On optimal system, exact solutions and conservation laws of the Broer-Kaup system

2015 ◽  
Vol 130 (11) ◽  
Author(s):  
Zhonglong Zhao ◽  
Bo Han
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Optimal System

scholarly journals Exact Solutions and Conservation Laws of the (3 + 1)-Dimensional B-Type Kadomstev–Petviashvili (BKP)-Boussinesq Equation

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 97 ◽  
Author(s):  
Ben Gao ◽  
Yao Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Boussinesq Equation ◽  
Lie Symmetry ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Reduced Equations ◽  
Tanh Method ◽  
Similarity Reductions

In this paper, Lie symmetry analysis is presented for the (3 + 1)-dimensional BKP-Boussinesq equation, which seriously affects the dispersion relation and the phase shift. To start with, we derive the Lie point symmetry and construct the optimal system of one-dimensional subalgebras. Moreover, according to the optimal system, similarity reductions are investigated and we obtain exact solutions of reduced equations by means of the Tanh method. In the end, we establish conservation laws using Ibragimov’s approach.

scholarly journals Exact solutions of equal-width equation and its conservation laws

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 505-511 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie ◽  
Innocent Simbanefayi
Keyword(s):  
Wave Propagation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Optimal System ◽  
Invariant Solutions ◽  
Cnoidal Waves ◽  
Dispersion Process ◽  
One Dimensional ◽  
Linear Medium ◽  
D Wave

Abstract In this work we investigate the equal-width equation, which is used for simulation of (1-D) wave propagation in non-linear medium with dispersion process. Firstly, Lie symmetries are determined and then used to establish an optimal system of one-dimensional subalgebras. Thereafter with its aid we perform symmetry reductions and compute new invariant solutions, which are snoidal and cnoidal waves. Additionally, the conservation laws for the aforementioned equation are established by invoking multiplier method and Noether’s theorem.

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.

On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions

2019 ◽  
Vol 17 (01) ◽  
pp. 2050013 ◽  
Author(s):  
Jian-Gen Liu ◽  
Xiao-Jun Yang ◽  
Yi-Ying Feng ◽  
Hong-Yi Zhang
Keyword(s):  
Diffusion Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Fractional Differential Operator ◽  
Diffusion Phenomena ◽  
Fractional Differential ◽  
New Interpretation ◽  
Generalized Time

Under study in this paper is a time fractional generalized nonlinear diffusion equation which can be better to express diffusion phenomena than diffusion equation of integer order. Firstly, we apply the symmetry analysis method to find the symmetry of this considered equation. Then some conservation laws can also be constructed through the above obtained symmetry with the help of the Noether’s theorem. Next, we reduce this equation into an ordinary differential equation of fractional order in the symmetry with Erdélyi–Kober fractional differential operator under one-dimensional subalgebras optimal system framework. Finally, some exact solutions contain the invariant solutions have found for this given equation. The results give us a new interpretation of this type diffusion process.

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 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.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

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