New exact solutions for solving the initial-value-problem of the KdV–KP equation via the Lie group method

2015 ◽  
Vol 261 ◽  
pp. 408-418
Author(s):  
Mina B. Abd-el-Malek ◽  
Amr M. Amin
Keyword(s):  
Exact Solutions ◽  
Lie Group ◽  
Initial Value Problem ◽  
Group Method ◽  
Kp Equation ◽  
Initial Value ◽  
Lie Group Method ◽  
New Exact Solutions

New Exact Solutions to Variable Coefficient KP Equation

2012 ◽  
Vol 166-169 ◽  
pp. 3075-3078 ◽  
Author(s):  
Jun Yi Yin
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient ◽  
Function Method ◽  
Group Method ◽  
Kp Equation ◽  
Group Invariant ◽  
Lie Group Method ◽  
New Exact Solutions ◽  
Solitonic Solution ◽  
Group Invariant Solutions

Two kinds of new exact solutions were offered after studying the variable coefficient KP equation, of which, the group invariant solutions of KP equation was obtained by using Lie group method, while the solitonic solution of KP equation was obtained by using hyperbola function method.

scholarly journals Discrete Symmetries Analysis and Exact Solutions of the Inviscid Burgers Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Hongwei Yang ◽  
Yunlong Shi ◽  
Baoshu Yin ◽  
Huanhe Dong
Keyword(s):  
Exact Solutions ◽  
Discrete Symmetries ◽  
Burgers Equation ◽  
Similarity Solutions ◽  
Group Method ◽  
Lie Point Symmetries ◽  
Lie Group Method ◽  
New Exact Solutions ◽  
Inviscid Burgers Equation ◽  
Infinitesimal Transformations

We discuss the Lie point symmetries and discrete symmetries of the inviscid Burgers equation. By employing the Lie group method of infinitesimal transformations, symmetry reductions and similarity solutions of the governing equation are given. Based on discrete symmetries analysis, two groups of discrete symmetries are obtained, which lead to new exact solutions of the inviscid Burgers equation.

scholarly journals Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 224
Author(s):  
Ghaylen Laouini ◽  
Amr M. Amin ◽  
Mohamed Moustafa
Keyword(s):  
Lie Group ◽  
Traveling Wave ◽  
Invariant Solution ◽  
Traveling Wave Solutions ◽  
Group Method ◽  
Lie Group Method ◽  
Reduced Equations ◽  
Wave Solutions ◽  
Negative Order ◽  
Petviashvili Equation

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

scholarly journals Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Winter Sinkala
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Second Order ◽  
Point Transformation ◽  
Group Method ◽  
Lie Group Method ◽  
Generic Solution ◽  
Point Transformations

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples.

scholarly journals Fractal Ion Acoustic Waves of the Space-Time Fractional Three Dimensional KP Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
M. A. Abdou ◽  
Saud Owyed ◽  
S. Saha Ray ◽  
Yu-Ming Chu ◽  
Mustafa Inc ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Acoustic Waves ◽  
Three Dimensional ◽  
Fractional Operators ◽  
Trigonometric Functions ◽  
Fractional Partial Differential Equations ◽  
Kp Equation ◽  
Ion Acoustic Waves ◽  
New Exact Solutions ◽  
Conformable Derivatives

Methods known as fractional subequation and sine-Gordon expansion (FSGE) are employed to acquire new exact solutions of some fractional partial differential equations emerging in plasma physics. Fractional operators are employed in the sense of conformable derivatives (CD). New exact solutions are constructed in terms of hyperbolic, rational, and trigonometric functions. Computational results indicate the power of the method.

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