  We discuss Green's-function solutions of the equation for a geometrically
thin, axisymmetric Keplerian accretion disc with a viscosity prescription "\nu
~ R^n". The mathematical problem was solved by Lynden-Bell & Pringle (1974) for
the special cases with boundary conditions of zero viscous torque and zero mass
flow at the disc center. While it has been widely established that the
observational appearance of astrophysical discs depend on the physical size of
the central object(s), exact time-dependent solutions with boundary conditions
imposed at finite radius have not been published for a general value of the
power-law index "n". We derive exact Green's-function solutions that satisfy
either a zero-torque or a zero-flux condition at a nonzero inner boundary
R_{in}>0, for an arbitrary initial surface density profile. Whereas the
viscously dissipated power diverges at the disc center for the previously known
solutions with R_{in}=0, the new solutions with R_{in}>0 have finite
expressions for the disc luminosity that agree, in the limit t=>infinity, with
standard expressions for steady-state disc luminosities. The new solutions are
applicable to the evolution of the innermost regions of thin accretion discs.
