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Harmonic analysis of random fractional diffusion–wave equations

V.V Anh, N.N Leonenko
2003 Applied Mathematics and Computation  
We refer to Eq. (1.1) as a 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.  ... 
doi:10.1016/s0096-3003(02)00322-3 fatcat:grffaphhmvhilff3rxbh2nmvpi

Wave Propagation in Excitable Media Through Randomly Distributed Heterogeneities: Simulations and Comparison to the Effective Medium Theory

Sergio Alonso, Markus Bär, M. Belhaq, P. Lafitte, T. Lelièvre
2013 ESAIM: Proceedings and Surveys  
The resulting velocities 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.  ... 
doi:10.1051/proc/201339002 fatcat:d5fxwjh3ynfspco57an5aypdl4

Occurrence of anomalous diffusion and non-local response in highly-scattering acoustic periodic media

Salvatore Buonocore, Mihir Sen, Fabio Semperlotti
2019 New Journal of Physics  
We show that such behavior is well captured by a 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.  ... 
doi:10.1088/1367-2630/aafb7d fatcat:veklauzvyrf45cllrgssykwf7e

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.  ... 

Page 748 of Mathematical Reviews Vol. , Issue 82b [page]

1982 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.  ... 

Time-Fractional Heat Conduction in a Half-Line Domain due to Boundary Value of Temperature Varying Harmonically in Time

Yuriy Povstenko
2016 Mathematical Problems in Engineering  
The Dirichlet problem for the time-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  ... 
doi:10.1155/2016/8605056 fatcat:hvspcc2ofve4tfomwtvdxgs46a

The space-time fractional diffusion equation with Caputo derivatives

F. Huang, F. Liu
2005 Journal of Applied Mathematics and Computing  
We further explain the similarity property by discussing the scale-invariance 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.  ... 
doi:10.1007/bf02935797 fatcat:ezjrurxyufannoyn2on4amr7fy

Characteristics of Nonthermal Dupree Diffusion on Space-Charge Wave in a Kappa Distribution Plasma Column with Turbulent Diffusion

Myoung-Jae Lee, Young-Dae Jung
2020 Entropy  
The nonthermal 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.  ... 
doi:10.3390/e22020257 pmid:33286030 pmcid:PMC7516704 fatcat:sp42z55zsnazfntwt3m5uofyjq

Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics

Mingyu Xu, Wenchang Tan
2006 Science in China Series G  
From point 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  ... 
doi:10.1007/s11433-006-0257-2 fatcat:h4lp7rpvi5axbpcfmwz6zll4a4

Page 7278 of Mathematical Reviews Vol. , Issue 97M [page]

1997 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.  ... 

Fractional-time quantum dynamics

Alexander Iomin
2009 Physical Review E  
In particular, the quantum dynamics is considered in the framework 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.  ... 
doi:10.1103/physreve.80.022103 pmid:19792181 fatcat:fwtmnbuybber5g2olh3txkxoam

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  ... 

Fractional Laplacian, Levy stable distribution, and time-space models for linear and nonlinear frequency-dependent lossy media [article]

W. Chen, S. Holm
2002 arXiv   pre-print
In this study, we developed a linear integro-differential 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  ... 
arXiv:math-ph/0212075v1 fatcat:tgzvm4kwmfezrefkugkoajlocm

Human brain networks function in connectome-specific harmonic waves

Selen Atasoy, Isaac Donnelly, Joel Pearson
2016 Nature Communications  
In this new frequency-specific representation 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)  ... 
doi:10.1038/ncomms10340 pmid:26792267 pmcid:PMC4735826 fatcat:2ivttcegcbdmheyod4d22yj67u

Attenuation of long interfacial waves over a randomly rough seabed

2007 Journal of Fluid Mechanics  
For time-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.  ... 
doi:10.1017/s0022112007007112 fatcat:sy7aeo4cpzcbdhdmqbjxduulwy
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