Classification Of Ordinary Differential Equations Research Articles

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

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.

Classification of Certain Class of Ordinary Differential Equations of the First Order

We study problem of global classification of ordinary differential equations with the linear-fractional right-hand side with rational coefficients with respect to a symmetry group. We find the field of differential invariants and obtain the equivalence criterion for two such equations. We adduce certain examples for applying of this criterion. These examples were obtained by means of computer.

On Differential Invariants and Classification of Ordinary Differential Equations of the Form y'' = A(x, y)y' + B(x, y)

The class of second-order ordinary differential equations y'' = A(x, y)y' + B(x, y) is studied by methods of the geometry of jet spaces and the geometric theory of differential equations. The symmetry group of this class of equations is calculated, and the field of differential invariants of its action on equations is described. These results are used to state and prove a criterion for the local equivalence of two nondegenerate ordinary differential equations of the form y'' = A(x, y)y' + B(x, y), inwhich the coefficients A and B are rational in x and y.

On Differential Invariants and Classification of Ordinary Differential Equations of the Form $y-A(x,y)y'+B(x,y)$

В работе с помощью методов геометрии пространств джетов и геометрической теории дифференциальных уравнений изучается класс обыкновенных дифференциальных уравнений второго порядка вида $y-A(x,y)y'+B(x,y)$. Вычислена группа симметрий этого класса уравнений, описано поле дифференциальных инвариантов ее действия на уравнениях. С помощью этих результатов формулируется и доказывается критерий локальной эквивалентности двух невырожденных ОДУ вида $y-A(x,y)y'+B(x,y)$ с рациональными по $x$ и $y$ коэффициентами $A$ и $B$. Библиография: 24 названия.

On Lie’s problem and differential invariants of ODEs y″ = F(x, y)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.

Differential invariants and contact classification of ordinary differential equations

In this work a new approach for classification of ordinary differential equations is suggested. The field of differential invariants is described, basic differential invariants are found, and contact classification of ordinary differential equations in terms of these invariants is obtained.

Group classification of ODE [formula omitted

In his extensive work of 1884 on the group classification of ordinary differential equations Lie performed, inter alia, the group classification of the particular type of the second-order equations y″=F(x,y). In the present paper we extend Lie’s classification to the third-order equations y‴=F(x,y,y′).

Classification of ordinary differential equations by conditional linearizability and symmetry

Lie’s invariant criteria for determining whether a second order scalar equation is linearizable by point transformation have been extended to third and fourth order scalar ordinary differential equations (ODEs). By differentiating the linearizable by point transformation scalar second order ODE (respectively third order ODE) and then requiring that the original equation holds, what is called conditional linearizability by point transformation of third and fourth order scalar ODEs, is discussed. The result is that the new higher order nonlinear ODE has only two arbitrary constants available in its solution. One can use the same procedure for the third and fourth order extensions mentioned above to get conditional linearizability by point or other types of transformation of higher order scalar equations. Again, the number of arbitrary constants available will be the order of the original ODE. A classification of ODEs according to conditional linearizability by transformation and classifiability by symmetry are proposed in this paper.

Symmetry group classification of ordinary differential equations: Survey of some results

AbstractAfter the initial seminal works of Sophus Lie on ordinary differential equations, several important results on point symmetry group analysis of ordinary differential equations have been obtained. In this review, we present the salient features of point symmetry group classification of scalar ordinary differential equations: linear nth‐order, second‐order equations as well as related results. The main focus here is the contributions of Peter Leach, in this area, in whose honour this paper is written on the occasion of his 65th birthday celebrations. Copyright © 2007 John Wiley & Sons, Ltd.

Group Classification of Equations of the Form y = f(x,y)

The problem of classification of ordinary differential equations of the form y″ = f(x,y) by admissible local Lie groups of transformations is solved. “Standard” equations are listed on the basis of the equivalence concept. The classes of equations admitting a one‐parameter group and obtained from the “standard” equations by invariant extension are described.

An Elementary Demonstration of the Existence of $s\ell(3,R)$ Symmetry for all Second-Order Linear Ordinary Differential Equations

Previous article Next article An Elementary Demonstration of the Existence of $s\ell(3,R)$ Symmetry for all Second-Order Linear Ordinary Differential EquationsK. S. Govinder and P. G. L. LeachK. S. Govinder and P. G. L. Leachhttps://doi.org/10.1137/S0036144597316838PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractAll second-order linear ordinary differential equations are shown in an elementary way to possess the symmetry algebra ${s}\ell(2,R)$.[1] M. Aguirre and , J. Krause, SL(3,R) as the group of symmetry transformations for all one‐dimensional linear systems, J. Math. Phys., 29 (1988), 9–15 10.1063/1.528139 89g:58184 CrossrefISIGoogle Scholar[2] Robert Anderson and , Suzanne Davison, A generalization of Lie’s “counting” theorem for second‐order ordinary differential equations, J. Math. Anal. Appl., 48 (1974), 301–315 51:971 CrossrefISIGoogle Scholar[3] Nedialko Nedialkov and , Kenneth Jackson, An interval Hermite‐Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation, Kluwer Acad. Publ., Dordrecht, 1999, 289–310 1744280 Google Scholar[4] L. M. Berković, Glyden–Mes˘c˘erskii problem, Celest. Mech., 24 (1981), pp. 407–429. csm CLMCAV 0008-8714 Celest. Mech. CrossrefISIGoogle Scholar[5] Google Scholar[6] F. Mahomed and , P. Leach, The Lie algebra sl(3,R) and linearization, Quaestiones Math., 12 (1989), 121–139 91d:58221 CrossrefGoogle Scholar[7] F. Mahomed and , P. Leach, Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl., 151 (1990), 80–107 91h:58091 CrossrefISIGoogle Scholar[8] C. Wulfman and , B. Wybourne, The Lie group of Newton’s and Lagrange’s equations for the harmonic oscillator, J. Phys. A, 9 (1976), 507–518 10.1088/0305-4470/9/4/007 54:479 CrossrefGoogle ScholarKeywordslinear differential equations$s\ell(3R)$ Previous article Next article FiguresRelatedReferencesCited byDetails Analysis of modified Painlevé–Ince equations28 May 2020 | Ricerche di Matematica, Vol. 71, No. 1 Cross Ref Maximally superintegrable systems in flat three-dimensional space are linearizableJournal of Mathematical Physics, Vol. 62, No. 1 Cross Ref Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimesInternational Journal of Geometric Methods in Modern Physics, Vol. 12, No. 03 Cross Ref Symmetry group classification of ordinary differential equations: Survey of some results1 January 2007 | Mathematical Methods in the Applied Sciences, Vol. 30, No. 16 Cross Ref Lie symmetry algebra of one-dimensional nonconservative dynamical systems11 September 2007 | Chinese Physics, Vol. 16, No. 9 Cross Ref Integrability analysis of the Emden-Fowler equation21 January 2013 | Journal of Nonlinear Mathematical Physics, Vol. 14, No. 3 Cross Ref Symmetries of First Integrals and Their Associated Differential EquationsJournal of Mathematical Analysis and Applications, Vol. 235, No. 1 Cross Ref Volume 40, Issue 4| 1998SIAM Review History Published online:02 August 2006 InformationCopyright © 1998 Society for Industrial and Applied MathematicsKeywordslinear differential equations$s\ell(3R)$MSC codes34A30PDF Download Article & Publication DataArticle DOI:10.1137/S0036144597316838Article page range:pp. 945-946ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

22.—A Critical Class of Examples concerning the Integrable-square Classification of Ordinary Differential Equations

SynopsisLet the coefficient q be real-valued on the half-line [0, ∞) and let q′ be locally absolutely continuous on [0, ∞). The ordinary symmetric differential expressions M and M2 are determined byIt has been shown in a previous paper by the authors that if for non-negative numbers k and X the coefficient q satisfies the conditionthen M is limit-point and M2 is limit–2 at ∞.This paper is concerned with showing that for powers of the independent variable x the condition (*) is best possible in order that both M and M2 should have the classification at ∞ given above.