New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods

Güner, Özkan

doi:10.1007/s11082-017-1311-1

Cited by 9 publications

(3 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[27] to obtain a variety of exact solutions to equation (1). Moreover, using the ansatz method and the functional variable method, Guner [28] examined exact analytic solutions for equation (1) and constructed singular soliton solutions. In the two latter studies, the fractional complex transform method is used to reduce equation (1) into ordinary differential equation (ODE).…”

Section: Introductionmentioning

confidence: 99%

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Al-Ghafri

Rezazadeh

2019

Applied Mathematics and Nonlinear Sciences

View full text Add to dashboard Cite

In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given.

show abstract

Section: Introductionmentioning

confidence: 99%

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Al-Ghafri

Rezazadeh

2019

Applied Mathematics and Nonlinear Sciences

View full text Add to dashboard Cite

show abstract

“…Furthermore, some scientists have devoted substantial efforts to finding exact solutions of the (3 + 1)-dimensional mKdV-ZK equation in the sense of the classical partial, conformable, and Jumarie's modified Riemann-Liouville derivatives using different and reliable approaches. In the past few years, the investigation of exact traveling wave solutions for such mKdV-ZK equations has been discussed in [41][42][43][48][49][50][51]. To the best of the authors' knowledge, there are no research scholars who have found explicit exact solutions for Equations ( 1) and ( 2) using the G /G 2 -expansion method.…”

Section: Introductionmentioning

confidence: 99%

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

et al. 2021

View full text Add to dashboard Cite

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

show abstract

“…Numerous methods have been applied to obtain different solutions of fractional partial differential equations. Some of them are; the functional variable method [7], the extended tanh function method [8], the first integral method [9], the extended direct algebraic method [10], the modified simple equation method [11], the modified trial equation method [12], the ( ′ / ) -expansion method [13], the extended trial equation method [14], the kudryashov method [15] and so on [16][17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

On the Physical Behaviours of the Conformable Fractional Modified Camassa– Holm Equation Using Two Efficient Methods

Elma

Mısırlı

2021

ijmph

View full text Add to dashboard Cite

In recent years, many authors have researched about fractional partial differential equations. Physical phenomena, which arise in engineering and applied science, can be defined more accurately by using FPDEs. Thus, obtaining exact solutions of the FPDEs equations have become more important to understand physical problems. In this article, we have reached the new traveling wave solutions of the conformable fractional modified Camassa -Holm equation via two efficient methods such as first integral method and the functional variable method. The wave transformation and conformable fractional derivative have been used to convert FPDE to the ordinary differential equation. The Camassa -Holm equation is physical model of shallow water waves with non-hydrostatic pressure. Thanks to these powerful methods, some comparisons, such as type of solutions and physical behaviours, have been made. Additionally, mathematica program have been used with the aim of checking of solutions. Investigating results of the fractional differential equations can help understanding complex phenomena in applied mathematics and physics.

show abstract

New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods

Cited by 9 publications

References 40 publications

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

On the Physical Behaviours of the Conformable Fractional Modified Camassa– Holm Equation Using Two Efficient Methods

Contact Info

Product

Resources

About