Equivalence transformations for one-dimensional wave equations of balance form


Equivalence transformations associated with the most general one-dimensional wave equation of balance form are investigated. Equivalence transformations considered here constitute a group and are thus much more general than symmetry groups, in the sense that they map equations involving arbitrary functions or parameters into equations of the same family. The formalism adopted in this work is based on exterior calculus, and the problem is simply reduced to determine isovector fields in the tangent space of an extended differentiable manifold dictated by the structure of the differential equation whose orbits induce transformations which leave an ideal of an exterior algebra over the manifold generated by certain contact and balance forms invariant. A general form of the isovector field is obtained. The components of the independent isovector fields are none other than the infinitesimal generators of the corresponding equivalence groups. Some special cases are also treated.

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Şuhubi, E.S. Equivalence transformations for one-dimensional wave equations of balance form. ARI 50, 151–160 (1998). https://doi.org/10.1007/s007770050009

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Key words

  • Equivalence transformations
  • Equivalence groups
  • Symmetry transformations
  • Symmetry groups
  • One-dimensional quasi-linear wave equations
  • Balance equations