and the projective Riccati equations

AbstractSolving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.

Structure of optical soliton solution for nonliear resonant space-time Schrödinger equation in conformable sense with full nonlinearity term

Nonclassical quantum mechanics along with dispersive interactions of free particles, long-range boson stars, hydrodynamics, harmonic oscillator, shallow-water waves, and quantum condensates can be modeled via the nonlinear fractional Schrödinger equation. In this paper, various types of optical soliton wave solutions are investigated for perturbed, conformable space-time fractional Schrödinger model competed with a weakly nonlocal term. The fractional derivatives are described by means of conformable space-time fractional sense. Two different types of nonlinearity are discussed based on Kerr and dual power laws for the proposed fractional complex system. The method employed for solving the nonlinear fractional resonant Schrödinger model is the hyperbolic function method utilizing some fractional complex transformations. Several types of exact analytical solutions are obtained, including bright, dark, singular dual-power-type soliton and singular Kerr-type soliton solutions. Moreover, some graphical simulations of those solutions are provided for understanding the physical phenomena.

Traveling wave solution and Painleve’ analysis of generalized fisher equation and diffusive Lotka–Volterra model

In this paper, we have obtained the traveling wave solution for generalized Fisher equation and Lotka–Volterra (L-V) model with diffusion using hyperbolic function method. The Painleve’ analysis has been used to check both of the system’s integrability. Obtained solutions have also been plotted to represent their spatio-temporal dependence. The three dimensional plot shows a monotonic profile of the solutions.

Analytical Behaviour of Travelling Wave Solutions to the Van der Waals Model

In this article, our work is related to the most popular model that occurred naturally and also industrially, known as the Van der Waals normal form for the fluidized matter. After using a wave transformation that converts the nonlinear partial differential equation into nonlinear ordinary differential equations, we employed two novel analytical methods namely the hyperbolic function method the modified Kudryashov method. As a consequence, many new exact solutions are established coupled with 2D and 3D graphical patterns for clarity and accuracy.

A series of abundant new optical solitons to the conformable space-time fractional perturbed nonlinear Schrödinger equation

In this work, we are investigating a series of new optical soliton solutions to the perturbed nonlinear Schrödinger equation (PNLSE) having the form of kerr law nonlinearity with conformable space-time fractional. Thereby, two relevant integration tools known as new extended direct algebraic method and extended hyperbolic function method are applied to obtain varieties of optical soliton solutions. The series of soliton solutions with fractional derivative order obtained by these methods can be classified as complex trigonometric and hyperbolic functions as well as other elementary functions. Furthermore, conditions for validity of the obtained analytical solutions, graphical illustration (2-D, 3-D) point out the impact of the fractional-order used.

Investigation for Optical Soliton Solutions of Two Nonlinear Schrödinger Equations via Two Concrete Finite Series Methods

This article investigates the two cubic nonlinear Schrodinger equations which point out the evolution of disturbances in dynamics. These are $$(2+1)$$ -dimension Heisenberg ferromagnetic spin chains equation and the $$(1+1)$$ -dimension compressional dispersive Alfven envelop equation. The investigation is taken in a straight forward way through two recent finite series methods namely the $$exp_a$$ function and the hyperbolic function methods. The new explicit exact soliton solutions including some free parameters are obtained as fallouts of these methods. These solutions show that the proposed schemes are more simple and effective as compared to many other schemes. Additionally, the effects of the free parameters in these solutions are discussed graphically for physical interests and potential applications.

Time-Space Fractional Model for Complex Cylindrical Ion-Acoustic Waves in Ultrarelativistic Plasmas

In this paper, the fractional order models are used to study the propagation of ion-acoustic waves in ultrarelativistic plasmas in nonplanar geometry (cylindrical). Firstly, according to the control equations, (2 + 1)-dimensional (2D) cylindrical Kadomtsev–Petviashvili (CKP) equation and 2D cylindrical-modified Kadomtsev–Petviashvili (CMKP) equation are derived by using multiscale analysis and reduced perturbation methods. Secondly, using the semi-inverse method and the fractional variation principle, the abovementioned equations are derived the time-space fractional equations (TSF-CKP and TSF-CMKP). Furthermore, based on the fractional order transformation, the 1-decay mode solution of the TSF-CKP equation is obtained by using the simplified homogeneous balance method, and using the generalized hyperbolic-function method, the exact analytic solution of TSF-CMKP equation is obtained. Finally, the effects of the phase speedλ, electron number density (throughβ3) and the fractional orderα,β,ωon the propagation of ion-acoustic waves in ultrarelativistic plasmas are analyzed.

On finite series solutions of conformable time-fractional Cahn-Allen equation

AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.

Analytical Solution of the Non-Linear Equation in Biodegradation of N-Butanol in a Biofilter

In this paper, mathematical models of biofilm mixtures of n-butanol biofilters were discussed. The model is based on the mass transfer in the biofilm interface and chemical oxidation in the biofilm phase and gas phase. An approximate analytical expression of concentration profiles of n-butanol in the biofilm phase and gas phase has been derived using the homotopy perturbation method and hyperbolic function method for all possible values of parameters. Furthermore, in this work, the numerical simulation of the problem is also reported using the Matlab program. Good agreement between the analytical and numerical results is noted. Graphical results are presented and discussed quantitatively to illustrate the solution. The analytical results will be useful in finding the yields of biomass and oxygen consumption, the specific biomass surface area, activation energy and saturation constant for the Michaelis-Menten kinetics.

A method for constructing rolling bearing lifetime health indicator based on multi-scale convolutional neural networks

The degradation of rolling bearings is complex, and traditional methods of degenerating feature degradation are highly dependent on previous research and expertise. However, the traditional methods’ ability for learning the complex relationship between degraded features and large amounts of measured data is limited. So, it is difficult to construct a single health indicator (HI) to predict the degradation state of the bearing. In order to solve this problem, a multi-scale convolutional neural network (MSCNN) is proposed, which uses the frequency signal as input to construct HI. In combination with the “bathtub curve,” the HI we proposed is constructed by the inverse hyperbolic tangent function (atanh) method which is different from the proportional HI construction method. Five kinds of processing methods, such as original vibration signal, frequency signal and three kinds of modal decomposition preprocessed signals, are tested with PHM 2012 bearing lifetime data. And the proposed method is compared with other three HI construction methods. The experimental results show that under the HI construction method of atanh, the MSCNN model with the input of frequency signal can better represent the degradation state of the bearing compared with other methods. Meanwhile, the remaining life prediction is more accurate compared.

New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation

This study focuses on acquiring new complex hyperbolic and trigonometric function solutions to the generalized conformable fractional Gardner equation by using two distinct integration schemes, namely: the modified Kudryashov method and the hyperbolic function method with the help of symbolic computation package. We generate new closed-form solitary wave solutions to the governing model. Implemented schemes are pragmatically effective for solving various types of nonlinear partial differential equations with integer or fractional order arising in various fields of applied sciences.