Investigating the correspondence between systems of partial
differential equations and their analytic solutions using a formal
approach, this monograph presents algorithms to determine the set
of analytic solutions of such a system and conversely to find
differential equations whose set of solutions coincides with a
given parametrized set of analytic functions. After giving a
detailed introduction to Janet bases and Thomas decomposition, the
problem of finding an implicit description of certain sets of
analytic functions in terms of differential equations is addressed.
Effective methods of varying generality are developed to solve the
differential elimination problems that arise in this context. In
particular, it is demonstrated how the symbolic solution of partial
differential equations profits from the study of the
implicitization problem. For instance, certain families of exact
solutions of the Navier-Stokes equations can be computed.
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