Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation

Giusti, Andrea

doi:10.1063/1.5001555

Cited by 29 publications

(15 citation statements)

References 32 publications

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“…Besides, it is worth noting that fractional calculus plays a central role also in many other fields of science, see e.g. [8][9][10][11] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Scott-Blair models with time-varying viscosity

Colombaro

Garra

Giusti

et al. 2018

Applied Mathematics Letters

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In a recent paper, Zhou et al. [12] studied the time-dependent properties of Glass Fiber Reinforced Polymers composites by employing a new rheological model with a time-dependent viscosity coefficient. This rheological model is essentially based on a generalized Scott-Blair body with a time-dependent viscosity coefficient. Motivated by this study, in this note we suggest a different generalization of the Scott-Blair model based on the application of Caputo-type fractional derivatives of a function with respect to another function. This new mathematical approach can be useful in viscoelasticity and diffusion processes in order to model systems with time-dependent features. In this paper we also provide the general solution of the creep experiment for our improved Scott-Blair model together with some explicit examples and illuminating plots.

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“…Besides, it is worth noting that fractional calculus plays a central role also in many other fields of science, see e.g. [8][9][10][11] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Scott-Blair models with time-varying viscosity

Colombaro

Garra

Giusti

et al. 2018

Applied Mathematics Letters

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“…The telegraph equation has also been used to model various problems in wave propagation and signal analysis more generally [34,35], random walk theory [31], transport in heterogeneous porous media [32] and pulse transmission through a nerve axon [36,37]. It also has applications in other fields, in particular, it has been used to model transport processes in physical, biological, social, and ecological systems [34,[38][39][40][41][42][43]. Understanding the problem of critical size in those wide-ranging specific contexts is likely to result in exciting new research.…”

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confidence: 99%

Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

Alharbi

Petrovskii

2018

Mathematics

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A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction-telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain's critical size coincides with the critical size of the corresponding reaction-diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction-telegraph equation is larger than the critical domain size of the reaction-diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions.

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“…There is a wide literature about the applications and generalizations of the telegraph process, we refer to the recent monograph [21] for a complete review about this topic. We also observe that the telegraph equation, whose origin comes back to the classical equations of electromagnetism, has been also suggested by Davydov, Cattaneo and Vernotte as an alternative to the classical heat equation for diffusion processes with finite velocity of propagation, overcoming the so-called paradox of the infinite velocity of heat propagation (we refer to the classical review [18] and [15] about this topic).…”

Section: Non-homogeneous Telegraph Process With Time-varying Parametersmentioning

confidence: 65%

“…The space-fractional derivative appearing in (15) is the Riesz derivative [20] ∂ 2α f ∂|x| 2α = − 1 2 cos απ…”

Section: The Space-fractional Telegraph Equation With Time-varying Comentioning

confidence: 99%

“…In Section 3 we generalize the above results to the case of the space-fractional telegraph equation with time-dependent velocity and rate. Indeed, in the recent literature space and time-fractional generalizations of the telegraph equations have attracted the interest of different authors, see for example [6,10,11,15,25,26]. In [10] the relationship between space-time fractional telegraph equations and time-changed processes has been discussed.…”

Section: Introductionmentioning

confidence: 99%

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Probability distributions for the run-and-tumble models with variable speed and tumbling rate

Angelani¹,

Garra²

2018

Modern Stochastics: Theory and Applications

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In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity c(t) and changing direction at instants distributed according to a non-stationary Poisson distribution with rate λ(t). We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applications to run-and-tumble models.

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Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation

Cited by 29 publications

References 32 publications

Scott-Blair models with time-varying viscosity

Scott-Blair models with time-varying viscosity

Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

Probability distributions for the run-and-tumble models with variable speed and tumbling rate

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