scholarly journals ON THE COMPLEX MIXED DARK-BRIGHT WAVE DISTRIBUTIONS TO SOME CONFORMABLE NONLINEAR INTEGRABLE MODELS

Fractals ◽  
2021 ◽  
pp. 2240018
Author(s):  
ARMANDO CIANCIO ◽  
GULNUR YEL ◽  
AJAY KUMAR ◽  
HACI MEHMET BASKONUS ◽  
ESIN ILHAN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Expansion Method ◽  
Research Paper ◽  
Trigonometric Function ◽  
Integrable Models ◽  
Hyperbolic Function ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Wave Solutions

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.

scholarly journals F-Expansion Method and Its Application for Finding Exact Analytical Solutions of Fractional order RLW Equation

2021 ◽  
Author(s):  
Attia Rani ◽  
Qazi Mahmood Ul-Hassan ◽  
Muhammad Ashraf ◽  
Jamshad Ahmad
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Nonlinear Partial Differential Equation ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Partial Differential ◽  
Wave Solutions ◽  
Rlw Equation

Exact nonlinear partial differential equation solutions are critical for describing new complex characteristics in a variety of fields of applied science. The aim of this research is to use the F-expansion method to find the generalized solitary wave solution of the regularized long wave (RLW) equation of fractional order. Fractional partial differential equations can also be transformed into ordinary differential equations using fractional complex transformation and the properties of the modified Riemann–Liouville fractional-order operator. Because of the chain rule and the derivative of composite functions, nonlinear fractional differential equations (NLFDEs) can be converted to ordinary differential equations. We have investigated various set of explicit solutions with some free parameters using this approach. The solitary wave solutions are derived from the moving wave solutions when the parameters are set to special values. Our findings show that this approach is a very active and straightforward way of formulating exact solutions to nonlinear evolution equations that arise in mathematical physics and engineering. It is anticipated that this research will provide insight and knowledge into the implementation of novel methods for solving wave equations.

scholarly journals New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-35 ◽  
Author(s):  
Jalil Manafian ◽  
Mehrdad Lakestani
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Time Dependent ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Function Solution ◽  
Wave Solutions ◽  
Nagumo Equation

An improvement of the expansion methods, namely, the improved tan⁡Φξ/2-expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new tan⁡Φξ/2-expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.

scholarly journals Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Haiyong Qin ◽  
Mostafa M. A. Khater ◽  
Raghda A. M. Attia
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Longitudinal Strain ◽  
Rational Functions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Strain Wave ◽  
Integer Order ◽  
Wave Solutions

A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.

scholarly journals Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 952
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert ◽  
Surattana Sungnul
Keyword(s):  
Partial Differential Equations ◽  
Solitary Wave ◽  
Differential Equations ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Solitary Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions

In this paper, the ( G ′ / G , 1 / G ) -expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics. The considered equations comprise the time-fractional, (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation, and the space-time-fractional generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) system in the sense of the conformable fractional derivative. Applying traveling wave transformations to the equations, we obtain the corresponding ordinary differential equations in which each of them provides a system of nonlinear algebraic equations when the method is used. As a result, many analytical exact solutions obtained of these equations are expressed in terms of hyperbolic function solutions, trigonometric function solutions, and rational function solutions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, solitary wave solutions of kink type, and so on. The method is very efficient, powerful, and reliable for solving the proposed equations and other nonlinear fractional partial differential equations with the aid of a symbolic software package.

scholarly journals Study on Date–Jimbo–Kashiwara–Miwa Equation with Conformable Derivative Dependent on Time Parameter to Find the Exact Dynamic Wave Solutions

2021 ◽  
Vol 6 (1) ◽  
pp. 4
Author(s):  
Md Ashik Iqbal ◽  
Ye Wang ◽  
Md Mamun Miah ◽  
Mohamed S. Osman
Keyword(s):  
Expansion Method ◽  
Vital Role ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Conformable Derivative ◽  
Wave Solutions ◽  
Soliton Wave Solutions ◽  
Kink Soliton ◽  
Variable G ◽  
Dynamic Wave

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.

scholarly journals Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative

2021 ◽  
Vol 5 (4) ◽  
pp. 238
Author(s):  
Li Yan ◽  
Gulnur Yel ◽  
Ajay Kumar ◽  
Haci Mehmet Baskonus ◽  
Wei Gao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rational Function ◽  
Analytical Approach ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Analytical Scheme

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

scholarly journals Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Rational Solutions ◽  
Partial Differential ◽  
Wave Solutions

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

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