Lie symmetry analysis and reduction for exact solution of (2+1)-dimensional Bogoyavlensky–Konopelchenko equation by geometric approach

2018 ◽  
Vol 32 (11) ◽  
pp. 1850127 ◽  
Author(s):  
S. Saha Ray
Keyword(s):  
Exact Solution ◽  
Differential Forms ◽  
Geometric Approach ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Invariance Group ◽  
Similarity Reduction ◽  
Lie Symmetry Method ◽  
Explicit Exact Solution ◽  
Symmetry Method

In this paper, the symmetry analysis and similarity reduction of the (2[Formula: see text]+[Formula: see text]1)-dimensional Bogoyavlensky–Konopelchenko (B–K) equation are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Using the extended Harrison and Estabrook’s differential forms approach, the infinitesimal generators for (2[Formula: see text]+[Formula: see text]1)-dimensional B–K equation are obtained. Firstly, the vector field associated with the Lie group of transformation is derived. Then the symmetry reduction and the corresponding explicit exact solution of (2[Formula: see text]+[Formula: see text]1)-dimensional B–K equation is obtained.

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.

scholarly journals Parametric conditions and exact solution for the Duffing-Van der Pol class of equations

2018 ◽  
Vol 40 (3) ◽  
pp. 251-264
Author(s):  
Dao Huy Bich ◽  
Nguyen Dang Bich
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Lower Order ◽  
Main Idea ◽  
Lie Symmetry ◽  
First Integrals ◽  
Original Equation ◽  
Van Der Pol ◽  
Lie Symmetry Method ◽  
Symmetry Method

This paper presents a methodology to find the exact solution and respective parametric conditions to the Duffing-Van der Pol class of equations. The supposed method in this paper is different from the Prelle and Singer method and the Lie symmetry method. The main idea of the supposed method is implemented in finding the first integrals of the original equation and leading this equation to a solved equation of lower order to which the exact solution can be obtained. As results the parametric conditions and the exact solutions in parametric forms are indicated. The algorithm for determining integral constants and the investigation of solution characteristics are considered.

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 nonlocal Boussinesq equation: multiple-soliton solutions and symmetry analysis

2021 ◽  
Author(s):  
Xi-zhong Liu ◽  
Jun Yu
Keyword(s):  
Exact Solutions ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Method ◽  
Multiple Soliton Solutions ◽  
Symmetry Method

Abstract A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2, 3, 4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Similarity Reduction and Exact Solutions of a Boussinesq-like Equation

2018 ◽  
Vol 73 (4) ◽  
pp. 357-362 ◽  
Author(s):  
Bo Zhang ◽  
Hengchun Hu
Keyword(s):  
Exact Solutions ◽  
Direct Method ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Invariant Solutions ◽  
Similarity Reduction ◽  
Group Invariant ◽  
Lie Symmetry Method ◽  
Group Invariant Solutions ◽  
Symmetry Method

AbstractThe similarity reduction and similarity solutions of a Boussinesq-like equation are obtained by means of Clarkson and Kruskal (CK) direct method. By using Lie symmetry method, we also obtain the similarity reduction and group invariant solutions of the model. Further, we compare the results obtained by the CK direct method and Lie symmetry method, and we demonstrate the connection of the two methods.

scholarly journals Lie symmetries of Generalized Equal Width wave equations

2021 ◽  
Vol 6 (11) ◽  
pp. 12148-12165
Author(s):  
Mobeen Munir ◽  
◽  
Muhammad Athar ◽  
Sakhi Sarwar ◽  
Wasfi Shatanawi ◽  
...  
Keyword(s):  
Wave Equations ◽  
Travelling Wave ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Lie Group Theory ◽  
Lie Symmetry Method ◽  
Solution Methods ◽  
The One ◽  
Symmetry Method

<abstract><p>Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.</p></abstract>

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.

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