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Analytical and Numerical Solutions of the Space and Time Fractional Bloch-Torrey Equation

[+] Author Affiliations
Q. Yu, F. Liu, I. Turner

Queensland University of Technology, Brisbane, QLD, Australia

K. Burrage

Queensland University of Technology, Brisbane, QLD, Australia; Oxford University, Oxford, UK

Paper No. DETC2011-47613, pp. 201-210; 10 pages
doi:10.1115/DETC2011-47613
From:
  • ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 3: 2011 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications, Parts A and B
  • Washington, DC, USA, August 28–31, 2011
  • Conference Sponsors: Design Engineering Division and Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5480-8
  • Copyright © 2011 by ASME

abstract

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et at, J. Magnetic Resonance, 190 (2008) 255–270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Copyright © 2011 by ASME
Topics: Spacetime , Equations

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