Linear Partial Differential Equations and Linear Partial Differential Operators in Computer Algebra

Abstract

AbstractIn this survey paper we describe our recent contributions to symbolic algorithmic problems in the theory of Linear Partial Differential Operators (LPDOs). Such operators are derived from Linear Partial Differential Equations in the usual way. The theory of LPDOs has a long history, dealing with problems such as the determination of differential invariants, factorization, and exact methods of integration. The study of constructive factorization have led us to the notion of obstacles to factorization, to the construction of a full generating set of invariants for bivariate LPDOs of order 3, to necessary and sufficient conditions for the existence of a factorization in terms of generating invariants, and a result concerning multiple factorizations of LPDOs. We give links to our further work on generalizations of these results to n-variate LPDOs of arbitrary order.KeywordsLinear Partial Differential Operators (LPDOs)Invariant LaplacianLaplace Transformation MethodIncomplete FactorizationFormal AdjointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.