  Recent gamma ray observations show that middle aged supernova remnants
interacting with molecular clouds can be sources of both GeV and TeV emission.
Models involving re-acceleration of pre-existing cosmic rays in the ambient
medium and direct interaction between supernova remnant and molecular clouds
have been proposed to explain the observed gamma ray emission. For the
re-acceleration process, standard DSA theory in the test particle limit
produces a steady state particle spectrum that is too flat compared to
observations, which suggests that the high energy part of the observed spectrum
has not yet reached a steady state. We derive a time dependent DSA solution in
the test particle limit for situations involving re-acceleration of
pre-existing cosmic rays in the preshock medium. Simple estimates with our time
dependent DSA solution plus a molecular cloud interaction model can reproduce
the overall shape of the spectra of IC 443 and W44 from GeV to TeV energies
through pure $\pi^0$-decay emission. We allow for a power law momentum
dependence of the diffusion coefficient, finding that a power law index of 0.5
is favored.
