scholarly journals A Note on the Integration of Scalar Fourth-Order Ordinary Differential Equations with Four-Dimensional Symmetry Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Said Waqas Shah ◽  
F. M. Mahomed ◽  
H. Azad
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Second Order ◽  
Two Dimensional ◽  
Lie Point Symmetries ◽  
Complete Integration ◽  
Symmetry Algebras

The complete integration of scalar fourth-order ODEs with four-dimensional symmetry algebras is performed by utilizing Lie’s method which was invoked to integrate scalar second-order ODEs admitting two-dimensional symmetry algebras. We obtain a complete integration of all scalar fourth-order ODEs that possess four Lie point symmetries.

DETERMINING THE DELAY TIME OF THE TWO-DIMENSIONAL RECONSTRUCTION FOR ORDINARY DIFFERENTIAL EQUATIONS

1995 ◽  
Vol 09 (19) ◽  
pp. 1185-1198 ◽  
Author(s):  
YANG ZHI-AN ◽  
CHEN SHI-GANG ◽  
WANG GUANG-RUI
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delay Time ◽  
Second Order ◽  
Two Dimensional ◽  
Absolute Value ◽  
Dimensional Reconstruction ◽  
The Absolute

The second order Jacobian J has been calculated and the delay time τ for the two-dimensional reconstruction of ordinary differential equations has been determined according to the first maximum of the absolute value of J. This method reveals more features about reconstruction than others, and is used in the Rössler and the forced Brusselator equation.

scholarly journals Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. P. Markakis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Closed Form Solution ◽  
First Integrals ◽  
Form Solution ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Abel Equations

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

scholarly journals Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1180
Author(s):  
Muhammad Safdar ◽  
Asghar Qadir ◽  
Muhammad Umar Farooq
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
First Integrals ◽  
Second Order ◽  
Complex Method ◽  
Two Dimensional ◽  
Noether Symmetries ◽  
The Real ◽  
Complex Methods

Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order.

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