trigonometric function
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2021 ◽  
Vol 6 (1) ◽  
pp. 4
Author(s):  
Md Ashik Iqbal ◽  
Ye Wang ◽  
Md Mamun Miah ◽  
Mohamed S. Osman

In this article, we construct the exact dynamical wave solutions to the Date–Jimbo–Kashiwara–Miwa equation with conformable derivative by using an efficient and well-established approach, namely: the two-variable G’/G,  1/G-expansion method. The solutions of the Date–Jimbo–Kashiwara–Miwa equation with conformable derivative play a vital role in many scientific occurrences. The regular dynamical wave solutions of the abovementioned equation imply three different fundamental functions, which are the trigonometric function, the hyperbolic function, and the rational function. These solutions are classified graphically into three categories, such as singular periodic solitary, kink soliton, and anti-kink soliton wave solutions. Furthermore, the effect of the fractional parameter on these solutions is discussed through 2D plots.


Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Ibrahim Alkhairy ◽  
M. Nagy ◽  
Abdisalam Hassan Muse ◽  
Eslam Hussam

The purpose of this paper is to investigate a new family of distributions based on an inverse trigonometric function known as the arctangent function. In the context of actuarial science, heavy-tailed probability distributions are immensely beneficial and play an important role in modelling data sets. Actuaries are committed to finding for such distributions in order to get an excellent fit to complex economic and actuarial data sets. The current research takes a look at a popular method for generating new distributions which are excellent candidates for dealing with heavy-tailed data. The proposed family of distributions is known as the Arctan-X family of distributions and is introduced using an inverse trigonometric function. For the specific purpose of the show of strength, we studied the Arctan-Weibull distribution as a special case of the developed family. To estimate the parameters of the Arctan-Weibull distribution, the frequentist approach, i.e., maximum likelihood estimation, is used. A rigorous Monte Carlo simulation analysis is used to determine the efficiency of the obtained estimators. The Arctan-Weibull model is demonstrated using a real-world insurance data set. The Arctan-Weibull is compared to well-known two-, three-, and four-parameter competitors. Among the competing distributions are Weibull, Kappa, Burr-XII, and beta-Weibull. For model comparison, we used the most precise tests used to know whether the Arctan-Weibull distribution is more useful than competing models.


2021 ◽  
Author(s):  
Kun Zheng ◽  
Yu Miao ◽  
Tong Li ◽  
Shuang-Long Yang ◽  
Li Xi ◽  
...  

Abstract The anti-trigonometric function is used to strictly solve the uniaxial anisotropic Stoner-Wohlfarth (SW) model, which can obtain the relation of the angle α (θ) between the magnetization (the anisotropy field) and the applied magnetic field. Using this analytic solution, the hysteresis loops of uniaxial anisotropic SW particles magnetized in typical directions could be numerically calculated. Then, the hysteresis loops are obtained in randomly distributed SW particle ensembles while ignoring the dipole interaction among them with the analytic solution. Finally, the correctness of the analytic solution is verified by the exact solutions of remanence, switching field, and coercivity from SW model. The analytic solution provides an important reference for understanding the magnetizing and magnetization reversal processes of magnetic materials.


Author(s):  
Wanbo Yu ◽  
Siyu Gong

In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space.


CAUCHY ◽  
2021 ◽  
Vol 7 (1) ◽  
pp. 84-96
Author(s):  
Juhari Juhari

This study discusses the analysis of the modification of Newton-Secant method and solving nonlinear equations having a multiplicity of  by using a modified Newton-Secant method. A nonlinear equation that has a multiplicity   is an equation that has more than one root. The first step is to analyze the modification of the Newton-Secant method, namely to construct a mathematical model of the Newton-Secant method using the concept of the Newton method and the concept of the Secant method. The second step is to construct a modified mathematical model of the Newton-Secant method by adding the parameter . After obtaining the modified formula for the Newton-Secant method, then applying the method to solve a nonlinear equations that have a multiplicity . In this case, it is applied to the nonlinear equation which has a multiplicity of . The solution is done by selecting two different initial values, namely  and . Furthermore, to determine the effectivity of this method, the researcher compared the result with the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified. The obtained results from the analysis of modification of Newton-Secant method is an iteration formula of the modified Newton-Secant method. And for the result of  using a modified Newton-Secant method with two different initial values, the root of  is obtained approximately, namely  with less than iterations. whereas when using the Newton-Raphson method, the Secant method, and the Newton-Secant method, the root  is also approximated, namely  with more than  iterations. Based on the problem to find the root of the nonlinear equation  it can be concluded that the modified Newton-Secant method is more effective than the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified


Fractals ◽  
2021 ◽  
pp. 2240018
Author(s):  
ARMANDO CIANCIO ◽  
GULNUR YEL ◽  
AJAY KUMAR ◽  
HACI MEHMET BASKONUS ◽  
ESIN ILHAN

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.


2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Sami Injrou ◽  
Ramez Karroum ◽  
Nadia Deeb

In this paper, the subequation method and the sine-cosine method are improved to give a set of traveling wave solutions for the time-fractional generalized Fitzhugh–Nagumo equation with time-dependent coefficients involving the conformable fractional derivative. Various structures of solutions such as the hyperbolic function solutions, the trigonometric function solutions, and the rational solutions are constructed. These solutions may be useful to describe several physical applications. The results show that these methods are shown to be affective and easy to apply for this type of nonlinear fractional partial differential equations (NFPDEs) with time-dependent coefficients.


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