  Within an algebraic framework, used to construct the induced-matter-theory
(IMT) setting, in $(D+1)$-dimensional Brans-Dicke (BD) scenario, we obtain a
modified BD theory (MBDT) in $D$ dimensions. Being more specific, from the
$(D+1)$-dimensional field equations, a $D$--dimensional BD theory, bearing new
features, is extracted by means of a suitable dimensional reduction onto a
hypersurface orthogonal to the extra dimension. In particular, the BD scalar
field in such $D$-dimensional theory has a self-interacting potential, which
can be suitably interpreted as produced by the extra dimension. Subsequently,
as an application to cosmology, we consider an extended spatially flat FLRW
geometry in a $(D+1)$-dimensional space-time. After obtaining the power-law
solutions in the bulk, we proceed to construct the corresponding physics, by
means of the induced MBDT procedure, on the $D$-dimensional hypersurface. We
then contrast the resulted solutions (for different phases of the universe)
with those usually extracted from the conventional GR and BD theories in view
of current ranges for cosmological parameters. We show that the induced perfect
fluid background and the induced scalar potential can be employed, within some
limits, for describing different epochs of the universe. Finally, we comment on
the observational viability of such a model.
