scholarly journals The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Chein-Shan Liu ◽  
Wun-Sin Jhao
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group

Lie Group Analysis for a Mixed Convective Flow and Heat Mass Transfer Over a Permeable Stretching Surface with Soret and Dufour Effects

2013 ◽  
Vol 30 (1) ◽  
pp. 67-75 ◽  
Author(s):  
Reda G. Abdel-Rahman ◽  
Ahmed M. Megahed
Keyword(s):  
Mass Transfer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Group Analysis ◽  
Group Method ◽  
Stretching Surface ◽  
Partial Differential ◽  
Soret And Dufour Effects

ABSTRACTThe Lie group transformation method is applied for solving the problem of mixed convection flow with mass transfer over a permeable stretching surface with Soret and Dufour effects. The application of Lie group method reduces the number of independent variables by one and consequently the system of governing partial differential equations reduces to a system of ordinary differential equations with appropriate boundary conditions. Further, the reduced non-linear ordinary differential equations are solved numerically by using the shooting method. The effects of various parameters governing the flow and heat transfer are shown through graphs and discussed. Our aim is to detect new similarity variables which transform our system of partial differential equations to a system of ordinary differential equations. In this work a special attention is given to investigate the effect of the Soret and Dufour numbers on the velocity, temperature and concentration fields above the sheet.

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 Lie group classification of first-order delay ordinary differential equations

2018 ◽  
Vol 51 (20) ◽  
pp. 205202 ◽  
Author(s):  
Vladimir A Dorodnitsyn ◽  
Roman Kozlov ◽  
Sergey V Meleshko ◽  
Pavel Winternitz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Group Classification ◽  
First Order ◽  
Lie Group Classification

Geometric Integrators

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Differential Equations ◽  
Fluid Mechanics ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Short Review ◽  
Research Problem ◽  
Navier Stokes ◽  
Lie Theory ◽  
Magnus Expansion ◽  
Geometric Integrators

Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric Integrators that respect these symmetries. Extending thesemethods to Euler and Navier-Stokes is an outstanding research problem in fluid mechanics. Therefore, a short review of geometric integrators for ODEs is given in this last chapter. Exponential coordinates on a Lie group are explained; the formula for differentiating a matrix exponential is given and used to derive the first few terms of the Magnus expansion. Geometric integrators corresponding to the Euler and trapezoidal methods for ODEs are given.

scholarly journals Lie Group Method of the Diffusion Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jian-Qiang Sun ◽  
Rong-Fang Huang ◽  
Xiao-Yan Gu ◽  
Ling Yu
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Diffusion Equations ◽  
Group Method ◽  
Kutta Method ◽  
Lie Group Method ◽  
Runge Kutta Method ◽  
Runge Kutta

The diffusion equation is discretized in spacial direction and transformed into the ordinary differential equations. The ordinary differential equations are solved by Lie group method and the explicit Runge-Kutta method. Numerical results showed that Lie group method is more stable than the corresponding explicit Runge-Kutta method.

Elementary Lie Group Analysis and Ordinary Differential Equations

2000 ◽  
Vol 84 (499) ◽  
pp. 186
Author(s):  
Steve Abbott ◽  
Nail H. Ibragimov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Group Analysis ◽  
Lie Group Analysis

scholarly journals Reduction for Ordinary and Partial Differential Equations by Using Lie Group

2019 ◽  
Vol 27 (2) ◽  
pp. 252-260
Author(s):  
Eman Ali Hussain ◽  
Zainab Mohammed Alwan
Keyword(s):  
Exact Solution ◽  
Group Theory ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Partial Differential ◽  
Lie Group Theory ◽  
Set Up

In this publication, We have done Lie group theory is applied to reduce the order of ordinary differential equations (ODEs) with 1-parameter  and reduce a PDEs to ODEs . Also, set up algorithm to solve ODEs and PDEs to obtain the exact solution.

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