Trigonometric Form Research Articles

Solution of stochastic Allen–Cahn equation in the framework of soliton theoretical approach

In this paper, the Allen–Cahn equation with time noise is under consideration. The extended fan-sub technique is used to find the exact solutions. The solutions are successfully extracted in the form of hyperbolic, trigonometric and mixed forms of solitons. Importantly, the physical unique value problems of the solutions are discussed using the different values of parameters. The 2D, 3D, and their corresponding contour behaviors of these solutions are depicted by choosing the different values of parameters. The stability is controlled through the Borel coefficient in the noise term.

Periodic and breather solutions for miscellaneous soliton in metamaterials model by computational schemes

In this paper, the novel exact solitary wave solutions for the generalized nonlinear Schrödinger equation with parabolic nonlinear (NL) law employing the improved [Formula: see text] function scheme and the combined [Formula: see text] function scheme are found. Diverse collections of hyperbolic and trigonometric function solutions acquired rely on a map between the considered equation and an auxiliary ODE. Received solutions are recast in several hyperbolic, rational and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solution. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model utilizing mathematical software. The main subject of this work is that one can visualize and update the knowledge to overcome the most common techniques and defeat to solve the ODEs and PDEs. We demonstrated that these solutions validated the program using Maple and found them correct. The proposed methodology for solving the metamaterials model has been designed to be effectual, unpretentious, expedient and manageable. Applications of the solutions by the mentioned techniques will be useful to investigate the signals properties of optical fibers, plasma physics phenomena, electromagnetic fields occurrences and various types of nonlinear metamaterials models.

Observer-Based Adaptive Control for Single-Phase UPS Inverter Under Nonlinear Load

This article proposed a current-sensorless control scheme for the single-phase uninterruptible power supply (UPS) inverter under nonlinear load. Different from the existing current-sensorless literature, the information of load current can be obtained by the designed adaptive laws. It is meaningful for the system’s security and reliability. Under the assumption that the load is a periodic ideal-independent current source, the Fourier series in trigonometric form is expanded to model the unknown disturbance of load current. The observer-based feedback control rate and update laws are derived to adjust online the Fourier coefficients of the load current modeling. Furthermore, the stability of the system is rigorously analyzed via the Lyapunov approach. Under the same limitation of not using the current sensor, comparative experiments are designed. Simulation and experimental results have verified the effectiveness of the control approaches.

A NUMERICAL SCHEME BASED ON TWO- AND THREE-STEP NEWTON INTERPOLATION POLYNOMIALS FOR FRACTAL–FRACTIONAL VARIABLE ORDERS CHAOTIC ATTRACTORS

The terms fractional differentiation and fractal differentiation are recently combined to form a new fractional differentiation operator. Several kernels, such as the power-law and the Mittag-Leffler function, are employed to investigate these novel operators. Both fractional-order and fractal dimensions are found in these developed operators. In the present investigation, we have applied these new operators to study certain chaotic attractors with Mittag-Leffler and power-law kernels. These chaotic models are solved using an effective numerical procedure called the Adams–Bashforth technique, based on two- and three-step Newton interpolation polynomials. In this analysis, the order of the fractional derivative is assumed to be the exponential, trigonometric, and hyperbolic forms of variables. The solutions obtained by the numerical procedure with two different kernels are portrayed in terms of chaotic plots.

Elastic bending analysis exact solution of plate using alternative I refined plate theory

Plate materials play a significant role in civil engineering and its application is found in building, bridges, marine, nuclear and structural engineering. Plates are three-dimensional (3-D) structural elements and normal practice in its analysis as two-dimensional (2-D) element has produced unreliable design which is a major problem in the construction industry. In this paper, a 3-D exact elastic plate theory (alternative I refined plate theory) is developed and applied to the bending analysis of an isotropic rectangular plate with free support at the third edge and the other edges clamped (CCFC) under uniformly distributed transverse load. The energy equation is formulated using the three-dimensional constitutive relations thereafter, the compatibility equation was obtained through general variation. The solution of the compatibility equation gave an exact deflection function of the plate as presented in the trigonometric form. This deflection function which is a product of the coefficient of deflection and shape function of the plate are substituted back into the energy equation, and simplified through minimization with respect to the coefficients of deflection and shear deformation to get a realistic formula for calculating the deflection, rotations and stresses of thick rectangular plate. The results of deflections and stresses obtained at aspect ratio of 2 were compared with those of previous studies available in the literature. It was observed that the present theory varied more with those of 2-D refined plate theory (RPT) with an assumed deflection function by 6.2% whereas it varied with exact 2-D RPT by 3.1%. This shows the inefficiency of assuming the deflection function in the analysis of thick plate. Meanwhile, the recorded total percentage differences showed that 2-D RPT over predicted the bending characteristics of the plate with 4.7%. This finding provides in-depth insight about the coarseness of 2-D RPT in the thick plate analysis.

Soliton behavior of algae growth dynamics leading to the variation in nutrients concentration

This paper provides the effects of nutrients concentrations on the algae growth via solitary wave analysis and classical existence theory. There is a well known Diffusive Nutrient-Algae model based on the Sanyang wetland, is under consideration. The new families of exact solutions which represent nutrient concentration and algae density in different form are observed using the ϕ6-model expansion. Furthermore, the existence of these solutions are also discussed under different restrictive conditions and variables of nutrient concentration and algae density are represented in hyperbolic, trigonometric and rational forms. The graphics of these solutions are also sketched in 3D and 2D representations.

Performance of basic arithmetic actions with complex numbers, which are presented in interval hyperbolic form

The goal of the work. Development of methods for performing basic arithmetic operations with interval complex numbers, which are presented in hyperbolic form, their modulus and argument. Results. The paper considers the method of extending interval numbers defined in hyperbolic form (hyperbolic interval numbers) to the field of complex numbers. To do this, the real and imaginary part of a complex number is presented in the form of a hyperbolic interval number. The connections between the representation of interval numbers in the classical form, the CENTER-RADIUS system and the hyperbolic form are established. Methods of performing basic arithmetic operations with hyperbolic complex numbers are proposed, namely: addition, subtraction, multiplication and division. A method of raising the positive interval number of a complex interval number defined in a hyperbolic form to an integer positive degree is proposed. Methods for calculating the modulus and argument of a complex number defined in hyperbolic form are proposed. A method for determining the root of a degree from an interval complex number represented in hyperbolic form is proposed. Using the connections between hyperbolic and trigonometric functions, a form of representation of an interval number in trigonometric form is proposed. It is established that it is most expedient to perform addition and subtraction actions with complex interval numbers, which have a classical form or are defined in the CENTER-RADIUS system. The operations of multiplication, division and elevation to an integer power are most expedient to perform with complex interval numbers which are defined in hyperbolic form. The operation of calculating the root of a degree from an interval complex number, presented in hyperbolic form, is most expedient to perform with the combined use of the representation of the interval number in the system CENTER-RADIUS and in hyperbolic form.

Nonparaxial pulse propagation in a planar waveguide with Kerr–like and quintic nonlinearities; computational simulations

This article investigates some novel solitary wave solutions of the cubic-quintic nonlinear Helmholtz equation by implementing two recent analytical (Khater II and generalized exponential methods) techniques. As a result of seeing the nonparaxial effect, this model was developed to show how a pulse propagates in a planar waveguide with Kerr-like and quintic nonlinearities as well as spatial dispersion. When the traditional approximation of a slowly changing envelope fails, this effect takes over. Some novel soliton wave solutions are obtained and formulated in distinct forms such as rational, hyperbolic, and trigonometric forms. The obtained solutions are represented through various figures’ types to show the physical and dynamical behavior of the nonparaxial pulse propagation. The model’s features are demonstrated, such as its intriguing bright, anti-bright, periodic, singular, gray, chirped anti-dark, and dark solitary waves depending upon the nature of nonlinearities. The solutions’ originality is discussed by comparing it with those constructed in recently published articles. All built results are checked their accuracy by putting them back into the original model by employing Mathematica 12.

Extracting the exact solitons of time-fractional three coupled nonlinear Maccari’s system with complex form via four different methods

This current research is considered some new exact soliton solutions to the time-fractional three coupled non-linear Maccari’s system in complex form with novel truncated M-fractional derivative. The obtained results may be used in the description of the model in fruitful way. The novel derivative operator is applied to study the aforementioned model. In addition, the achieved results are included in the form of dark, bright and combo optical solitons. The achieved solutions are also verified by using the MATHEMATICA software. Modified integration methods, namely, the Kudryashov method, the Expa function method, the extended (G′/G)−expansion method and the extended ShGEEM are applied to achieve the results. Obtained solutions are remodeled in several hyperbolic and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solutions. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model using mathematical software. These optical solitons suggest that these three methods are useful, easy to use and effective than other methods.

A variety of closed-form solutions, Painlevé analysis, and solitary wave profiles for modified KdV–Zakharov–Kuznetsov equation in (3+1)-dimensions

The major goal of the article is to use the generalized exponential rational function method to seek abundant closed-form solutions and dynamics of solitary wave profiles to the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov (MKdV–ZK) equation. We also apply the Painlevé analysis with Kruskal simplification to investigate the integrability of the said equation. The (3+1)-dimensional MKdV–ZK model is a significant model with great advantages in various disciplines of physics, such as nonlinear optics, fluid dynamics, plasma physics, mathematical physics, and quantum mechanics. By taking advantage of the generalized exponential rational function technique, we extracted various soliton solutions, including exponential form, trigonometric form, and hyperbolic form function solutions. From both mathematical and physical points of view, it is important to obtain bright–dark soliton forms and other solitonic wave profiles. Furthermore, the results have been graphically revealed using 3D, 2D, and contour plots to demonstrate the underlying dynamics of the interactive waves. The obtained findings indicate the method’s efficacy and dependability, allowing it to be widely applied in a variety of advanced nonlinear evolution equations.

Strong interaction of Jafari decomposition method with nonlinear fractional-order partial differential equations arising in plasma via the singular and nonsingular kernels

<abstract><p>This research utilizes the Jafari transform and the Adomian decomposition method to derive a fascinating explicit pattern for the outcomes of the KdV, mKdV, K(2,2) and K(3,3) models that involve the Caputo fractional derivative operator and the Atangana-Baleanu fractional derivative operator in the Caputo sense. The novel exact-approximate solutions are derived from the formulation of trigonometric, hyperbolic, and exponential function forms. Laser and plasma sciences may benefit from these solutions. It is demonstrated that this approach produces a simple and effective mathematical framework for tackling nonlinear problems. To provide additional context for these ideas, simulations are performed, employing a computationally packaged program to assist in comprehending the implications of solutions.</p></abstract>

기온과 토양 및 식물 표면 온도 변화

This study focuses on changes in air, soil surface and plant surface temperature. The principle results are as follows: (1) The ratio of change in soil and plant surface temperature to air temperature was 0.80-1.17 and 0.88-1.19oC/oC, respectively. (2) The difference in temperature between air temperature and soil and plant surface temperature was a linearly decreasing function of soil surface temperature and plant surface temperature, respectively. (3) A model for estimating air, soil surface and plant surface temperature was developed as a trigonometric function form (TFM) with Root mean square error (RMSE) of 1.75, 1.90-2.41 and 1.85-2.21oC, and NSE of 0.51, 0.37-0.58 and 0.39-0.61, respectively. (4) The longer change frequency the sooner inflection point for air, soil surface and plant surface temperature.

Parameter N Analysis on the Rational-Talbot Algorithm for Numerical Inversion of Laplace Transform

This article investigates the numerical inversion of the Laplace Transform by the Rational-Talbot method and analyzes the influence on the variation of the free parameter N established by the technique when applied to certain functions. The set of elementary functions, for which the method is tested, has exponential and oscillatory characteristics. Based on the results obtained, it was concluded that the Rational-Talbot method is e cient for the inversion of decreasing exponential functions. At the same time, to perform the inversion process effectively for trigonometric forms, the algorithm requires a greater amount of terms in the sum. For higher values of N, the technique works well. In fact, this is observed in inverting the functions transform, that combine trigonometric and polynomial factors. The method numerical results have a good precision for the treatment of decreasing exponential functions when multiplied by trigonometric functions.

Simulation of Wave Solutions of a Mathematical Model Representing Communication Signals

In this study, the Lonngren-wave equation is considered to be analyzed for its wave solutions. To implement this purpose the modified exponential function method is used and ultimately new hyperbolic, trigonometric and rational forms of the exact solutions are obtained. Furthermore, it was tested whether these forms satisfy the Lonngren-wave equation or not and it was seen that they verify the equation. Besides this, the two and three dimensional graphics together with the contour and density plots are presented.

Nonlinear dynamical study to time fractional Dullian–Gottwald–Holm model of shallow water waves

This paper studies the exact traveling wave solutions to the nonlinear Dullin–Gottwald–Holm model which has the application in shallow-water waves in which the fractional derivative is considered in the sense of conformable derivative. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtained using two integration norms. For this purpose [Formula: see text]-expansion method and reccati mapping techniques are used. The 3D plots and their corresponding contour graphs are also depicted. Being concise and straightforward, the calculations demonstrate the effectiveness and convenience of the method for solving other nonlinear partial differential equations.

Method of lines for solving linear equations of mathematical physics with the third and first types boundary conditions

The use of the method of lines in solving multidimensional problems of mathematical physics makes it possible to eliminate the discrepancies caused by the use of the sweep method in certain coordinates. As a result, the solution of the Poisson equation, for example, is obtained without using the relaxation method. In the article, the problem on the eigenvalues and vectors of the transition matrix is solved for boundary conditions of the third and first types, used to solve a one-dimensional equation of parabolic type by the method of lines. Due to the features of boundary conditions of the third type for determining the eigenvalues, a mixed method was proposed based on the Vieta theorem and the representation of the characteristic equation in trigonometric form typical for the method of lines. To solve the eigenvector problem, a simple sweep method was used with the algebraic compliments to the transition matrix. Discontinuous solutions of a one-dimensional parabolic equation were presented for various values of complex 1 -αl; the method for solving the characteristic equation was selected based on these values. The calculation results are in good agreement with the analytical solution.

Limit theorems for Jacobi ensembles with large parameters

Consider Jacobi random matrix ensembles with the distributions $$c_{k_1,k_2,k_3}\prod_{1\leq i< j \leq N}\left(x_j-x_i\right)^{k_3}\prod_{i=1}^N \left(1-x_i\right)^{\frac{k_1+k_2}{2}-\frac{1}{2}}\left(1+x_i\right)^{\frac{k_2}{2}-\frac{1}{2}} dx$$ of the eigenvalues on the alcoves $$A:=\{x\in\mathbb R^N| \> -1\leq x_1\le ...\le x_N\leq 1\}.$$ For $(k_1,k_2,k_3)=\kappa\cdot (a,b,1)$ with $a,b>0$ fixed, we derive a central limit theorem for the distributions above for $\kappa\to\infty$. The drift and the inverse of the limit covariance matrix are expressed in terms of the zeros of classical Jacobi polynomials. We also rewrite the CLT in trigonometric form and determine the eigenvalues and eigenvectors of the limit covariance matrices. These results are related to corresponding limits for $\beta$-Hermite and $\beta$-Laguerre ensembles for $\beta\to\infty$ by Dumitriu and Edelman and by Voit.

Solitary Wave Solutions of the (3+1)-dimensional Khokhlov–Zabolotskaya–Kuznetsov Equation by using the (G'/G,1/G)-Expansion Method

In this study, the (3+1)-dimensional Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which is a mathematical model of non-absorption and dispersion in the non-linear medium, which sheds light on the sound beam phenomenon, which has a physically important place, is examined. In order to find the exact solution of this equation, an effective and reliable method, (G^'/G,1/G)-expansion method, is used among analytical methods. The purpose of this method is to obtain more than one traveling wave solution classes depending on the conditions of the λ parameter. These classes are categorized into hyperbolic, trigonometric, complex trigonometric and rational forms. The graphics of the solitary waves represented by these successfully obtained solution classes are presented as 2-dimensional, 3-dimensional and contours. This article makes use of ready-made package programs for complex arithmetic operations and graphic drawings.

Research on the reciprocating load characteristics of a hydrodynamic bearing surface with sloped grooves

In this work, a hydrodynamic bearing surface with sloped grooves is designed, and its reciprocating load characteristics are studied theoretically by the mass-conserving cavitation algorithm. A modified smoothing trigonometric switch function form is applied in calculation, and the appropriate smoothing parameter is discussed. The effects of reciprocating motion, sloped groove’s shape and lubricant’s supply on reciprocating load characters are studied respectively. Experimental results show that the designed hydrodynamic bearing surface can provide about 200 N extra load-carrying capacity (LCC), maintaining the film thickness at about 40μm, and the vertical vibration amplitude is no more than 4μm. The experimental results show good agreement with theoretical simulations.

EPISTEMOLOGICAL OBSTACLE SISWA SMA DALAM MATERI TURUNAN TRIGONOMETRI

The learning system that was originally carried out at school became a long distance learning system at home which resulted in changes to student learning outcomes, especially in the derivative of trigonometic function. Students' limitations in understanding about derivative of trigonometric function became students’ learning obstacle during the pandemic. This study aims to analyze the epistemological obstacle of high school students in the derivative of trigonometric function during the pandemic. This study use qualitative method with case study design. The subjects of this study were 23 students of grade XII of Senior High School in Cirebon City and Garut Regency. Data collected by using test and semi-structured interview. Technical analysis of data using data reduction, data presentation and conclusion drawing. The results showed that students experienced an Epistemological Obstacle related to solving derivative problems of trigonometric functions, seen from students' errors in deriving basic trigonometric functions, derivatives in the form of multiplication, chain derivatives and derivatives involving mixed algebraic and trigonometric forms as well as errors in determining derivatives at a certain point.