Lie group method for constructing integrating factors of first-order ordinary differential equations

2021 ◽  
pp. 1-17
Author(s):  
Yuqiang Feng ◽  
Jicheng Yu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Group Method ◽  
First Order ◽  
Lie Group Method ◽  
Integrating Factors

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 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.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

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 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

scholarly journals Lie group and RK4 for solving nonlinear first order ODEs

2016 ◽  
Vol 5 (2) ◽  
pp. 117
Author(s):  
Sami H. Altoum
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Method ◽  
Lie Group ◽  
Analytical Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Group Method ◽  
Numerical Comparison ◽  
First Order ◽  
Lie Group Method

This paper deals with a numerical comparison between Lie group method and RK4 for solving an nonlinear ordinary differential equation. The Lie group method will be introduced as a analytical method and then compared to RK4 as a numerical method. Some examples will be considered and the global error we be computed numerically.

Strapdown Inertial Navigation Algorithms Based on Lie Group

2016 ◽  
Vol 70 (1) ◽  
pp. 165-183 ◽  
Author(s):  
Jun Mao ◽  
Junxiang Lian ◽  
Xiaoping Hu
Keyword(s):  
Differential Equations ◽  
Lie Group ◽  
Algorithm Design ◽  
Inertial Navigation ◽  
Direction Cosine ◽  
Group Method ◽  
Dual Quaternion ◽  
Lie Group Method ◽  
Direction Cosine Matrix ◽  
Strapdown Inertial Navigation

This paper presents a framework for a strapdown Inertial Navigation System (INS) algorithm design by using Lie group and Lie algebra. The general way to solve Lie group differential equations is introduced. Investigations reveal that this general Lie group method provides a simpler unified way to solve differential equations involving direction cosine matrix, quaternion and dual quaternion, which are widely used in INS algorithm design. Furthermore, we also present a new INS algorithm based on the Special Euclidean group se(3) under the guidelines of Lie group method. Analyses show that se(3) algorithm has the same accuracy as a dual quaternion algorithm, this is also justified by numerical simulations. Though the se(3) algorithm has no improvements in accuracy, the general Lie group method used in the design process shows its brevity and uniformity.

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