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Symmetry Algebra Classification of Scalar n th Order Ordinary Differential Equations
  • Waqas Shah,
  • Fazal Mahomed,
  • H. Azad
Waqas Shah
Government College University Department of Mathematics

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Fazal Mahomed
University of the Witwatersrand DSI-NRF Centre of Excellence in Human Development
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H. Azad
Government College University Department of Mathematics
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We obtain a complete classification of scalar nth 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. The n=1 ,2 ,3 cases are well-known in the literature. Further, it is known that there are three types of nth order equations depending upon the point symmetry algebra they possess, viz. first-order equations which admit an infinite dimensional Lie algebra of point symmetries, second-order equations possessing the maximum eight point symmetries and higher-order, n≥3, admitting the maximum n+4 dimensional point symmetry algebra. We show that scalar nth order equations for n>5 do not admit maximally an n+3 dimensional real Lie algebra of point symmetries. Moreover, we prove that for n>4 equations can admit two types of 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 dimensional and higher than n dimensional point symmetry algebras admissible for equations of order n≥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.
27 Jun 2023Submitted to Mathematical Methods in the Applied Sciences
27 Jun 2023Submission Checks Completed
27 Jun 2023Assigned to Editor
05 Jul 2023Review(s) Completed, Editorial Evaluation Pending
11 Jul 2023Reviewer(s) Assigned
08 Oct 2023Editorial Decision: Revise Minor
15 Oct 20231st Revision Received
15 Oct 2023Submission Checks Completed
15 Oct 2023Assigned to Editor
15 Oct 2023Review(s) Completed, Editorial Evaluation Pending