  A variety of boundary value problems in linear transport theory are expressed
as a diffusion equation of the two-way, or forward-backward, type. In such
problems boundary data are specified only on part of the boundary, which
introduces several technical challenges. Existence and uniqueness theorems have
been established in the literature under various assumptions; however,
calculating solutions in practice has proven difficult. Here we present one
possible means of practical calculation. By formulating the problem in terms of
projection operators, we derive a formal sum for the solution whose terms are
readily calculated. We demonstrate the validity of this approach for a variety
of physical problems, with focus on a periodic problem from the field of active
matter.
