Symmetries of Differential Equations: From Sophus Lie to Computer Algebra

SIAM Review ◽  
1988 ◽  
Vol 30 (3) ◽  
pp. 450-481 ◽  
Author(s):  
Fritz Schwarz
Keyword(s):  
Differential Equations ◽  
Computer Algebra ◽  
Symmetries Of Differential Equations

scholarly journals Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1018
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Differential Equations ◽  
Riemannian Space ◽  
Dimensional Subspace ◽  
Geometric Construction ◽  
Lie Point Symmetries ◽  
Geodesic Equations ◽  
Symmetries Of Differential Equations ◽  
Projective Collineations

We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study, we prove that the projective collineations of a n+1-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications.

scholarly journals Contact Symmetries of a Model in Optimal Investment Theory

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 217
Author(s):  
Daniel J. Arrigo ◽  
Joseph A. Van de Grift
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Governing Equation ◽  
Optimal Investment ◽  
Original Equation ◽  
Case Scenario ◽  
Investment Theory ◽  
Symmetries Of Differential Equations ◽  
Contact Symmetry ◽  
Terminal Value Problem

It is generally known that Lie symmetries of differential equations can lead to a reduction of the governing equation(s), lead to exact solutions of these equations and, in the best case scenario, lead to a linearization of the original equation. In this paper, we consider a model from optimal investment theory where we show the governing equation possesses an extensive contact symmetry and, through this, we show it is linearizable. Several exact solutions are provided including a solution to a particular terminal value problem.

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