scholarly journals Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 154 ◽  
Author(s):  
Roman Cherniha ◽  
Vasyl’ Davydovych
Keyword(s):  
Mathematical Model ◽  
Exact Solutions ◽  
System Modeling ◽  
Language Shift ◽  
Reaction Diffusion ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Traveling Fronts ◽  
Three Component Reaction ◽  
Symmetry Method

The known three-component reaction–diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by the Lie symmetry method; furthermore, exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift that has occurred in Ukraine during the Soviet times is suggested.

scholarly journals A hunter-gatherer–farmer population model: Lie symmetries, exact solutions and their interpretation

2018 ◽  
Vol 30 (2) ◽  
pp. 338-357 ◽  
Author(s):  
R. M. CHERNIHA ◽  
V. V. DAVYDOVYCH
Keyword(s):  
Exact Solutions ◽  
Population Model ◽  
Reaction Diffusion ◽  
Lie Symmetry ◽  
Lie Symmetries ◽  
System Modelling ◽  
Hunter Gatherers ◽  
Three Component Reaction ◽  
Biological Interpretation

The Lie symmetry classification of the known three-component reaction–diffusion system modelling the spread of an initially localized population of farmers into a region occupied by hunter-gatherers is derived. The Lie symmetries obtained for reducing the system in question to systems of ordinary differential equations (ODEs) and constructing exact solutions are applied. Several exact solutions of travelling front type are also found, their properties are identified and biological interpretation is discussed.

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 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.

scholarly journals Exact Solutions of Generalized Boussinesq-Burgers Equations and (2+1)-Dimensional Davey-Stewartson Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Isaiah Elvis Mhlanga ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Lie Symmetry ◽  
Coupled Systems ◽  
Burgers Equations ◽  
Lie Symmetry Method ◽  
Symmetry Method

We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+1)-dimensional Davey-Stewartson equations.

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.

Exact solutions of Zabolotskaya–Khokhlov equation using optimal system

2018 ◽  
Vol 15 (07) ◽  
pp. 1850111
Author(s):  
Mohsin Umair ◽  
Tooba Feroze
Keyword(s):  
Exact Solutions ◽  
Lie Symmetry ◽  
Optimal System ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this paper, using the Lie symmetry method, we obtain optimal system, group invariants and exact solutions of [Formula: see text] dimensional Zabolotskaya–Khokhlov equation.

New Exact Solutions of Date Jimbo Kashiwara Miwa Equation Using Lie Symmetry Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Khudija Bibi ◽  
Khalil Ahmad
Keyword(s):  
Exact Solutions ◽  
Linear Partial Differential Equation ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
New Exact Solutions ◽  
Independent Variable ◽  
Symmetry Generators ◽  
Logarithmic Functions ◽  
Djkm Equation ◽  
Symmetry Method

In this article, new exact solutions of 2 + 1 -dimensional Date Jimbo Kashiwara Miwa (DJKM) equation are constructed by applying the Lie symmetry method. By considering similarity variables obtained through Lie symmetry generators, considered 2 + 1 -dimensional DJKM equation is transformed into a linear partial differential equation with reduction of one independent variable. Afterwards by using Lie symmetry generators of this linear PDE, different invariant solutions involving exponential and logarithmic functions are explored which lead to the new exact solutions of the DJKM equation. Graphical representations of the obtained solutions are also presented to show the significance of the current work.

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.

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