We discuss Green’s function solutions of the equation for a geometrically thin, axisymmetric Keplerian accretion disc with a viscosity prescription ν∝Rn. The mathematical problem was solved by Lynden-Bell & Pringle for the special cases with boundary conditions of zero-viscous torque and zero mass flow at the disc centre. While it has been widely established that the observational appearance of astrophysical discs depends 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 non-zero inner boundary Rin > 0, for an arbitrary initial surface density profile. Whereas the viscously dissipated power diverges at the disc centre for the previously known solutions with Rin= 0, the new solutions with Rin > 0 have finite expressions for the disc luminosity that agree, in the limit t→∞, with standard expressions for steady-state disc luminosities. The new solutions are applicable to the evolution of the innermost regions of thin accretion discs.