Browsing by Author "Magin, Richard"

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Citation - WoS: 108
Citation - Scopus: 123
Fractional Bloch Equation With Delay
(Pergamon-elsevier Science Ltd, 2011) Daftardar-Gejji, Varsha; Baleanu, Dumitru; Magin, Richard; Bhalekar, Sachin
In this paper we investigate a fractional generalization of the Bloch equation that includes both fractional derivatives and time delays. The appearance of the fractional derivative on the left side of the Bloch equation encodes a degree of system memory in the dynamic model for magnetization. The introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation. The analysis of this system shows different stability behavior for the T-1 and the T-2 relaxation processes. The T-1 decay is stable for the range of delays tested (1-100 mu s), while the T-2 relaxation in this model exhibited a critical delay (typically 6 mu s) above which the system was unstable. Delays are expected to appear in NMR systems, in both the system model and in the signal excitation and detection processes. Therefore, by including both the fractional derivative and finite time delays in the Bloch equation, we believe that we have established a more complete and more realistic model for NMR resonance and relaxation. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 35
Citation - Scopus: 44
Generalized Fractional Order Bloch Equation With Extended Delay
(World Scientific Publ Co Pte Ltd, 2012) Daftardar-Gejji, Varsha; Baleanu, Dumitru; Magin, Richard; Bhalekar, Sachin
The fundamental description of relaxation (T-1 and T-2) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time-and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T-1 and T-2 relaxation. The T-1 decay is stable for the range of delays tested (1 mu sec to 200 mu sec), while the T-2 relaxation in this extended model exhibits a critical delay (typically 100 mu sec to 200 mu sec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation.
Citation - WoS: 147
Citation - Scopus: 168
Solving the Fractional Order Bloch Equation
(Wiley-hindawi, 2009) Feng, Xu; Baleanu, Dumitru; Magin, Richard
Nuclear magnetic resonance (NMR) is a physical phenomenon widely used in chemistry, medicine, and engineering to study complex materials. NMR is governed by the Bloch equation, which relates a macroscopic model of magnetization to applied radjofrequency, gradient and static magnetic fields. Simple models of materials are well described by the classical first order dynamics of precession and relaxation inherent in the vector form of the Bloch equation. Fractional order generalization of the Bloch equation presents an opportunity to extend its use to describe a wider range of experimental situations involving heterogeneous, porous, or composite materials. Here we describe the generalization of the Bloch equation in terms of Caputo fractional derivatives of order alpha (0 < alpha < 1) for a single spin system in a static magnetic field at resonance. The results are expressed in terms of the Mittag-Leffler function-a generalized exponential function that converges to the classical case when alpha = 1. (C) 2008 Wiley Periodicals, Inc. Concepts Magn Reson Part A 34A: 16-23, 2009.
Citation - WoS: 47
Citation - Scopus: 55
Transient Chaos in Fractional Bloch Equations
(Pergamon-elsevier Science Ltd, 2012) Daftardar-Gejji, Varsha; Baleanu, Dumitru; Magin, Richard; Bhalekar, Sachin
The Bloch equation provides the fundamental description of nuclear magnetic resonance (NMR) and relaxation (T-1 and T-2). This equation is the basis for both NMR spectroscopy and magnetic resonance imaging (MRI). The fractional-order Bloch equation is a generalization of the integer-order equation that interrelates the precession of the x, y and z components of magnetization with time- and space-dependent relaxation. In this paper we examine transient chaos in a non-linear version of the Bloch equation that includes both fractional derivatives and a model of radiation damping. Recent studies of spin turbulence in the integer-order Bloch equation suggest that perturbations of the magnetization may involve a fading power law form of system memory, which is concisely embedded in the order of the fractional derivative. Numerical analysis of this system shows different patterns in the stability behavior for alpha near 1.00. In general, when alpha is near 1.00, the system is chaotic, while for 0.98 >= alpha >= 0.94, the system shows transient chaos. As the value of alpha decreases further, the duration of the transient chaos diminishes and periodic sinusoidal oscillations emerge. These results are consistent with studies of the stability of both the integer and the fractional-order Bloch equation. They provide a more complete model of the dynamic behavior of the NMR system when non-linear feedback of magnetization via radiation damping is present. (C) 2012 Elsevier Ltd. All rights reserved.