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Construction of Mappings Relating Differential Equations

  • George W. Bluman
  • Alexei F. Cheviakov
  • Stephen C. Anco
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 168)

Abstract

A symmetry of a PDE is a transformation (mapping) of its solution manifold into itself, i.e., it is a transformation that maps any solution of the PDE into another solution of the same PDE. Invariant solutions (similarity solutions) are solutions that map into themselves. If a symmetry of a given PDE is a point symmetry, then invariant solutions arise constructively from a reduced differential equation with fewer independent variables [Ovsiannikov [(1962), (1982)]; Bluman & Cole (1974); Olver (1986); Bluman & Kumei (1989); Stephani (1989); Bluman & Anco (2002); Cantwell (2002)].

In this chapter, we consider the problem of determining whether there exists a mapping of a given PDE into a target PDE of interest and to construct such a mapping when it exists. A target PDE is either a specific PDE or a member of a class of PDEs. The target PDE is locally equivalent to the given PDE if the mapping is invertible. The invertible mapping is not necessarily unique if a target PDE is a member of a class of PDEs. It is shown that the situation for showing existence and then finding such a mapping is especially fruitful when the target PDE (or target class of PDEs) is completely characterized by a class of contact symmetries (which only exist as point symmetries in the case of a system of PDEs).

Keywords

Erential Equation Point Symmetry Determine Equation Point Transformation Invertible Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York 2010

Authors and Affiliations

  • George W. Bluman
    • 1
  • Alexei F. Cheviakov
    • 2
  • Stephen C. Anco
    • 3
  1. 1.Department of MathematicsThe University of British ColumbiaVancouverCanada
  2. 2.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada
  3. 3.Department of MathematicsBrock UniversitySt. CatharinesCanada

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