Spatiotemporal fractional‐derivative models (FDMs) have been increasingly used to simulate non‐Fickian diffusion, but methods have not been available to define boundary conditions for FDMs in bounded domains. This study defines boundary conditions and then develops a Lagrangian solver to approximate bounded, one‐dimensional fractional diffusion. Both the zero‐value and nonzero‐value Dirichlet, Neumann, and mixed Robin boundary conditions are defined, where the sign of Riemann‐Liouville fractional derivative (capturing nonzero‐value spatial‐nonlocal boundary conditions with directional superdiffusion) remains consistent with the sign of the fractional‐diffusive flux term in the FDMs. New Lagrangian schemes are then proposed to track solute particles moving in bounded domains, where the solutions are checked against analytical or Eulerian solutions available for simplified FDMs. Numerical experiments show that the particle‐tracking algorithm for non‐Fickian diffusion differs from Fickian diffusion in relocating the particle position around the reflective boundary, likely due to the nonlocal and nonsymmetric fractional diffusion. For a nonzero‐value Neumann or Robin boundary, a source cell with a reflective face can be applied to define the release rate of random‐walking particles at the specified flux boundary. Mathematical definitions of physically meaningful nonlocal boundaries combined with bounded Lagrangian solvers in this study may provide the only viable techniques at present to quantify the impact of boundaries on anomalous diffusion, expanding the applicability of FDMs from infinite domains to those with any size and boundary conditions.
- Award ID(s):
- 1818772
- NSF-PAR ID:
- 10175693
- Date Published:
- Journal Name:
- ESAIM: Control, Optimisation and Calculus of Variations
- Volume:
- 26
- ISSN:
- 1292-8119
- Page Range / eLocation ID:
- 20
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1051/cocv/2020005
