We discuss the definition and the construction of involutive bases, a special kind of Gröbner bases, for left ideals in the Weyl algebra. We consider not only term orders, for which the extension is straightforward, but also the more general case of multiplicative monomial orders.
Summary. We discuss the use of the formal theory of differential equations in the numerical analysis of general systems of partial differential equations. This theory provides us with a very powerful and natural framework for generalising many ideas from differential algebraic equations to partial differential equations. We study in particular the existence and uniqueness of (formal) solutions, the method of an underlying system, various index concepts and the effect of semi-discretisations.
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