scholarly journals Linearizations of ordinary differential equations by area preserving maps

1999 ◽  
Vol 156 ◽  
pp. 109-122 ◽  
Author(s):  
Tetsuya Ozawa ◽  
Hajime Sato
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Contact Form ◽  
Third Order ◽  
Area Preserving Maps ◽  
Geometric Explanation ◽  
Coordinate Changes ◽  
Area Preserving ◽  
Second Order Equations

We clarify the class of second and third order ordinary differential equations which can be tranformed to the simplest equations Y″ = 0 and Y‴ = 0. The coordinate changes employed to transform the equations are respectively area preserving maps for second order equations and contact form preserving maps for third order equations. A geometric explanation of the results is also given by using connections and associated covariant differentials both on tangent and cotangent spaces.

scholarly journals Wang-Ball Polynomials for the Numerical Solution of Singular Ordinary Differential Equations

2021 ◽  
pp. 941-949
Author(s):  
Ahmed Kherd ◽  
Azizan Saaban ◽  
Ibrahim Eskander Ibrahim Fadhel
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Operational Matrix ◽  
Second Order ◽  
Operational Matrices ◽  
Second Order Equations

This paper presents a new numerical method for the solution of ordinary differential equations (ODE). The linear second-order equations considered herein are solved using operational matrices of Wang-Ball Polynomials. By the improvement of the operational matrix, the singularity of the ODE is removed, hence ensuring that a solution is obtained. In order to show the employability of the method, several problems were considered. The results indicate that the method is suitable to obtain accurate solutions.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 609-629 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Nonlinear Odes ◽  
Second Order Equations ◽  
Integrating Factors

Coupled second-order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations (ODEs), we focus our attention on the method of deriving a general solution for two coupled second-order nonlinear ODEs through the extended Prelle–Singer procedure. We describe a procedure to obtain integrating factors and the required number of integrals of motion so that the general solution follows straightforwardly from these integrals. Our method tackles both isotropic and non-isotropic cases in a systematic way. In addition to the above-mentioned method, we introduce a new method of transforming coupled second-order nonlinear ODEs into uncoupled ones. We illustrate the theory with potentially important examples.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. V. Linearization of coupled second-order equations

2009 ◽  
Vol 465 (2108) ◽  
pp. 2369-2389 ◽  
Author(s):  
V. K. Chandrasekar ◽  
M. Senthilvelan ◽  
M. Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Straightforward Method ◽  
The Given ◽  
Second Order Equations

Linearization of coupled second-order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper, we describe a simple and straightforward method to derive linearizing transformations for a class of two coupled SNODEs. Our procedure gives several new types of linearizing transformations of both invertible and non-invertible kinds. In both cases, we provide algorithms to derive the general solution of the given SNODE. We illustrate the theory with potentially important examples.

scholarly journals The Complete Solution of some Kinds of Linear Third Order Partial Differential Equations with Three Independent Variables and variable coefficients

2016 ◽  
Vol 12 (10) ◽  
pp. 6705-6713
Author(s):  
Rusul Hassan Naser ◽  
Wafaa Hadi Hanoon ◽  
Layla Abd Al-Jaleel Mohsin
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complete Solution ◽  
Variable Coefficients ◽  
Second Order ◽  
Third Order ◽  
Partial Differential ◽  
Independent Variables

In this paper we find the complete solution of some kinds of linear third order partial differential equations of variable coefficients with three independent variables which have the general form  Where A,B,…,T are variable coefficients . By use the some assumptions will transform the above equation to thenonlinear second order ordinary differential equations

High Accuracy Simulation for the Weakly Coupled System of Two Second Order Ordinary Differential Equations

2011 ◽  
Vol 467-469 ◽  
pp. 377-382
Author(s):  
Xin Cai
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Asymptotic Properties ◽  
Coupled System ◽  
Second Order ◽  
Third Order ◽  
Multi Scale ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
Component Systems

Coupled system of two second order ordinary differential equations with small parameter was considered. This is a multi-scale system. The solution of the system will change rapidly near both sides of the boundary layer. The system was decomposed into several systems in order to have fourth order asymptotic decomposition firstly. The asymptotic properties of all these systems were discussed secondly. The third order numerical methods were constructed for left side and right side singular component systems thirdly. The error estimation for the system was given finally.

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