Subordination Approach to Space-Time Fractional Diffusion

Bazhlekova, Emilia; Bazhlekov, Ivan B.

doi:10.3390/math7050415

Cited by 12 publications

(18 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where η(t) = e −rt and η(s) = 1 s+r . This equation can be solved by using a subordination approach [1,59,[61][62][63]. Equation (21) in Laplace space reads…”

Section: One-dimensional Diffusion-advection Equation With Stochastic Resettingmentioning

confidence: 99%

Diffusion–Advection Equations on a Comb: Resetting and Random Search

et al. 2021

View full text Add to dashboard Cite

This review addresses issues of various drift–diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both a Brownian diffusion search with drift and an inhomogeneous advection search on the comb structures are analyzed. The analytical results are verified by numerical simulations in terms of coupled Langevin equations for the comb structure. The subordination approach is one of the main technical methods used here, and we demonstrated how it can be effective in the study of various random search problems with and without resetting.

show abstract

“…where η(t) = e −rt and η(s) = 1 s+r . This equation can be solved by using a subordination approach [1,59,[61][62][63]. Equation (21) in Laplace space reads…”

Section: One-dimensional Diffusion-advection Equation With Stochastic Resettingmentioning

confidence: 99%

Diffusion–Advection Equations on a Comb: Resetting and Random Search

et al. 2021

View full text Add to dashboard Cite

show abstract

“…Moreover, we refer to recent papers devoted to the analytical and theoretical studies of the time-fractional diffusion equation [30] , [31] , [32] , [33] .…”

Section: Introductionmentioning

confidence: 99%

Computation of solution to fractional order partial reaction diffusion equations

Gul

Alrabaiah

Ali

et al. 2020

Journal of Advanced Research

View full text Add to dashboard Cite

show abstract

“…Although (31) cannot be written in a simple closed form, it can be written in closed form using the Fox H-function [19,63]. In two-dimensions, the closed form of ( 25) is given by e.g., [46][47][48],…”

Section: The Quasi-diffusion Propagatormentioning

confidence: 99%

“…The quasi-diffusion propagator has also been investigated using subordination principles [45][46][47] (see [5] for a general description of subordination processes in the CTRW model). Subordination principles in stochastic processes involve the definition of a stochastic process in time (the subordinating function) that is within another stochastic process (the subordinated stochastic process).…”

Section: The Quasi-diffusion Propagatormentioning

confidence: 99%

“…Here we consider the mathematics of quasi-diffusion and quasi-diffusion imaging. The fundamental solution of the space-time diffusion-wave equation within the CTRW diffusion model has been extensively studied [43][44][45][46][47], as have the properties of the quasidiffusion model [47][48][49][50]. In particular, quasi-diffusion is referred to as α-fractional diffusion by [48,49] who suggest this special case of the CTRW diffusion model represents a natural fractionalisation of the diffusion process [49].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

et al. 2021

View full text Add to dashboard Cite

Quasi-diffusion imaging (QDI) is a novel quantitative diffusion magnetic resonance imaging (dMRI) technique that enables high quality tissue microstructural imaging in a clinically feasible acquisition time. QDI is derived from a special case of the continuous time random walk (CTRW) model of diffusion dynamics and assumes water diffusion is locally Gaussian within tissue microstructure. By assuming a Gaussian scaling relationship between temporal () and spatial () fractional exponents, the dMRI signal attenuation is expressed according to a diffusion coefficient, (in mm2 s−1), and a fractional exponent, . Here we investigate the mathematical properties of the QDI signal and its interpretation within the quasi-diffusion model. Firstly, the QDI equation is derived and its power law behaviour described. Secondly, we derive a probability distribution of underlying Fickian diffusion coefficients via the inverse Laplace transform. We then describe the functional form of the quasi-diffusion propagator, and apply this to dMRI of the human brain to perform mean apparent propagator imaging. QDI is currently unique in tissue microstructural imaging as it provides a simple form for the inverse Laplace transform and diffusion propagator directly from its representation of the dMRI signal. This study shows the potential of QDI as a promising new model-based dMRI technique with significant scope for further development.

show abstract

Subordination Approach to Space-Time Fractional Diffusion

Cited by 12 publications

References 34 publications

Diffusion–Advection Equations on a Comb: Resetting and Random Search

Diffusion–Advection Equations on a Comb: Resetting and Random Search

Computation of solution to fractional order partial reaction diffusion equations

The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

Contact Info

Product

Resources

About