On the Generalizations of the Kummer–Schwarz Equation

Bozhkov, Yuri; Conceição, Pammela Ramos da

doi:10.1016/j.na.2019.111691

Cited by 8 publications

(9 citation statements)

References 15 publications

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“…Let's remember that ξ = c 1 (x), then for the other cases of f in (12b) the following differential equation must be satisfied c 1 (x)f x + 2c 1,x f + c 1,xxx = 0, so, if the functions c 1 (x) and f (x) satisfy Eq. (12b) then we have the following operator, additional to (9): X c1 = c 1 (x)∂ x , therefore following [3] we have the solution is given in the terms of two linearly independent solutions of the associated to (12b) homogeneous second-order Sturm-Liouville equation…”

Section: F(x) Condition Infinitesimal Generators Of the Groupmentioning

confidence: 99%

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Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

Hernández¹,

Duque²,

Acevedo³

et al. 2021

Rev Integr temas mat

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Section: F(x) Condition Infinitesimal Generators Of the Groupmentioning

confidence: 99%

“…that is z 1 and z 2 are linearly independent solutions of (13) with Wronskian = 1. Thus therefore following [3] we have the solution…”

Section: F(x) Condition Infinitesimal Generators Of the Groupmentioning

confidence: 99%

Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

Hernández¹,

Duque²,

Acevedo³

et al. 2021

Rev Integr temas mat

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show abstract

“…In my memory, researchers always pay attention to PDEs, well there are not enough research results on ODEs, especially on the symmetries of DEs in literature. In this paper, we mainly consider a class of ODEs appeared in the fields of physics and engineering, named the generalizations of the Kummer-Schwarz equations [14]:…”

Section: Introductionmentioning

confidence: 99%

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

Tian¹,

Pang²

2022

Sound&Vibration

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This is the first paper on symmetry classification for ordinary differential equations (ODEs) based on Wu's method. We carry out symmetry classification of two ODEs, named the generalizations of the Kummer-Schwarz equations which involving arbitrary function. First, Lie algorithm is used to give the determining equations of symmetry for the given equations, which involving arbitrary functions. Next, differential form Wu's method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets, which are easy to be solved relatively. Each branch of the decomposition yields a class of symmetries and associated parameters. The algorithm makes the classification become direct and systematic. Yuri Dimitrov Bozhkov, and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020. From this paper, we can find that the differential form Wu's method for symmetry classification of ODEs with arbitrary function (parameter) is effective, and is an alternative method.

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“…The classification of the Lie algebra has been carry out in different works. For example in [8] it was presented that the Lie algebra associated to generalizations of the Kummer-Schwarz equation in a particular case is ⊕ . Moreover, in [9] the authors classify the Lie algebra corresponding for a Generalization of the Levinson-Smith equation, and it is proved that this Lie algebra is a solvable nonnilpotent five dimensional Lie algebra which is isomorphic to g5,35 in the MubarAkzyanov's classification.…”

Section: Introductionmentioning

confidence: 99%

Lie algebra classification for the Chazy equation and further topics related with this algebra

et al. 2021

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It is known that the classification of the Lie algebras is a classical problem. Due to Levi’s Theorem the question can be reduced to the classification of semi-simple and solvable Lie algebras. This paper is devoted to classify the Lie algebra generated by the Lie symmetry group of the Chazy equation. We also present explicitly the one parame-ter subgroup related to the infinitesimal generators of the Chazy symmetry group. Moreover the classification of the Lie algebra associated to the optimal system is investigated. La clasificación de las álgebras de Lie es un problema clásico. Acorde al teorema de Levi la cuestión puede reducirse a la clasificación de álgebras de Lie semi-simples y solubles. Este artículo está dedicado a clasificar el álgebra de Lie generada por el grupo de simetría de Lie para la ecuación de Chazy. También presentamos explícitamente los subgrupos a un parámetro relacionados con los generadores de las simetrías del grupo de Chazy. Además, la clasificación de la álgebra de Lie asociada al sistema optimo es investigada.

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On the Generalizations of the Kummer–Schwarz Equation

Cited by 8 publications

References 15 publications

Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

Lie algebra classification for the Chazy equation and further topics related with this algebra

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