 1  Author
 D.A Becker, E. W. Richter  Requires cookie*   Title
 Partially Invariant Solutions for Twodimensional Ideal MHD Equations    Abstract
 A generalization of the usual method of similarity analysis of differential equations, the method of partially invariant solutions, was introduced by Ovsiannikov. The degree of noninvariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of firstorder partial differential equations. We apply the algorithm to the nonlinear equations of the twodimensional nonstationary ideal MHD with a magnetic field perpendicular to the plane of motion.   
Reference
 Z. Naturforsch. 45a, 1219—1229 (1990); received July 4 1990   
Published
 1990   
Keywords
 Magnetohydrodynamics, Compressible flows, Group theory   
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2  Author
 H. Kötz  Requires cookie*   Title
 A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro)Differential Equations. I. Optimal Systems of Solvable Lie Subalgebras    Abstract
 Lie group analysis is a powerful tool for obtaining exact similarity solutions of nonlinear (integro) differential equations. In order to calculate the groupinvariant solutions one first has to find the full Lie point symmetry group admitted by the given (integro)differential equations and to determine all the subgroups of this Lie group. An effective, systematic means to classify the similarity solutions afterwards is an "optimal system", i.e. a list of groupinvariant solutions from which every other such solution can be derived. The problem to find optimal systems of similarity solutions leads to that to "construct" the optimal systems of subalgebras for the Lie algebra of the known Lie point symmetry group. Our aim is to demonstrate a practicable technique for determining these optimal subalgebraic systems using the invariants relative to the group of the inner automorphisms of the Lie algebra in case of a finitedimensional Lie point symmetry group. Here, we restrict our attention to optimal subsystems of solvable Lie subalgebras. This technique is applied to the ninedimensional real Lie point symmetry group admitted by the twodimensional nonstationary ideal magnetohydrodynamic equations.   
Reference
 Z. Naturforsch. 47a, 1161—1173 (1992); received July 9 1992   
Published
 1992   
Keywords
 Group theory, Magnetohydrodynamics, Compressible flows   
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3  Author
 DirkA Becker  Requires cookie*   Title
 A Simplified Algorithm for Finding Partially Invariant Solutions of Quasilinear Systems    Abstract
 The method of partially invariant solutions of PDE systems was introduced by Ovsiannikov as a generalization of the classical similarity analysis. It offers a possibility to calculate exact solutions possessing a higher degree of freedom than similarity solutions. Ovsiannikov's algorithm, however, is somewhat hard to apply because one has to deal with three equation systems derived from the original PDE system. By means of the twodimensional Euler equations, we show how the algorithm can be essentially simplified if classical similarity solutions are already known. Further, we prove a necessary criterion for the simplified algorithm to be senseful.   
Reference
 Z. Naturforsch. 49a, 458—464 (1994); received October 7 1993   
Published
 1994   
Keywords
 Hydrodynamics, Nonlinear systems, Group theory   
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4  Author
 H. Teuscher, P. Kramer  Requires cookie*   Title
 D    Abstract
 e c o m p o s itio n o f P l a n e W a v e s in to I r r e d u c i b le R e p r e s e n t a t i o n s o f S p a c e G r o u p s Using a relation between representation theory of crystallographic space groups and a Dirichlet type of boundary problem for the Laplacian, we derive the solutions for the Dirichlet problem, as well as for a similar Neumann boundary problem, by a complete decomposition of plane waves into irreducible representations of a particular space group. This decomposition corresponds to a basis transformation in L2(Q) and yields a new set of basis functions adapted to the symmetry of the lattice considered.   
Reference
 Z. Naturforsch. 50a, 577—583 (1995); received December 16 1994   
Published
 1995   
Keywords
 Group Theory, Representation Theory, BoundaryValue problem, Basis Set for Band Calculations, Free Particle in a Box   
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