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

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... 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 π^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.