Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains
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Description
Fractional differential equations are powerful tools to model the non-locality and spatial heterogeneity evident in many real-world problems. Although numerous numerical methods have been proposed, most of them are limited to regular domains and uniform meshes. For irregular convex domains, the treatment of the space fractional derivative becomes more challenging and the general methods are no longer feasible. In this work, we propose a novel numerical technique based on the Galerkin finite element method (FEM) with an unstructured mesh to deal with the space fractional derivative on arbitrarily shaped convex and non-convex domains, which is the most original and significant contribution of this paper. Moreover, we present a second order finite difference scheme for the temporal fractional derivative. In addition, the stability and convergence of the method are discussed and numerical examples on different irregular convex domains and non-convex domains illustrate the reliability of the method. We also extend the theory and develop a computational model for the case of a multiply-connected domain. Finally, to demonstrate the versatility and applicability of our method, we solve the coupled two-dimensional fractional Bloch-Torrey equation on a human brain-like domain and exhibit the effects of the time and space fractional indices on the behaviour of the transverse magnetization.
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ID Code: | 116276 | ||||||||
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Item Type: | Contribution to Journal (Journal Article) | ||||||||
Refereed: | Yes | ||||||||
ORCID iD: |
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Measurements or Duration: | 23 pages | ||||||||
Keywords: | Finite element method, Unstructured mesh, Riesz fractional derivative, Irregular domains, Two-dimensional, Time-space fractional diffusion equation | ||||||||
DOI: | 10.1016/j.apm.2018.01.044 | ||||||||
ISSN: | 0307-904X | ||||||||
Pure ID: | 40841801 | ||||||||
Divisions: | Past > Institutes > Institute for Future Environments Past > QUT Faculties & Divisions > Science & Engineering Faculty Current > Research Centres > ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS) |
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Funding: | |||||||||
Copyright Owner: | Consult author(s) regarding copyright matters | ||||||||
Copyright Statement: | This work is covered by copyright. Unless the document is being made available under a Creative Commons Licence, you must assume that re-use is limited to personal use and that permission from the copyright owner must be obtained for all other uses. If the document is available under a Creative Commons License (or other specified license) then refer to the Licence for details of permitted re-use. It is a condition of access that users recognise and abide by the legal requirements associated with these rights. If you believe that this work infringes copyright please provide details by email to qut.copyright@qut.edu.au | ||||||||
Deposited On: | 19 Feb 2018 02:37 | ||||||||
Last Modified: | 26 Jul 2024 22:16 |
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