  We propose a novel approach to the problem of cosmological perturbations in a
braneworld model with induced gravity, which leads to a closed system of
equations on the brane. We focus on a spatially closed brane that bounds the
interior four-ball of the bulk space. The background cosmological evolution on
the brane is now described by the normal branch, and the boundary conditions in
the bulk become the regularity conditions for the metric everywhere inside the
four-ball. In this approach, there is no spatial infinity or any other boundary
in the bulk space since the spatial section is compact, hence, we term this
setup as a no-boundary proposal. Assuming that the bulk cosmological constant
is absent and employing the Mukohyama master variable, we argue that the
effects of nonlocality on brane perturbations may be ignored if the brane is
marginally closed. In this case, there arises a relation that closes the system
of equations for perturbations on the brane. Perturbations of pressureless
matter and dark radiation can now be described by a system of coupled
second-order differential equations. Remarkably, this system can be exactly
solved in the matter-dominated and de Sitter regimes. In this case, apart from
the usual growing and decaying modes, we find two additional modes that behave
monotonically on super-Hubble spatial scales and exhibit rapid oscillations
with decaying amplitude on sub-Hubble spatial scales.
