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Formal Algorithmic Elimination for PDEs
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This monograph investigates the correspondence between systems of partial differential equations and their analytic solutions from a formal viewpoint. Algorithms are presented to determine the set of analytic solutions to such a system and 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 monograph addresses the problem of finding implicit descriptions in terms of differential equations of certain sets of analytic functions. Effective methods of different generality are developed to solve the differential elimination problems that arise in this context. In particular, it is demonstrated how symbolic solving 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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