You can call solution(t), then call the IDCT on that, but this could be expensive if you want to do it at a large number of times t.
In principle, internal to solution(t) is probably a piecewise polynomial in time, and since polynomial evaluation is linear in the coefficients, you could call the IDCT on the coefficients to get a callable object spatialsolution(t) that gives you the spatial solution at any desired t (by evaluating the transformed coefficients). I’m not sure if DifferentialEquations.jl gives you enough access to the solution internals to do this, though?