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Licensed Unlicensed Requires Authentication Published by De Gruyter December 31, 2019

Mass-conserving tempered fractional diffusion in a bounded interval

Anna Lischke, James F. Kelly and Mark M. Meerschaert

Abstract

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. A reflecting boundary condition enforces mass conservation on a bounded interval. In this work, explicit and implicit Euler schemes for tempered fractional diffusion with discrete reflecting or absorbing boundary conditions are constructed. Discrete reflecting boundaries are formulated such that the Euler schemes conserve mass. Conditional stability of the explicit Euler methods and unconditional stability of the implicit Euler methods are established. Analytical steady-state solutions involving the Mittag-Leffler function are derived and shown to be consistent with late-time numerical solutions. Several numerical examples are presented to demonstrate the accuracy and usefulness of the proposed numerical schemes.


This paper is dedicated to the memory of late Professor Wen Chen


Acknowledgments

We would like to thank Harish Sankanarayanan (Department of Statistics and Probability, Michigan State University) for the fruitful discussions and suggestions for improving the manuscript. This work was supported by the OSD/ARO/MURI on “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications” (W911NF-15-1-0562). Kelly acknowledges support of the Chief of Naval Research via the base 6.1 support program.

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Received: 2019-05-13
Published Online: 2019-12-31
Published in Print: 2019-12-18

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