Constructing Solutions to Two-Way Diffusion Problems

Abstract

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.

arXiv commentPublished in J. Phys. A: Math. Theor

References in corpus (4)

• Self-Starting Micromotors in a Bacterial Bath
• Minimal model of active colloids highlights the role of mechanical interactions in controlling the emergent behavior of active matter
• Tunable long range forces mediated by self-propelled colloidal hard spheres
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Cited by in corpus (4)

• Steady states of active Brownian particles interacting with boundaries
• Close encounters of the sticky kind: Brownian motion at absorbing boundaries
• Trapping of an active Brownian particle at a partially absorbing wall
• On the effect of boundaries on noninteracting weakly active particles in different geometries