scholarly journals Equivalence classes of second-order ordinary differential equations with only a three-dimensional Lie algebra of point symmetries and linearisation

2003 ◽  
Vol 284 (1) ◽  
pp. 31-48 ◽  
Author(s):  
P.G.L. Leach
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Second Order ◽  
Equivalence Classes

scholarly journals GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

scholarly journals Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
M. Safdar ◽  
Asghar Qadir ◽  
S. Ali
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Canonical Form ◽  
Linear Equations ◽  
Two Dimensions ◽  
Second Order ◽  
Equivalence Classes ◽  
Order Ordinary Differential Equation ◽  
Change Of Variables ◽  
Symmetry Structure

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An “optimal (or simplest) canonical form” of linear systems had been established to obtain the symmetry structure, namely, with 5-, 6-, 7-, 8-, and 15-dimensional Lie algebras. For those systems that arise from a scalar complex second-order ordinary differential equation, treated as a pair of real ordinary differential equations, we provide a “reduced optimal canonical form.” This form yields three of the five equivalence classes of linearizable systems of two dimensions. We show that there exist 6-, 7-, and 15-dimensional algebras for these systems and illustrate our results with examples.

Pushing the limit for the grid-based treatment of Schrödinger's equation: a sparse Numerov approach for one, two and three dimensional quantum problems

2016 ◽  
Vol 18 (46) ◽  
pp. 31521-31533 ◽  
Author(s):  
Ulrich Kuenzer ◽  
Jan-Andrè Sorarù ◽  
Thomas S. Hofer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Second Order ◽  
Special Focus ◽  
Schrödinger’S Equation ◽  
Numerov Method ◽  
Schrödinger's Equation ◽  
Grid Based

The general Numerov method employed to numerically solve ordinary differential equations of second order was adapted with a special focus on matrix sparsity and applications in higher dimensions.

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