scholarly journals Exact Solutions and Conservation Laws of the Drinfel’d-Sokolov-Wilson System

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Catherine Matjila ◽  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Water Waves ◽  
Expansion Method

We study the Drinfel'd-Sokolov-Wilson system, which was introduced as a model of water waves. Firstly we obtain exact solutions of this system using the(G′/G)-expansion method. In addition to exact solutions we also construct conservation laws for the underlying system using Noether's approach.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Emrullah Yaşar ◽  
Sait San ◽  
Yeşim Sağlam Özkan
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Waves ◽  
Function Method ◽  
Boussinesq Equation ◽  
Lie Point Symmetries ◽  
Shallow Water Waves ◽  
Non Linear ◽  
Ill Posed

AbstractIn this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

Lie symmetry analysis, exact solutions, conservation laws of variable-coefficients Calogero–Bogoyavlenskii–Schiff equation

2021 ◽  
Author(s):  
Feng Zhang ◽  
Yuru Hu ◽  
Xiangpeng Xin ◽  
Hanze Liu
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Interaction Solution ◽  
Group Invariant Solutions ◽  
Dimensional Variable

In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.

scholarly journals Symmetry Analysis and Conservation Laws of a Generalized Two-Dimensional Nonlinear KP-MEW Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Khadijo Rashid Adem ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Two Dimensional ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Similarity Reductions

Lie symmetry analysis is performed on a generalized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation. The symmetries and adjoint representations for this equation are given and an optimal system of one-dimensional subalgebras is derived. The similarity reductions and exact solutions with the aid ofG′/G-expansion method are obtained based on the optimal systems of one-dimensional subalgebras. Finally conservation laws are constructed by using the multiplier method.

scholarly journals On the Conservation Laws and Exact Solutions to the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 765
Author(s):  
Arzu Akbulut ◽  
Hassan Almusawa ◽  
Melike Kaplan ◽  
Mohamed S. Osman
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Graphical Representations ◽  
Rational Solutions ◽  
The Given ◽  
Conservation Theorem

In this paper, we consider conservation laws and exact solutions of the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov equation. Firstly, we construct conservation laws of the given equation with the help of the conservation theorem; the developed conservation laws are modified conservation laws. Then, we obtain exact solutions of the given equation via the (G′/G,1/G)-expansion method. The obtained solutions are classified as trigonometric solutions, hyperbolic solutions and rational solutions. Furthermore, graphical representations of the obtained solutions are given.

scholarly journals Some Exact Solutions and Conservation Laws of the Coupled Time-Fractional Boussinesq-Burgers System

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 77 ◽  
Author(s):  
Dandan Shi ◽  
Yufeng Zhang ◽  
Wenhao Liu ◽  
Jiangen Liu
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Water Waves ◽  
Power Series Expansion ◽  
Expansion Method ◽  
Power Series Solution ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Invariant Properties ◽  
Burgers System

In this paper, we investigate the invariant properties of the coupled time-fractional Boussinesq-Burgers system. The coupled time-fractional Boussinesq-Burgers system is established to study the fluid flow in the power system and describe the propagation of shallow water waves. Firstly, the Lie symmetry analysis method is used to consider the Lie point symmetry, similarity transformation. Using the obtained symmetries, then the coupled time-fractional Boussinesq-Burgers system is reduced to nonlinear fractional ordinary differential equations (FODEs), with E r d e ´ l y i - K o b e r fractional differential operator. Secondly, we solve the reduced system of FODEs by using a power series expansion method. Meanwhile, the convergence of the power series solution is analyzed. Thirdly, by using the new conservation theorem, the conservation laws of the coupled time-fractional Boussinesq-Burgers system is constructed. In particular, the presentation of the numerical simulations of q-homotopy analysis method of coupled time fractional Boussinesq-Burgers system is dedicated.

Conservation laws and Exact Solutions of Phi-Four (Phi-4) Equation via the (G′/G, 1/G)-Expansion Method

2016 ◽  
Vol 71 (5) ◽  
pp. 439-446 ◽  
Author(s):  
Arzu Akbulut ◽  
Melike Kaplan ◽  
Filiz Tascan
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Adjoint Equation ◽  
Expansion Method ◽  
Lie Point Symmetries ◽  
Invariance Condition ◽  
Conservation Theorem

AbstractIn this article, we constructed formal Lagrangian of Phi-4 equation, and then via this formal Lagrangian, we found adjoint equation. We investigated if the Lie point symmetries of the equation satisfy invariance condition or not. Then we used conservation theorem to find conservation laws of Phi-4 equation. Finally, the exact solutions of the equation were obtained through the (G′/G, 1/G)-expansion method.

scholarly journals Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Feng Zhang ◽  
Yuru Hu ◽  
Xiangpeng Xin
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Optimal System ◽  
Differential Invariants ◽  
Group Method ◽  
Equivalence Transformations ◽  
3 Dimensional ◽  
Dimensional Variable

In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.

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

2021 ◽  
Vol 5 (3) ◽  
pp. 88
Author(s):  
Supaporn Kaewta ◽  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert ◽  
Surattana Sungnul
Keyword(s):  
Exact Solutions ◽  
Fiber Optics ◽  
Soliton Solution ◽  
Water Waves ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
New Exact Solutions

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.

scholarly journals Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier–Stokes equations

2021 ◽  
Vol 477 (2250) ◽  
pp. 20210220
Author(s):  
Xiaoyu Cheng ◽  
Lizhen Wang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Power Series Expansion ◽  
Expansion Method ◽  
Optimal System ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Fractional Differential Operator

In this paper, we investigate the exact solutions and conservation laws of (2 + 1)-dimensional time fractional Navier–Stokes equations (TFNSE). Specifically, Lie symmetries and corresponding one-dimensional optimal system for TFNSE in Riemann–Liouville sense are obtained. Then, based on the admitted symmetries and optimal system, we reduce these equations to one-dimensional equations or (1 + 1)-dimensional fractional partial differential equations (PDEs) with the help of Erdélyi–Kober fractional differential operator and compound variable transformation. In addition, we solve the reduced PDEs applying power series expansion method and invariant subspace method, respectively. Furthermore, the conservation laws of TFNSE are derived using new Noether theorem.

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