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1Author    D.-A Becker, E. W. RichterRequires cookie*
 Title    Partially Invariant Solutions for Two-dimensional 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 non-invariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of first-order partial differential equations. We apply the algorithm to the non-linear equations of the two-dimensional non-stationary 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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 TEI-XML for    default:Reihe_A/45/ZNA-1990-45a-1219.pdf 
 Identifier    ZNA-1990-45a-1219 
 Volume    45 
2Author    H. KötzRequires 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 group-invariant 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 group-invariant 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 finite-dimensional Lie point symmetry group. Here, we restrict our attention to optimal subsystems of solvable Lie subalgebras. This technique is applied to the nine-dimensional real Lie point symmetry group admitted by the two-dimensional non-stationary ideal magneto-hydrodynamic equations. 
  Reference    Z. Naturforsch. 47a, 1161—1173 (1992); received July 9 1992 
  Published    1992 
  Keywords    Group theory, Magnetohydrodynamics, Compressible flows 
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 TEI-XML for    default:Reihe_A/47/ZNA-1992-47a-1161.pdf 
 Identifier    ZNA-1992-47a-1161 
 Volume    47 
3Author    Dirk-A BeckerRequires 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 two-dimensional 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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 TEI-XML for    default:Reihe_A/49/ZNA-1994-49a-0458.pdf 
 Identifier    ZNA-1994-49a-0458 
 Volume    49 
4Author    H. Teuscher, P. KramerRequires cookie*
 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, Boundary-Value problem, Basis Set for Band Calculations, Free Particle in a Box 
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 TEI-XML for    default:Reihe_A/50/ZNA-1995-50a-0577.pdf 
 Identifier    ZNA-1995-50a-0577 
 Volume    50