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Harmonic analysis of random fractional diffusion–wave equations

2003
*
Applied Mathematics and Computation
*

We refer to Eq. (1.1) as a

doi:10.1016/s0096-3003(02)00322-3
fatcat:grffaphhmvhilff3rxbh2nmvpi
*fractional**diffusion**equation*when 0 < β ≤ 1 and as a*fractional**wave**equation*when 1 < β ≤ 2. ... To the best*of*our knowledge, the Green function (2.5) or (2.7)*of*the*fractional**diffusion*-*wave**equation*(1.1) is the most general in the existing literature. ...##
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Wave Propagation in Excitable Media Through Randomly Distributed Heterogeneities: Simulations and Comparison to the Effective Medium Theory

2013
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ESAIM: Proceedings and Surveys
*

The resulting velocities

doi:10.1051/proc/201339002
fatcat:d5fxwjh3ynfspco57an5aypdl4
*of*the traveling*waves*found by numerical simulations*of*the*random*media are compared with the predictions*of*the effective medium theory. ... The propagation*of*traveling*waves*in excitable media with randomly distributed*diffusion*coefficient is studied. ... which corresponds to the weighted*harmonic*mean*of*the*diffuse*coefficients. ...##
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Occurrence of anomalous diffusion and non-local response in highly-scattering acoustic periodic media

2019
*
New Journal of Physics
*

We show that such behavior is well captured by a

doi:10.1088/1367-2630/aafb7d
fatcat:veklauzvyrf45cllrgssykwf7e
*fractional**diffusive*transport model whose order can be obtained by the*analysis**of*the heavy tails. ... Previous studies had correlated the occurrence*of*anomalous*diffusion*to either the*random*properties*of*the scattering medium or to the presence*of*localized disorder. ... distribution, the governing*equation*is a generalization to the*fractional*order*of*the classical*diffusion**equation*. ...##
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Page 10164 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

method for simulating a

*random*walk governed by a distributed order time*fractional**diffusion**equation*. ...*random*variable whose proba- bility density obeys a distributed order time*fractional**diffusion**equation*. ...##
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Page 748 of Mathematical Reviews Vol. , Issue 82b
[page]

1982
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Mathematical Reviews
*

The authors show that phenomenological modelling

*of*a damped*harmonic*oscillator with*random*frequency fluctuations by a Stratonovié stochastic differential*equation*results in the same*diffusion*process ... Stochastic differential*equations*and*wave*propagation in*random*media constitute two broad fields*of*research developed in the last decades. ...##
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Time-Fractional Heat Conduction in a Half-Line Domain due to Boundary Value of Temperature Varying Harmonically in Time

2016
*
Mathematical Problems in Engineering
*

The Dirichlet problem for the time-

doi:10.1155/2016/8605056
fatcat:hvspcc2ofve4tfomwtvdxgs46a
*fractional*heat conduction*equation*in a half-line domain is studied with the boundary value*of*temperature varying*harmonically*in time. ... Different formulations*of*the considered problem for the classical heat conduction*equation*and for the*wave**equation*describing ballistic heat conduction are discussed. ... For example, we can assume the Dirichlet boundary condition varying*harmonically*in time: ( , ) = 0 = 0. (5) Similar*analysis*can be also carried out in the case*of*the boundary value*of*heat flux varying ...##
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The space-time fractional diffusion equation with Caputo derivatives

2005
*
Journal of Applied Mathematics and Computing
*

We further explain the similarity property by discussing the scale-invariance

doi:10.1007/bf02935797
fatcat:ezjrurxyufannoyn2on4amr7fy
*of*the space-time*fractional**diffusion**equation*. ... We deal with the Cauchy problem for the space-time*fractional**diffusion**equation*, which is obtained from standard*diffusion**equation*by replacing the second-order space derivative with a Caputo (or Riemann-Liouville ... Acknowledgements: This research has been supported by the National Natural Science Foundation*of*China 10271098. ...##
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Characteristics of Nonthermal Dupree Diffusion on Space-Charge Wave in a Kappa Distribution Plasma Column with Turbulent Diffusion

2020
*
Entropy
*

The nonthermal

doi:10.3390/e22020257
pmid:33286030
pmcid:PMC7516704
fatcat:sp42z55zsnazfntwt3m5uofyjq
*diffusion*effects on the dispersion*equations**of*ion-acoustic space-charge*wave*(SCW) in a nonthermal plasma column composed*of*nonthermal turbulent electrons and cold ions are investigated ... based on the*analysis**of*normal modes and the separation*of*variables. ... mathematical complexity*of*complex*analysis*. ...##
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Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics

2006
*
Science in China Series G
*

From point

doi:10.1007/s11433-006-0257-2
fatcat:h4lp7rpvi5axbpcfmwz6zll4a4
*of*view*of*physics, especially*of*mechanics, we briefly introduce*fractional*operators (with emphasis on*fractional*calculus and*fractional*differential*equations*) used for describing intermediate ... The applications*of*FO to the theory*of**random*walk and anomalous*diffusion*Classical (integer order) partial differential*equation**of**diffusion*and*wave*has been extended to the*equation*with*fractional*... After the*fractional*constitutive relation and the generalized concept*of**random*walk have been established, two different points*of*view gave out the unified form*of**fractional**diffusion**equation*[87 ...##
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Page 7278 of Mathematical Reviews Vol. , Issue 97M
[page]

1997
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Mathematical Reviews
*

Goncharuk,

*Fractional*step method for a class*of*quasilinear stochastic dif- ferential*equations**of*parabolic type (221-235); B. ... Turicchia, Nonlinear simulation*of*the electrocortical activity (551-553); A. R. Osborne, Periodic inverse scattering transform*analysis**of*Adriatic Sea surface*waves*(554-562); J.-F. Paque- rot, M. ...##
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Fractional-time quantum dynamics

2009
*
Physical Review E
*

In particular, the quantum dynamics is considered in the framework

doi:10.1103/physreve.80.022103
pmid:19792181
fatcat:fwtmnbuybber5g2olh3txkxoam
*of*the*fractional*time Schrödinger*equation*(SE), which differs from the standard SE by the*fractional*time derivative: / t→^α/ t^α. ... Application*of*the*fractional*calculus to quantum processes is presented. ... In the quantum case a "*random*entrapping" is due to the reversible leakage probability*of*the*wave*function. ...##
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Page 6762 of Mathematical Reviews Vol. , Issue 2001I
[page]

2001
*
Mathematical Reviews
*

Summary: “We study transport in

*random*unidirectional*wave*-like velocity fields with nonlinear dispersion relations. ... Juhani Pitkdranta (FIN-HUT-IM; Espoo) 20011:82060 82C70 76F25 Fannjiang, Albert (1-CAD; Davis, CA); Komorowski, Tomasz*Fractional*Brownian motions and enhanced*diffusion*in a unidirectional*wave*-like turbulence ...##
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Fractional Laplacian, Levy stable distribution, and time-space models for linear and nonlinear frequency-dependent lossy media
[article]

2002
*
arXiv
*
pre-print

In this study, we developed a linear integro-differential

arXiv:math-ph/0212075v1
fatcat:tgzvm4kwmfezrefkugkoajlocm
*equation**wave*model for the anomalous attenuation by using the space*fractional*Laplacian operation, and the strategy is then extended to the nonlinear ... A new definition*of*the*fractional*Laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hyper-singularity in the conventional*fractional*... In what follows, the new definition*of*the*fractional*Laplacian is introduced first in section I, followed by a presentation and*analysis**of*the linear*fractional*Laplacian thermoviscous models*of**wave*...##
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Human brain networks function in connectome-specific harmonic waves

2016
*
Nature Communications
*

In this new frequency-specific representation

doi:10.1038/ncomms10340
pmid:26792267
pmcid:PMC4735826
fatcat:2ivttcegcbdmheyod4d22yj67u
*of*cortical activity, that we call 'connectome*harmonics*', oscillatory networks*of*the human brain at rest match*harmonic**wave*patterns*of*certain frequencies ... We demonstrate a neural mechanism behind the self-organization*of*connectome*harmonics*with a continuous neural field model*of*excitatory-inhibitory interactions on the connectome. ... Data used in the preparation*of*this work were obtained and made available by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) ...##
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Attenuation of long interfacial waves over a randomly rough seabed

2007
*
Journal of Fluid Mechanics
*

For time-

doi:10.1017/s0022112007007112
fatcat:sy7aeo4cpzcbdhdmqbjxduulwy
*harmonic**waves*, modecoupling*equations*are derived and examined for the competition between*diffusion*by*random*scattering, steepening by nonlinearity and frequency dispersion for a broad range ... and multiplescale*analysis*. ... Hence*randomness*gives rise to*diffusion*. The last term alters the dispersiveness. Thus, the governing integro-differential*equation*combines the features*of*KdV and Burgers*equations*. ...
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