Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivalence Problem for Non-Linearizable Third-Order ODEs with Four-Dimensional Lie Symmetry Subalgebras under Point Transformations

Omar A. Abuloha, Marwan Aloqeili, Ahmad Y. Al-Dweik, F. M. Mahomed

Abstract

Cartan's equivalence method is applied to explicitly construct invariant coframes for four branches, which are used to characterize all non-linearizable third-order ODEs with a four-dimensional Lie symmetry subalgebra under point transformations. Additionally, we present a method for constructing the point transformations based on the derived invariant coframes. Examples are provided to illustrate our approach.

Equivalence Problem for Non-Linearizable Third-Order ODEs with Four-Dimensional Lie Symmetry Subalgebras under Point Transformations

Abstract

Cartan's equivalence method is applied to explicitly construct invariant coframes for four branches, which are used to characterize all non-linearizable third-order ODEs with a four-dimensional Lie symmetry subalgebra under point transformations. Additionally, we present a method for constructing the point transformations based on the derived invariant coframes. Examples are provided to illustrate our approach.
Paper Structure (16 sections, 6 theorems, 85 equations, 2 tables)

This paper contains 16 sections, 6 theorems, 85 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Utilization of Cartan's equivalence approach
  3. Branch $I_{4}\ne 0$ and $I_{6}\neq 0$:
  4. The canonical form $u"'= e^{-u"}.$
  5. The canonical form $u"'= u"^{\frac{b-2}{b-1}}, b\ne-1,0,\frac{1}{2},1,2.$
  6. Branch $I_{6}= 0$ and $I_4\ne0, I_{5}\neq 0$:
  7. The canonical form $u"'=\alpha\frac{u"^2}{u'},\alpha\ne0,\frac{3}{2},3.$
  8. The canonical form $u"'=u"^2.$
  9. Branch $I_{4}= 0$ and $I_6\ne0$:
  10. The canonical form $u"'=3\frac{u"^2}{u'}+\alpha\frac{(2xu"+u')^{3/2}}{x^2\sqrt{u'}},~\alpha\ne0$.
  11. The canonical form $u"'=u"^{\frac{3}{2}}.$
  12. The canonical form $u"'= u"^3$.
  13. Branch $I_{4}= 0$, $I_6=0$ and $I_5\ne0$:
  14. Main theorems
  15. Construction of Point transformations induced by invariant coframes
  16. ...and 1 more sections

Key Result

Theorem 3.1

A third-order ODE $u^{\prime\prime\prime}=f(x,u,p,q)$ is equivalent to one of the canonical forms (i) $u"'=e^{-u"},$ (ii) $u"'= u"^{\frac{b-2}{b-1}}, b\ne-1,0,\frac{1}{2},1,2,$ with four symmetries under point transformation if and only if they belong to the branch $I_4\ne 0, I_6\ne0$ and the exteri have identical constant structure equations for appropriate choices of $J_4$ and $J_6$, where More

Theorems & Definitions (17)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 7 more