scholarly journals Various Exact Solutions for the Conformable Time-Fractional Generalized Fitzhugh–Nagumo Equation with Time-Dependent Coefficients

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Sami Injrou ◽  
Ramez Karroum ◽  
Nadia Deeb
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Trigonometric Function ◽  
Time Dependent ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Nagumo Equation

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.

scholarly journals A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fanwei Meng
Keyword(s):  
Variable Coefficient ◽  
Toda Lattice ◽  
Traveling Wave Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Lattice Equation ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Toda Lattice Equation

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

scholarly journals Solutions of Fractional Konopelchenko-Dubrovsky and Nizhnik-Novikov-Veselov Equations Using a Generalized Fractional Subequation Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yanqin Liu ◽  
Limei Yan
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
Coupled Equations ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Relationship Of ◽  
The Relationship

A new generalized fractional subequation method based on the relationship of fractional coupled equations is proposed. This method is applied to the space-time fractional coupled Konopelchenko-Dubrovsky equations and Nizhnik-Novikov-Veselov equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. It is observed that the proposed approach provides a simple and reliable tool for solving many other fractional coupled differential equations.

scholarly journals Exact solutions of space–time fractional KdV–MKdV equation and Konopelchenko–Dubrovsky equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 871-880
Author(s):  
Bo Tang ◽  
Jiajia Tao ◽  
Shijun Chen ◽  
Junfeng Qu ◽  
Qian Wang ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Space Time ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Rational Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Physical Features

Abstract In the present study, we deal with the space–time fractional KdV–MKdV equation and the space–time fractional Konopelchenko–Dubrovsky equation in the sense of the conformable fractional derivative. By means of the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method, many exact solutions are obtained, which include hyperbolic function solutions, trigonometric function solutions and rational solutions. The results show that the extend \left(\tfrac{G^{\prime} }{G}\right) -expansion method is an efficient technique for solving nonlinear fractional partial equations. We also provide some graphical representations to demonstrate the physical features of the obtained solutions.

scholarly journals Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Tianyong Han ◽  
Zhao Li
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Boussinesq Equations ◽  
Hyperbolic Function ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
System Method ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Complete Discrimination System

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.

scholarly journals Bifurcation and Solitary-Like Solutions for Compound KdV-Burgers-Type Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Yin Li ◽  
Ruiying Wei ◽  
Guoming Jian
Keyword(s):  
Dynamical Systems ◽  
Type Equation ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Bifurcation Method ◽  
Dynamical Characteristics ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Ode Method ◽  
Parameter Condition

Firstly, based on the improved sub-ODE method and the bifurcation method of dynamical systems, we investigate the bifurcation of solitary waves in the compound KdV-Burgers-type equation. Secondly, numbers of solitary patterns solutions are given for each parameter condition and numerical simulations are used to display the dynamical characteristics. Finally, we obtain twelve solitary patterns solutions under some parameter conditions, such as the trigonometric function solutions and the hyperbolic function solutions.

Non-TravellingWave Solutions of the (2+1)-Dimensional Dispersive Long Wave System

2009 ◽  
Vol 64 (9-10) ◽  
pp. 540-552
Author(s):  
Mamdouh M. Hassan
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Wave System ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Rational Solutions ◽  
Long Wave ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

With the aid of symbolic computation and the extended F-expansion method, we construct more general types of exact non-travelling wave solutions of the (2+1)-dimensional dispersive long wave system. These solutions include single and combined Jacobi elliptic function solutions, rational solutions, hyperbolic function solutions, and trigonometric function solutions.

scholarly journals The Improvedexp⁡-Φξ-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Guiying Chen ◽  
Xiangpeng Xin ◽  
Hanze Liu
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.

Exact solutions of Wick-type stochastic Korteweg – de Vries equation

2012 ◽  
Vol 90 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Sheng Zhang
Keyword(s):  
Vries Equation ◽  
Similarity Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
De Vries ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Hermite Transformation ◽  
Similarity Reductions

An algorithm is devised for using the compatibility method to find symmetries and similarity reductions of Wick-type stochastic partial differential equations. To illustrate the validity and advantages of the devised algorithm, the Wick-type stochastic Korteweg – de Vries equation is considered. With the aid of Hermite transformation and white noise theory, we obtain a symmetry and corresponding reduction of the Wick-type stochastic Korteweg – de Vries equation. Furthermore, some new similarity solutions are obtained including Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions.

scholarly journals On the exact solutions of the fractional (2+1)- dimensional Davey - Stewartson equation system

2017 ◽  
Vol 7 (3) ◽  
pp. 265-270
Author(s):  
Gülnur Yel ◽  
Zeynep Fidan Koçak
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Equation System ◽  
Trigonometric Function ◽  
Equation Method ◽  
Exact Traveling Wave Solutions ◽  
Trial Equation Method ◽  
Wave Solutions ◽  
Trigonometric Function Solutions

In this work, we construct the exact traveling wave solutions of the fractional (2+1)-dimensional Davey-Stewartson equation system (D-S) that is complex equation system using the Modified Trial Equation Method (MTEM). We obtained trigonometric function solutions by this method that are new in literature.

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.

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