Numerical simulation of the fractional Bloch equations

Yu Q., Liu F., Turner I. and Burrage K. (2014) Numerical simulation of the fractional Bloch equations. Journal of Computational and Applied Mathematics, 255 635-651. doi:10.1016/j.cam.2013.06.027

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Author Yu Q.
Liu F.
Turner I.
Burrage K.
Title Numerical simulation of the fractional Bloch equations
Journal name Journal of Computational and Applied Mathematics   Check publisher's open access policy
ISSN 0377-0427
Publication date 2014-01-01
Year available 2013
Sub-type Article (original research)
DOI 10.1016/j.cam.2013.06.027
Open Access Status File (Author Post-print)
Volume 255
Start page 635
End page 651
Total pages 17
Place of publication Amsterdam, Netherlands
Publisher Elsevier BV * North-Holland
Collection year 2015
Language eng
Subject 2605 Computational Mathematics
2604 Applied Mathematics
Formatted abstract
In physics and chemistry, specifically in NMR (nuclear magnetic resonance) or MRI (magnetic resonance imaging), or ESR (electron spin resonance) the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M=(Mx,My,Mz) as a function of time when relaxation times T1 and T2 are present. Recently, some fractional models have been proposed for the Bloch equations, however, effective numerical methods and supporting error analyses for the fractional Bloch equation (FBE) are still limited. In this paper, the time-fractional Bloch equations (TFBE) and the anomalous fractional Bloch equations (AFBE) are considered. Firstly, we derive an analytical solution for the TFBE with an initial condition. Secondly, we propose an effective predictor-corrector method (PCM) for the TFBE, and the error analysis for PCM is investigated. Furthermore, we derive an effective implicit numerical method (INM) for the anomalous fractional Bloch equations (AFBE), and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the AFBE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.
Keyword Anomalous fractional
Bloch equations
Convergence
Implicit numerical method
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: Non HERDC
Centre for Advanced Imaging Publications
 
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