scholarly journals Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 170 ◽  
Author(s):  
Amjad Hussain ◽  
Shahida Bano ◽  
Ilyas Khan ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Conservation Laws ◽  
Vector Fields ◽  
Lie Symmetry ◽  
Two Dimensional ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Transport Reaction ◽  
Lie Symmetry Method ◽  
Huxley Equation ◽  
Symmetry Method

In this paper, we investigate a spatially two-dimensional Burgers–Huxley equation that depicts the interaction between convection effects, diffusion transport, reaction gadget, nerve proliferation in neurophysics, as well as motion in liquid crystals. We have used the Lie symmetry method to study the vector fields, optimal systems of first order, symmetry reductions, and exact solutions. Furthermore, using the power series method, a set of series solutions are obtained. Finally, conservation laws are derived using optimal systems.

scholarly journals Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Maria Ihsane El Bahi ◽  
Khalid Hilal
Keyword(s):  
Differential Equation ◽  
Conservation Laws ◽  
Nonlinear Partial Differential Equation ◽  
Lie Symmetry ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Symmetry Operators ◽  
Generalized Time ◽  
Symmetry Method ◽  
Conservation Theorem

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.

scholarly journals Comments on the Paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation”

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 900
Author(s):  
Roman Cherniha
Keyword(s):  
Conservation Laws ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Explicit Solutions ◽  
Two Dimensional ◽  
Lie Symmetry Analysis ◽  
Huxley Equation

This comment is devoted to the paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation” (Symmery, 2020, vol.12, 170), in which several results are either incorrect, or incomplete, or misleading.

Symmetry analysis and invariant solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili equation

2018 ◽  
Vol 15 (08) ◽  
pp. 1850125 ◽  
Author(s):  
Vishakha Jadaun ◽  
Sachin Kumar
Keyword(s):  
Lie Algebra ◽  
Zeta Functions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Petviashvili Equation ◽  
Symmetry Method ◽  
Similarity Reductions

Based on Lie symmetry analysis, we study nonlinear waves in fluid mechanics with strong spatial dispersion. The similarity reductions and exact solutions are obtained based on the optimal system and power series method. We obtain the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reductions for the [Formula: see text]-dimensional Kadomtsev–Petviashvili equation. For different Lie algebra, Lie symmetry method reduces Kadomtsev–Petviashvili equation into various ordinary differential equations (ODEs). Some of the solutions of [Formula: see text]-dimensional Kadomtsev–Petviashvili equation are of the forms — traveling waves, Weierstrass’s elliptic and Zeta functions and exponential functions.

A (2+1)-dimensional Korteweg–de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640001 ◽  
Author(s):  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Water Waves ◽  
Function Method ◽  
Type Equation ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
Shallow Water Waves ◽  
Lie Symmetry Method ◽  
De Vries ◽  
Symmetry Method

We consider a (2+1)-dimensional Korteweg–de Vries type equation which models the shallow-water waves, surface and internal waves. In the analysis, we use the Lie symmetry method and the multiple exp-function method. Furthermore, conservation laws are computed using the multiplier method.

scholarly journals Analysis of the Symmetries and Conservation Laws of the Nonlinear Jaulent-Miodek Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mehdi Nadjafikhah ◽  
Mostafa Hesamiarshad
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Symbolic Computation ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Reduced Equations ◽  
Symmetry Generators ◽  
Symmetry Method

Lie symmetry method is performed for the nonlinear Jaulent-Miodek equation. We will find the symmetry group and optimal systems of Lie subalgebras. The Lie invariants associated with the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. And conservation laws of the J-M equation are presented with two steps: firstly, finding multipliers for computation of conservation laws and, secondly, symbolic computation of conservation laws will be applied.

scholarly journals Lie Symmetry Analysis of Kudryashov-Sinelshchikov Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Vahid Shirvani-Sh
Keyword(s):  
Nonlinear Evolution ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Lie Symmetry Method ◽  
Group Invariant Solutions ◽  
Fifth Order ◽  
Symmetry Method ◽  
Preliminary Classification

The Lie symmetry method is performed for the fifth-order nonlinear evolution Kudryashov-Sinelshchikov equation. We will find ones and two-dimensional optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group-invariant solutions is investigated.

Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham–Broer–Kaup–Like Equations

2017 ◽  
Vol 72 (3) ◽  
pp. 269-279 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Shou-Fu Tian ◽  
Chun-Yan Qin ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Vector Fields ◽  
Wave Equations ◽  
Boussinesq Equations ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
Bidirectional Propagation

AbstractIn this article, a generalised Whitham–Broer–Kaup–Like (WBKL) equations is investigated, which can describe the bidirectional propagation of long waves in shallow water. The equations can be reduced to the dispersive long wave equations, variant Boussinesq equations, Whitham–Broer–Kaup–Like equations, etc. The Lie symmetry analysis method is used to consider the vector fields and optimal system of the equations. The similarity reductions are given on the basic of the optimal system. Furthermore, the power series solutions are derived by using the power series theory. Finally, based on a new theorem of conservation laws, the conservation laws associated with symmetries of this equations are constructed with a detailed derivation.

scholarly journals Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

2019 ◽  
Vol 27 (2) ◽  
pp. 171-185 ◽  
Author(s):  
Astha Chauhan ◽  
Rajan Arora
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Analytic Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Liouville Fractional Derivative ◽  
Symmetry Generators ◽  
Symmetry Method

AbstractIn this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

Lie symmetry analysis and some exact solutions for modified Zakharov–Kuznetsov equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850383 ◽  
Author(s):  
Xuan Zhou ◽  
Wenrui Shan ◽  
Zhilei Niu ◽  
Pengcheng Xiao ◽  
Ying Wang
Keyword(s):  
Exact Solutions ◽  
Detailed Analysis ◽  
Symmetry Group ◽  
Nonlinear Waves ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Infinitesimal Generators ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this study, the Lie symmetry method is used to perform detailed analysis on the modified Zakharov–Kuznetsov equation. We have obtained the infinitesimal generators, commutator table of Lie algebra and symmetry group. In addition to that, optimal system of one-dimensional subalgebras up to conjugacy is derived and used to construct distinct exact solutions. These solutions describe the dynamics of nonlinear waves in isothermal multicomponent magnetized plasmas.

scholarly journals Lie symmetry and μ-symmetry methods for nonlinear generalized Camassa–Holm equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. Jafari ◽  
K. Goodarzi ◽  
M. Khorshidi ◽  
V. Parvaneh ◽  
Z. Hammouch
Keyword(s):  
Conservation Laws ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Holm Equation ◽  
Symmetry Method

AbstractIn this paper, a Lie symmetry method is used for the nonlinear generalized Camassa–Holm equation and as a result reduction of the order and computing the conservation laws are presented. Furthermore, μ-symmetry and μ-conservation laws of the generalized Camassa–Holm equation are obtained.

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