Algorithmic methods for Fuchsian systems of linear partial differential equations
Toshinori Ôaku · Journal of the Mathematical Society of Japan · 1995
A generalization of the notion of regular singularity for linear ordinary differential equations to (single) partial differential equations was introduced by Baouendi and Goulaouic [1].They called such equations Fuchsian partial dif- ferential equations with respect to a hypersurface.In [12], [21], Kashiwara and Oshima called the same equations ones with regular singularities in a weak sense along a hypersurface and studied the boundary value problem for such equations.Recently, the notion of Fuchsian partial differential equation of [1] has been generalized to that of Fuchsian system of linear partial differential equa- tions along a submanifold $Y$ of arbitrary codimension by Laurent and Monteiro Fernandes [13].EsPecially, it has been proved in [13] for Fuchsian systems that any power series solution which converges with respect to the variables tangent to $Y$ and formal with respect to the variable(s) normal to $Y$ converges with respect to all the variables.It is also known that the holonomic system with regular singularities in the sense of Kashiwara and Kawai is Fuchsian along any submanifold (cf.[11], [13]).Thus Fuchsian systems constitute a nice and substantially wide class of systems containing many interesting examples.Suppose that a system of linear partial differential equations $\mathscr{M}:P_{1}u=\ldots=P_{s}u=0$ for an unknown function $u$ in an open subset of $C^{n+1}$ and a non-singular complex analytic hypersurface $Y$ are given.(For example, if $\mathscr{M}$ is holonomic, then we take as $Y$ an irreducible component of the "singular locus" of .St.)Then, from the computational point of view, we have the following basic problems about .St: A. IS .St Fuchsian along $Y$ ?B. If so, find the structure of the space of multi-valued analytic (or hyper-