Glauco Lopez Diaz,
"Symbolic Methods for Factoring Linear Differential Operators"
Symbolic Methods for Factoring Linear Differential Operators
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A survey of symbolic methods for factoring linear differential operators is given. Starting from basic notions -- ring of operators, differential Galois theory -- methods for finding rational and exponential solutions that can provide first order right-hand factors are considered. Subsequently several known algorithms for factorization are presented. These include Singer's eigenring factorization algorithm, factorization via Newton polygons, van Hoeij's methods for local factorization, and an adapted version of Pade approximation. In addition a procedure based on pure algebraic methods for factoring second order linear partial differential operators is developed. Splitting an operator of this kind reduces to solving a system of linear algebraic equations. Those solutions which satisfy a certain differential condition, immediately produce linear factors of the operator. The method applies also to operators of third order, thereby resulting in a more complicated system of equations. In contrast to the second order case, differential equations must also be solved, which, in particular cases, are simplified with the aid of characteristic sets. Finally, complete decomposition into linear factors of ordinary differential operators of arbitrary order is discussed. A splitting formula is developed, provided that a linear basis of solutions is available. This theoretical representation is valuable in understanding the nature of the classical Beke algorithm and its variants like the algorithm LODEF by Schwarz and the Beke-Bronstein algorithm.
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Research Institute for Symbolic Computation
Sponsor: DET Project, Fonds zur Förderung der wiss. Forschung