Browsing by Author "Magin, Richard L."

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Citation - WoS: 387
Citation - Scopus: 426
Anomalous Diffusion Expressed Through Fractional Order Differential Operators in the Bloch-Torrey Equation
(Academic Press inc Elsevier Science, 2008) Abdullah, Osama; Baleanu, Dumitru; Zhou, Xiaohong Joe; Magin, Richard L.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on d iffusi on -weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (turn (2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology. (c) 2007 Elsevier Inc. All rights reserved.
Citation - WoS: 78
Chaos in the Fractional Order Nonlinear Bloch Equation With Delay
(Elsevier Science Bv, 2015) Magin, Richard L.; Bhalekar, Sachin; Daftardar-Gejji, Varsha; Baleanu, Dumitru; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (alpha) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, tau = 0, we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at alpha = 0.8548 with subsequent period doubling that leads to chaos at alpha = 0.9436. A periodic window is observed for the range 0.962 < alpha < 0.9858, with chaos arising again as a nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value alpha = 0.8532, and the transition from two to four cycles at alpha = 0.9259. With further increases in the fractional order, period doubling continues until at alpha = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at alpha = 0.8441, and alpha = 0.8635, respectively. However, the system exhibits chaos at much lower values of a (alpha - 0.8635). A periodic window is observed in the interval 0.897 < alpha < 0.9341, with chaos again appearing for larger values of a. In general, as the value of a decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a delay-dependent model that predicts instability in this non-linear fractional order system consistent with the experimental observations of spin turbulence. (C) 2015 Elsevier B.V. All rights reserved.