Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring

TL;DR

This paper introduces fractional-derivatives series solutions for linear PDEs and extends the concept to non-holonomic D-modules, providing algorithms for factorization of differential operators.

Contribution

It develops a new fractional-derivatives series concept and extends solution methods to non-holonomic D-modules, enabling factorization of complex differential operators.

Findings

Existence of fractional-derivatives series solutions for PDEs

Extension of solutions to non-holonomic D-modules

Algorithms for factoring linear partial differential operators

Abstract

We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.