Three approaches are considered to solve the equation which describes the timedependent diffusive shock acceleration of test particles at the non-relativistic shocks. At first, the solution of Drury (1983) for the particle distribution function at the shock is generalized to any relation between the acceleration time-scales upstream and downstream and for the time-dependent injection efficiency. Three alternative solutions for the spatial dependence of the distribution function are derived. Then, the two other approaches to solve the time-dependent equation are presented, one of which does not require the Laplace transform. At the end, our more general solution is discussed, with a particular attention to the time-dependent injection in supernova remnants. It is shown that, comparing to the case with the dominant upstream acceleration time-scale, the maximum momentum of accelerated particles shifts toward the smaller momenta with increase of the downstream acceleration time-scale. The timedependent injection affects the shape of the particle spectrum. In particular, i) the power-law index is not solely determined by the shock compression, in contrast to the stationary solution; ii) the larger the injection efficiency during the first decades after the supernova explosion, the harder the particle spectrum around the high-energy cutoff at the later times. This is important, in particular, for interpretation of the radio and gamma-ray observations of supernova remnants, as demonstrated on a number of examples.

Abstract. By the method of classical potential theory, we obtain the integral representation of the two-parameter operator semigroup that describes the inhomogeneous Feller process on a closed interval ‫ݎ[‬ ଵ , ‫ݎ‬ ଶ ] that is a result of pasting together two diffusion processes given on ‫ݎ(‬ ଵ , ‫)ݎ‬ and ‫,ݎ(‬ ‫ݎ‬ ଶ ), respectively, where −∞ < ‫ݎ‬ ଵ < ‫ݎ‬ < ‫ݎ‬ ଶ < ∞.

By analytical methods we construct the two-parameter Feller semigroup associated with Markov process on a line with moving membrane such that at points on both sides of the membrane it coincides with the ordinary diffusion processes given there, and its behavior after visiting the membrane is determined by one of variants of nonlocal Feller-Wentzell conjugation condition.