The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semi-geostrophic equations in such a way that a well-defined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an 'infinite linear cascade'.

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A special class of symmetry reductions called nonclassical equivalence transformations is discussed in connection to a class of parameter identification problems represented by partial differential equations. These symmetry reductions relate the forward and inverse problems, reduce the dimension of the equation, yield special types of solutions, and may be incorporated into the boundary conditions as well. As an example, we discuss the nonlinear stationary heat conduction equation and show that this approach permits the study of the model on new types of domains. Our MAPLE routine GENDEFNC which uses the package DESOLV (authors Carminati and Vu) has been updated for this propose and its output is the nonlinear partial differential equation system of the determining equations of the nonclassical equivalence transformations.

A new approach to parameter identification problems from the point of view of symmetry analysis theory is given. A mathematical model that arises in the design of car windshield represented by a linear second order mixed type PDE is considered. Following a particular case of the direct method (due to Clarkson and Kruskal), we introduce a method to study the group invariance between the parameter and the data. The equivalence transformations associated with this inverse problem are also found. As a consequence, the symmetry reductions relate the inverse and the direct problem and lead us to a reduced order model.