  We introduce a class of singular partial differential equations, the
second-order hyperbolic Fuchsian systems, and we investigate the associated
initial value problem when data are imposed on the singularity. First of all,
we analyze a class of equations in which hyperbolicity is not assumed and we
construct asymptotic solutions of arbitrary order. Second, for the proposed
class of second-order hyperbolic Fuchsian systems, we establish the existence
of solutions with prescribed asymptotic behavior on the singularity. Our proof
is based on a new scheme which is also suitable to design numerical
approximations. Furthermore, as shown in a follow-up paper, the second-order
Fuchsian framework is appropriate to handle Einstein's field equations for
Gowdy symmetric spacetimes and allows us to recover (and slightly generalize)
earlier results by Rendall and collaborators, while providing a direct approach
leading to accurate numerical solutions. The proposed framework is also robust
enough to encompass matter models arising in general relativity.
