NDLI: Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

Content Provider	Springer Nature Link
Author	Fels, M.E.
Copyright Year	2006
Abstract	The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.
Starting Page	417
Ending Page	454
Page Count	38
File Format	PDF
ISSN	16153375
Journal	Foundations of Computational Mathematics
Volume Number	7
Issue Number	4
e-ISSN	16153383
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-12-28
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Numerical Analysis Computer Science Math Applications in Computer Science Linear and Multilinear Algebras, Matrix Theory Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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