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- Title
Symmetry algebra classification of scalar n$$ n $$th‐order ordinary differential equations.
- Authors
Waqas Shah, Said; Mahomed, F. M.; Azad, H.
- Abstract
We obtain a complete classification of scalar n$$ n $$th‐order ordinary differential equations for all subalgebras of vector fields in the real plane. While softwares like Maple can compute invariants of a given order, our results are for a general n$$ n $$. The n=1,2,3$$ n=1,2,3 $$ cases are well‐known in the literature. Further, it is known that there are three types of n$$ n $$th‐order equations depending upon the point symmetry algebra they possess, namely, first‐order equations which admit an infinite dimensional Lie algebra of point symmetries, second‐order equations possessing the maximum 8‐point symmetries, and higher‐order, n≥3$$ n\ge 3 $$, admitting the maximum n+4$$ n+4 $$ dimensional point symmetry algebra. We show that scalar n$$ n $$th‐order equations for n>5$$ n>5 $$ do not admit maximally an n+3$$ n+3 $$ dimensional real Lie algebra of point symmetries. Moreover, we prove that for n>4$$ n>4 $$, equations can admit two types of n+2$$ n+2 $$ dimensional real Lie algebra of point symmetries: one type resulting in nonlinear equations which are not linearizable via a point transformation and the second type yielding linearizable (via point transformation) equations. Furthermore, we present the types of maximal real n$$ n $$ dimensional and higher than n$$ n $$‐dimensional point symmetry algebras admissible for equations of order n≥4$$ n\ge 4 $$ and their canonical forms. The types of lower‐dimensional point symmetry algebras which can be admitted are shown, and the equations are constructible as well. We state the relevant results in tabular form and in theorems.
- Subjects
INFINITE dimensional Lie algebras; NILPOTENT Lie groups; ALGEBRA; SYMMETRY; LIE algebras; NONLINEAR equations
- Publication
Mathematical Methods in the Applied Sciences, 2024, Vol 47, Issue 11, p8449
- ISSN
0170-4214
- Publication type
Article
- DOI
10.1002/mma.10025