Solving Second-Order Differential Equations with Lie Symmetries

• Published in: Acta applicandae mathematicae Vol. 60; no. 1; p. 39
• Main Author: Schwarz, Fritz
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.01.2000
• Subjects: Algebra; Algorithms; Applied mathematics; Mathematics; Ordinary differential equations; Partial differential equations; Studies; Symmetry; Theory
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1023/A:1006321609161

Lie"s theory for solving second-order quasilinear differential equations based on its symmetries is discussed in detail. Great importance is attached to constructive procedures that may be applied for designing solution algorithms. To this end Lie's original theory is supplemented by various results that have been obtained after his death one hundred years ago. This is true above all of Janet's theory for systems of linear partial differential equations and of Loewy's theory for decomposing linear differential equations into components of lowest order. These results allow it to formulate the equivalence problems connected with Lie symmetries more precisely. In particular, to determine the function field in which the transformation functions act is considered as part of the problem. The equation that originally has to be solved determines the base field, i.e. the smallest field containing its coefficients. Any other field occurring later on in the solution procedure is an extension of the base field and is determined explicitly. An equation with symmetries may be solved in closed form algorithmically if it may be transformed into a canonical form corresponding to its symmetry type by a transformation that is Liouvillian over the base field. For each symmetry type a solution algorithm is described, it is illustrated by several examples. Computer algebra software on top of the type system ALLTYPES has been made available in order to make it easier to apply these algorithms to concrete problems. [PUBLICATION ABSTRACT]