scholarly journals Studies of anomalous diffusion in the human brain using fractional order calculus

2010 ◽  
Vol 63 (3) ◽  
pp. 562-569 ◽  
Author(s):  
Xiaohong Joe Zhou ◽  
Qing Gao ◽  
Osama Abdullah ◽  
Richard L. Magin
Keyword(s):  
Human Brain ◽  
Fractional Order ◽  
Anomalous Diffusion ◽  
Fractional Order Calculus

scholarly journals Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120150 ◽  
Author(s):  
Qiang Yu ◽  
Fawang Liu ◽  
Ian Turner ◽  
Kevin Burrage
Keyword(s):  
Numerical Method ◽  
Fractional Order ◽  
Anomalous Diffusion ◽  
Fractional Laplacian ◽  
Dynamic Models ◽  
Stability And Convergence ◽  
Time And Space ◽  
Fractional Order Calculus ◽  
Space And Time ◽  
Implicit Numerical Method

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 magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is 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. 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) in both fractional Laplacian and Riesz derivative form 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 in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated. We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

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

2011 ◽  
Author(s):  
Q. Yu ◽  
F. Liu ◽  
I. Turner ◽  
K. Burrage
Keyword(s):  
Numerical Method ◽  
Fractional Order ◽  
Anomalous Diffusion ◽  
Dynamic Models ◽  
Numerical Solutions ◽  
Stability And Convergence ◽  
Time And Space ◽  
Fractional Order Calculus ◽  
Space And Time ◽  
Implicit Numerical Method

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.

Predicting the aggressiveness of peripheral zone prostate cancer using a fractional order calculus diffusion model

2021 ◽  
Vol 143 ◽  
pp. 109913
Author(s):  
Zhihua Li ◽  
Guangyu Dan ◽  
Vikram Tammana ◽  
Scott Johnson ◽  
Zheng Zhong ◽  
...  
Keyword(s):  
Prostate Cancer ◽  
Diffusion Model ◽  
Fractional Order ◽  
Peripheral Zone ◽  
Fractional Order Calculus

scholarly journals Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

Formulation of Fractional Derivative-Based De-Hazing Algorithm and Implementation on Mobile-Embedded Devices

2019 ◽  
Vol 19 (01) ◽  
pp. 1950003
Author(s):  
Uche A. Nnolim
Keyword(s):  
Image Processing ◽  
Fractional Derivative ◽  
Mobile Devices ◽  
Fractional Order ◽  
Spatial Filter ◽  
Embedded Devices ◽  
Processing Application ◽  
Fractional Order Calculus ◽  
Image Processing Application ◽  
Logarithmic Image Processing

This paper presents the modification of a previously developed algorithm using fractional order calculus and its implementation on mobile-embedded devices such as smartphones. The system performs enhancement on three categories of images such as those exhibiting uneven illumination, faded features/colors and hazy appearance. The key contributions include the simplified scheme for illumination correction, contrast enhancement and de-hazing using fractional derivative-based spatial filter kernels. These are achieved without resorting to logarithmic image processing, histogram-based statistics and complex de-hazing techniques employed by conventional algorithms. The simplified structure enables ease of implementation of the algorithm on mobile devices as an image processing application. Results indicate that the fractional order version of the algorithm yields good results relative to the integer order version and other algorithms from the literature.

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