Constructing solutions to two-way diffusion problems

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.

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