An introduction to fractional diffusion

Paper

Henry, B. I., Langlands, Trevor and Straka, P.. 2010. "An introduction to fractional diffusion." Dewar, Robert L. and Detering, Frank (ed.) 22nd Canberra International Physics Summer School. Canberra, Australia 08 - 19 Dec 2008 Singapore. https://doi.org/10.1142/9789814277327_0002
Paper/Presentation Title

An introduction to fractional diffusion

Presentation TypePaper
AuthorsHenry, B. I. (Author), Langlands, Trevor (Author) and Straka, P. (Author)
EditorsDewar, Robert L. and Detering, Frank
Journal or Proceedings TitleComplex Physical, Biophysical and Econophysical Systems
Journal Citation9, pp. 37-89
Number of Pages43
Year2010
Place of PublicationSingapore
ISBN9789814277327
Digital Object Identifier (DOI)https://doi.org/10.1142/9789814277327_0002
Web Address (URL) of Paperhttp://site.ebrary.com/lib/unisouthernqld/Doc?id= 1 0422486&ppg=48
Conference/Event22nd Canberra International Physics Summer School
Event Details
22nd Canberra International Physics Summer School
Event Date
08 to end of 19 Dec 2008
Event Location
Canberra, Australia
Abstract

The mathematical description of diffusion has a long history with many different formulations including phenomenological models based on conservation of mass and constitutive laws; probabilistic models based on random walks and central limit theorems; microscopic stochastic models based on Brownian motion and Langevin equations; and mesoscopic stochastic models based on master equations and Fokker-Planck equations. A fundamental result common to the different approaches is that the mean square displacement of a diffusing particle scales linearly with time. However there have been numerous experimental measurements in which the mean square displacement of diffusing particles scales as a fractional order power law in time. In recent years a great deal of progress has been made in extending the different models for diffusion to incorporate this fractional diffusion. The tools of fractional calculus have proven very useful in these developments, linking together fractional constitutive laws, continuous time random walks, fractional Langevin equations and fractional Brownian motions. These notes provide a tutorial style overview of standard and fractional diffusion processes.

Keywordsfractional calculus; anomalous subdiffusion; random walks; superdiffusion
ANZSRC Field of Research 2020490510. Stochastic analysis and modelling
519901. Complex physical systems
490410. Partial differential equations
Public Notes

Chapter 2. Electronic copy held USQ Library
For more information contact Dr Trevor Langlands Trevor.Langlands@usq.edu.au

Byline AffiliationsUniversity of New South Wales
SeriesWorld Scientfic Lecture Notes in Complex Systems
Book TitleComplex Physical, Biophysical and Econphysical Systems: Proceedings of the 22nd Canberra International Physics Summer School
Permalink -

https://research.usq.edu.au/item/9zxq3/an-introduction-to-fractional-diffusion

Log in to edit

Download files

Submitted Version
Henry_Langlands_Straka_2010_AV.pdf
File access level: Anyone

Other Documentation
Documentation.pdf
File access level: Anyone

  • 2108
    total views
  • 6690
    total downloads
  • 4
    views this month
  • 72
    downloads this month

Export as

Related outputs

Numerical investigation of two models of nonlinear fractional reaction subdiffusion equations
Osman, Sheelan and Langlands, Trevor. 2022. "Numerical investigation of two models of nonlinear fractional reaction subdiffusion equations." Fractional Calculus and Applied Analysis. 25 (6), pp. 2166-2192. https://doi.org/10.1007/s13540-022-00096-2
Connecting community online and through partnership: A reflective piece
Pickstone, Leigh, Sharma, Ekta, King, Rachel, Galligan, Linda and Langlands, Trevor. 2022. "Connecting community online and through partnership: A reflective piece." International Journal for Students as Partners. 6 (2), pp. 114-120. https://doi.org/10.15173/ijsap.v6i2.4825
Modern artificial intelligence model development for undergraduate student performance prediction: an investigation on engineering mathematics courses
Deo, Ravinesh C., Yaseen, Zaher Mundher, Al-Ansari, Nadhir, Nguyen-Huy, Thong, Langlands, Trevor and Galligan, Linda. 2020. "Modern artificial intelligence model development for undergraduate student performance prediction: an investigation on engineering mathematics courses." IEEE Access. 8, pp. 136697-136724. https://doi.org/10.1109/ACCESS.2020.3010938
An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations
Osman, S. A and Langlands, T. A. M.. 2019. "An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations." Applied Mathematics and Computation. 348, pp. 609-626. https://doi.org/10.1016/j.amc.2018.12.015
From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations
Angstmann, C. N., Donnelly, I. C., Henry, B. I., Jacobs, B. A., Langlands, T. A. M. and Nichols, J. A.. 2016. "From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations." Journal of Computational Physics. 307, pp. 508-534. https://doi.org/10.1016/j.jcp.2015.11.053
Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions
Langlands, Trevor, Henry, B. I. and Wearne, S. L.. 2009. "Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions." Journal of Mathematical Biology. 59 (6), pp. 761-808. https://doi.org/10.1007/s00285-009-0251-1
A mathematical model for the proliferation, accumulation and spread of pathogenic proteins along neuronal pathways with locally anomalous trapping
Angstmann, C. N., Donnelly, I. C., Henry, B. I. and Langlands, T. A. M.. 2016. "A mathematical model for the proliferation, accumulation and spread of pathogenic proteins along neuronal pathways with locally anomalous trapping." Mathematical Modelling of Natural Phenomena (MMNP). 11 (3), pp. 142-156. https://doi.org/10.1051/mmnp/20161139
Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations
Angstmann, C. N., Donnelly, I. C., Henry, B. I., Langlands, T. A. M. and Straka, P.. 2015. "Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations." SIAM Journal on Applied Mathematics. 75 (4), pp. 1445-1468. https://doi.org/10.1137/15M1011299
Continuous-time random walks on networks with vertex- and time-dependent forcing
Angstmann, C. N., Donnelly, I. C., Henry, B. I. and Langlands, T. A. M.. 2013. "Continuous-time random walks on networks with vertex- and time-dependent forcing." Physical Review E. 88 (2). https://doi.org/10.1103/PhysRevE.88.022811
Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions
Langlands, T. A. M., Henry, B. I. and Wearne, S. L.. 2011. "Fractional cable equation models for anomalous electrodiffusion in nerve cells: finite domain solutions." SIAM Journal on Applied Mathematics. 71 (4), pp. 1168-1203. https://doi.org/10.1137/090775920
Fractional diffusion in force fields, fractional electro-diffusion and fractional chemotaxis diffusion
Langlands, Trevor, Henry, Bruce and Straka, Peter. 2010. "Fractional diffusion in force fields, fractional electro-diffusion and fractional chemotaxis diffusion." Henry, Bruce and Roberts, John (ed.) Dynamics Days Asia Pacific 6 Conference (DDAP6). Sydney, Australia 12 - 14 Jul 2010 Sydney, Australia.
Anomalous subdiffusion with multispecies linear reaction dynamics
Langlands, Trevor, Henry, B. I. and Wearne, S. L.. 2008. "Anomalous subdiffusion with multispecies linear reaction dynamics." Physical Review B: Covering condensed matter and materials physics. 77, pp. 1-9. https://doi.org/10.1103/PhysRevE.77.021111
Fractional cable models for spiny neuronal dendrites
Henry, B. I., Langlands, Trevor and Wearne, S. L.. 2008. "Fractional cable models for spiny neuronal dendrites." Physical Review Letters. 100 (12), pp. 1-4. https://doi.org/10.1103/PhysRevLett.100.128103
Turing pattern formation with fractional diffusion and fractional reactions
Langlands, T. A. M., Henry, B. I. and Wearne, S. L.. 2006. "Turing pattern formation with fractional diffusion and fractional reactions." Journal of Physics: Condensed Matter. 19 (6), pp. 065115 -065134. https://doi.org/10.1088/0953-8984/19/6/065115
Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations
Henry, B. I., Langlands, Trevor and Wearne, S. L.. 2006. "Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations." Physical Review E. 74 (3), pp. 1-15. https://doi.org/10.1103/PhysRevE.74.031116
Solution of a modified fractional diffusion equation
Langlands, T. A. M.. 2006. "Solution of a modified fractional diffusion equation." Physica A: Statistical Mechanics and its Applications. 367, pp. 136-144. https://doi.org/10.1016/j.physa.2005.12.012
Fractional Fokker-Planck equations for subdiffusion with space-and time-dependent forces
Henry, B. I., Langlands, T. A. M. and Straka, P.. 2010. "Fractional Fokker-Planck equations for subdiffusion with space-and time-dependent forces." Physical Review Letters. 105 (17), pp. 17062-1-170602-4. https://doi.org/10.1103/PhysRevLett.105.170602
Fractional chemotaxis diffusion equations
Langlands, T. A. M. and Henry, B. I.. 2010. "Fractional chemotaxis diffusion equations." Physical Review E. 81 (5), pp. 1-12. https://doi.org/10.1103/PhysRevE.81.051102
Optimal targeting of hepatitis C virus treatment among injecting drug users to those not enrolled in methadone maintenance programs
Zeiler, Irmgard, Langlands, Trevor, Murray, John M. and Ritter, Alison. 2010. "Optimal targeting of hepatitis C virus treatment among injecting drug users to those not enrolled in methadone maintenance programs." Drug and Alcohol Dependence. 110 (3), pp. 228-233. https://doi.org/10.1016/j.drugalcdep.2010.03.006