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Invariants of third‐order ordinary differential equations y ′′′ = f ( x , y , y ′, y ′′ ) via point transformations
Author(s) -
AlDweik Ahmad Y.,
Mustafa M. T.,
Azad H.,
Mahomed F. M.
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3544
Subject(s) - mathematics , infinitesimal , invariant (physics) , differential operator , homogeneous space , ordinary differential equation , pure mathematics , differential equation , combinatorics , mathematical physics , mathematical analysis , geometry
A new systematic method to find the relative invariant differentiation operators is developed. We incorporate this new approach with Lie's infinitesimal method to study the general class y ′′′ = f ( x , y , y ′, y′′) under general point equivalence transformations in the generic case.As a result, all third‐order differential invariants, relative and absolute invariant differentiation operators are determined for third‐order ODEs y ′′′ = f ( x , y , y ′, y ′′), which are not quadratic in the second‐order derivative. These relative invariant differentiation operators are used to determine the fourth‐order differential invariants and absolute invariant differentiation operators in a degenerate case of interest. As an application, invariant descriptions of all the canonical forms in the complex planewith four infinitesimal symmetries for third‐order ODEs y ′′′ = f ( x , y , y ′, y′′), which are not quadratic in the second‐order derivative, are provided. Copyright © 2015 John Wiley & Sons, Ltd.