Fritz Schwarz

Automatically determining symmetries of partial differential equations

A REDUCE package for determining the group of Lie symmetries of an arbitrary system of partial differential equations is described. It may be used both interactively and in a batch mode. In many cases the system finds the full group completely automatically. In some other cases there are a few linear differential equations of the determining system left the solution of which cannot be found automatically at present. If it is provided by the user, the infinitesimal generators of the symmetry group are returned.ZusammenfassungEs wird ein REDUCE-Programm zur Bestimmung der Symmetrien beliebiger Systeme von partiellen Differentialgleichungen beschrieben. Es kann sowohl interaktiv als auch im Batch-Betrieb verwendet werden. In vielen Fällen findet es die volle Symmetriegruppe vollständig automatisch. In einigen anderen Fällen bleiben einige lineare Differentialgleichungen des bestimmenden Systems übrig, dessen Lösung im Augenblick nicht automatisch gefunden werden kann. Falls sie vom Benutzer eingegeben werden, antwortet das System mit den infinitesimalen Generatoren der Symmetriegruppe.

An algorithm for determining the size of symmetry groups

To determine the symmetry group of pointor Lie-symmetries of a differential equation is of great theoretical and practical importance, in particular for determining closed form solutions. There does not seem to exist an algorithm that finds this group in general. However, it is always possible to determine thesize of the symmetry group. In this article an algorithm is described that determines for any system of algebraic partial differential equations the number of parameters if the symmetry group is finite, and the number of unspecified functions and its arguments if it is infinite. To this end the so calleddetermining system is transformed into aninvolutive system by means of a critical-pair/completion algorithm similar like it is applied for computing Gröbner bases in polynomial ideal theory. The foundation for obtaining this form is the theory of Riquier and Janet for partial differential equations. The algorithmInvolution System has been implemented in several computer algebra systems as part of the packageSPDE. Various results that have been obtained by applying it are presented as well. If symmetry analysis is considered as part of the more general process of obtaining the best possible information on the solutions of a differential equation, the algorithm described in this article removes the heuristics which is usually involved in making the transition from analytical to numerical methods.ZusammenfassungDie Bestimmung der Symmetriegruppe von Punkt- oder Lie Symmetrien einer Differentialgleichung ist von großer theoretischer und praktischer Bedeutung, besonders um Lösungen in geschlossener Form zu finden. Es scheint keinen Algorithmus zu geben, der diese Gruppe im allgemeinen Fall findet. Es ist jedoch immer möglich, dieGröße der Symmetriegruppe zu finden. In diesem Artikel wird ein Algorithmus beschrieben, der für ein beliebiges System algebraischer Differentialgleichungen die Anzahl der Paramter für eine endliche Symmetriegruppe und die Anzahl der unbestimmten Funktionen und ihre Argumente für eine unendliche Gruppe bestimmt. Dazu wird das bestimmende System in involutive Form transformiert mit Hilfe eines sogenannten Vervollständigungsalgorithmus ähnlich wie bei der Berechnung von Gröbnerbasen in der Polynom- Idealtheorie. Die Grundlage dieses Algorithmus ist die Theorie partieller Differentialgleichungen von Riquier und Janet. Der AlgorithmusInvolution System ist in mehreren Computer-Algebra Systemen als Teil des Pakets SPDE implementiert. Verschiedene Ergebnisse, die damit erhalten wurden, werden ebenfalls beschrieben. Die Symmetrieanalyse ist Teil eines allgemeineren Prozesses, nämlich die bestmögliche Information über die Lösungen von Differentialgleichungen zu erhalten. In diesem Prozeß behebt der Algorithmus, der in dieser Arbeit beschrieben wird, zu einem großen Teil die Heuristik, die üblicherweise beim Übergang zu numerischen und graphischen Methoden involviert ist.

A reduce package for determining lie symmetries of ordinary and partial differential equations

Title of program: LIE0,LIE1,LIE2,LIE3,LIE4 Catalogue number: AAZB Program obtainable form: CPC Program library, Queens University in Belfast, N. Ireland (see application form in this issue) Computer: Siemens 7.760 Operating system: BS 2000 Programming language used: LISP High speed storage required: depends on the problem, minimum about 400 000 bytes No. of bits in a word: 32 Number of cards in combined program and test deck: 200

Equivalence classes, symmetries and solutions of linear third-order differential equations

Abstract The subject of this article are third-order differential equations that may be linearized by a variable change. To this end, at first the equivalence classes of linear equations are completely described. Thereafter it is shown how they combine into symmetry classes that are determined by the various symmetry types. An algorithm is presented allowing it to transform linearizable equations by hyperexponential transformations into linear form from which solutions may be obtained more easily. Several examples are worked out in detail.

Algorithmic solution of Abel's equation

The solution scheme for Abel’s equation proposed in this article avoids to a large extent thead hoc methods that have been discovered in the last two centuries since Abel introduced the equation named after him. On the one hand, it describes an algorithmic method for obtaining almost all closed form solutions known in the literature. It is based on Lie’s symmetry analysis. Secondly, for equations without a symmetry, a new method is proposed that allows to generate solutions of all equations within an equivalence class if a single representative has been solved before. It is based on functional decomposition of the absolute invariant of the equation at hand for which computer algebra algorithms have become available recently.

Addendum to: Automatically determining symmetries of partial differential equations

On apporte plusieurs ameliorations au programme REDUCE pour determiner les symetries des equations aux derivees partielles

Solving third order differential equations with maximal symmetry group

Abstract The largest group of Lie symmetries that a third-order ordinary differential equation (ode) may allow has seven parameters. Equations sharing this property belong to a single equivalence class with a canonical representative v′′′(u)=0. Due to this simple canonical form, any equation belonging to this equivalence class may be identified in terms of certain constraints for its coefficients. Furthermore a set of equations for the transformation functions to canonical form may be set up for which large classes of solutions may be determined algorithmically. Based on these steps a solution algorithm is described for any equation with this symmetry type which resembles a similar scheme for second order equations with projective symmetry group.

On 3rd order ordinary differential equations with maximal symmetry group

The following theorem is proved by investigating the Janet bases of determining systems: In order that a 3rd order quasilinear ordinary differential equation has a seven-parameter symmetry group it must have a certain structure, and a set of necessary and sufficient conditions for its coefficients must be satisfied. This theorem generalizes similar results for linear equations and for quasilinear equations of 2nd order. It is shown how this theorem facilitates the computation of closed form solutions.ZusammenfassungEs wird der folgende Satz bewiesen mit Hilfe der Eigenschaften von Janet Basen: Damit eine quasilineare gewöhnliche Differentialgleichung 3ter Ordnung eine sieben-parametrige Symmetriegruppe hat, muß sie eine bestimmte Skruktur haben und eine Menge notwendiger und hinreichender Bedingungen für die Koeffizienten muß erfüllt sein. Dieser Satz verallgemeinert ähnliche Ergebnisse für lineare Gleichungen und für quasilineare Gleichungen 2ter Ordnung. Es wird gezeigt, wie dieser Satz die Bestimmung geschlossener Lösungen erleichtert.

On positive function series

AbstractFunction series of the form