Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Jiang, H., Liu, F., Turner, I., & Burrage, K. (2012) Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. Journal of Mathematical Analysis and Applications, 389(2), pp. 1117-1127.

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Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

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ID Code: 51507
Item Type: Journal Article
Refereed: Yes
Keywords: Analytical solution, Fractional Laplacian operator, Multi-term time-space Caputo-Riesz fractional advection-diffusion equations, Multivariate Mittag-Leffler function, Nonhomogeneous initial-boundary-value problem
DOI: 10.1016/j.jmaa.2011.12.055
ISSN: 0022-247X
Divisions: Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Copyright Owner: Copyright 2012 Elsevier
Copyright Statement: This is the author’s version of a work that was accepted for publication in the Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Mathematical Analysis and Applications, Volume 389, Issue 2, (2012), Pages 1117–1127. DOI: 10.1016/j.jmaa.2011.12.055
Deposited On: 10 Jul 2012 05:23
Last Modified: 15 Sep 2013 01:32

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