Applications of Fractional Calculus in Physics

Hilfer, R.

doi:10.1142/3779

Cited by 4,604 publications

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“…where D 0 is the generalized diffusion coefficient (note that the units of D 0 are mm 2b /second a ), a (0 < a 1) is a fractional order derivative with respect to time, and b (0 < b 1), a fractional order derivative with respect to space (23). With this formalism, a fractional order generalization of the Bloch-Torrey equation can be written as …”

Section: Theorymentioning

confidence: 99%

“…Inspired by this possibility, we recently examined the connection between fractional order dynamics and diffusion by solving the Bloch-Torrey equation using fractional order calculus (21,22). Our previous studies showed that the stretched exponential model follows from a fundamental extension of the Bloch-Torrey equation through application of the operators of fractional calculus (FC) (23). More importantly, the model based on FC yields a new set of parameters to describe anomalous diffusion: diffusion coefficient D, fractional order derivative in space b, and a spatial parameter m (in units of micrometers).…”

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confidence: 99%

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

et al. 2010

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It is well known that diffusion-induced MR signal loss deviates from monoexponential decay, particularly at high b-values (e.g., >1500 sec/mm 2 for human brain tissues). A number of models have been developed to describe this anomalous diffusion behavior and relate the diffusion measurements to tissue structures. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (Magin et al., J Magn Reson 2008;190:255-270; Zhou et al., Proc Int'l Soc Magn Reson Med 2008). Using a spatial Laplacian $ 2b , this model yields a new set of parameters to describe anomalous diffusion: diffusion coefficient D, fractional order derivative in space b, and a spatial parameter m (in units of mm). In this study, we demonstrate that the fractional calculus model can be successfully applied to analyzing diffusion images of healthy human brain tissues in vivo. Five human volunteers were scanned on a commercial 3-T scanner using a customized single-shot echo-planar imaging diffusion sequence with 15 b values ranging from 0 to 4700 sec/mm 2 . The set of images was analyzed using the fractional calculus model, producing spatially resolved maps of D, b, and m. The b and m maps showed notable contrast between white and gray matter. The contrast has been attributed to the varying degree of complexity of the underlying tissue structures and microenvironment. Although the biophysical basis of b and m remains elusive, the potential utility of these parameters to characterize the environment for molecular diffusion, as a complement to apparent diffusion coefficient, may lead to a new way to investigate tissue structural changes in disease progression, intervention, and regression. Magn Reson Med 63:562-569,

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Section: Theorymentioning

confidence: 99%

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confidence: 99%

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

et al. 2010

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“…Caputo's definition, which is a modification of the Riemann-Liouville definition, has the advantage of dealing properly with initial value problems. The following Remark addresses some of the main properties of the fractional derivatives and integrals (see [12,[36][37][38][39]). …”

Section: Appendixmentioning

confidence: 99%

“…Fractional-order (or free-order) differential models have been successfully applied to system biology, physics, chemistry, and biochemistry, hydrology, medicine, and finance (see, e.g., [6][7][8][9][10][11][12] and the references therein). In many cases, they are more contestant with the real phenomena than the integer-order models, because the fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Salmonella Infection

Rihan¹

2017

Current Topics in Salmonella and Salmonellosis

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In this chapter, we propose a mathematical epidemic model, with integer and fractional order to describe the dynamics of Salmonella infection in animal herds. We investigate the qualitative behaviors of such model and find the conditions that guarantee the asymptotic stability of disease-free and endemic steady states. To assess the severity of the outbreak, as well as the strength of the medical and/or behavioral interventions necessary for control, we estimate basic reproduction number R 0 . This threshold parameter specifies the average number of secondary infections caused by one infected individual during his/her entire infectious period at the start of an outbreak. We also provide an unconditionally stable implicit scheme for the fractional-order epidemic model. The theoretical and computational results give insight into the modelers and infectious disease specialists.

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“…It has been demonstrated that these mathematical operators are useful mathematical tools in various disciplines of science and engineering including Electromagnetic theory [2][3][4][5]. Fractionalization of ordinary derivative and integral operators motivated the researchers in electromagnetics to explore the potential of fractionalization of other operators in the field [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Fields in Fractional Parallel Plate D'b', Db' and d'B Waveguides

Gulistan¹,

Majeed²,

Syed³

2013

PIER B

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Abstract-D B , DB and D B boundary conditions are used to investigate the resulting field patterns inside a parallel plate waveguide. The D B boundary conditions are accomodated by assigning the behavior of perfect magnetic conductor (PMC) for transverse electric mode (TE) and that of perfect electric conductor (PEC) for transverse magnetic (TM) mode, to the boundary, respectively. Likewise, DB boundary conditions are incorporated by assuming the behavior of boundary as PMC for both the TE mode and TM mode. Finally D B boundary conditions are realized by assigning PEC characteristic to the boundary for both TE and TM modes. A general wave propagating inside the parallel plate waveguide is assumed and decomposed into TE and TM modes for the purpose of analysis. Fractional curl operator has been used to study the fractional parallel plate D B , DB and D B waveguides for different values of fractional parameter α. Behavior of the field patterns in the waveguides are studied with respect to the fractional parameter α describing the order of the fractionalization.

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Applications of Fractional Calculus in Physics

Cited by 4,604 publications

References 0 publications

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

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

Dynamics of Salmonella Infection

Fields in Fractional Parallel Plate D'b', Db' and d'B Waveguides

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