Fractional Reaction-Diffusion Equations

Saxena, R. K.; Mathai, A. M.; Haubold, H. J.

doi:10.1007/s10509-006-9189-6

Cited by 84 publications

(61 citation statements)

References 31 publications

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“…In recent years, there has been a great deal of interest in fractional reaction-diffusion (FRD) systems [26][27][28][29][30][31][32][33][34][35] which from one side exhibit self-organization phenomena and from the other side introduce a new parameter to these systems, which is a fractional derivative index, and it gives a greater degree of freedom for diversity of self-organization phenomena. At the same time, the process of analyzing such FRD systems is much more complicated from the analytical and numerical point of view.…”

Section: Introductionmentioning

confidence: 99%

On the solutions of time-fractional reaction–diffusion equations

Rida

Elsayed

Arafa

2010

Communications in Nonlinear Science and Numerical Simulation

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Section: Introductionmentioning

confidence: 99%

On the solutions of time-fractional reaction–diffusion equations

Rida

Elsayed

Arafa

2010

Communications in Nonlinear Science and Numerical Simulation

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“…The interpretation of the parameter μ in Eqs. (21) and (26) deserves some discussion. Eventhough the term "−μφ" looks like a "damping", its role is exactly the opposite.…”

Section: Tempered Fractional Diffusionmentioning

confidence: 99%

“…[8], where it was shown that the truncation of the Lévy flights due to tempering leads to a transient front acceleration after which the front asymptotically reaches a terminal speed. Other works on anomalous transport in front propagation include studies on: bistable reaction processes and anomalous diffusion caused by Lévy flights [25]; analytic solutions of fractional reaction-diffusion equations [21]; reaction-diffusion systems with bistable reaction terms and directional anomalous diffusion [15]; construction of reaction-sub-diffusion equations [22]; fractional reproduction-dispersal equations and heavy tail dispersal kernels [1]; role of fluctuations in reaction-super-diffusion dynamics [2]; non-Markovian random walks and sub-diffusion in reaction-diffusion systems [13]; exact super-diffusive front propagation solutions with piecewise linear reaction kinetics functions [24]; and front dynamics in two-species competition models driven by Lévy flights [14], among others. The recent work in Ref.…”

Section: Introductionmentioning

confidence: 99%

Front propagation in reaction-diffusion systems with anomalous diffusion

del‐Castillo‐Negrete

2014

Bol. Soc. Mat. Mex.

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A numerical study of the role of anomalous diffusion in front propagation in reaction-diffusion systems is presented. Three models of anomalous diffusion are considered: fractional diffusion, tempered fractional diffusion, and a model that combines fractional diffusion and regular diffusion. The reaction kinetics corresponds to a Fisher-Kolmogorov nonlinearity. The numerical method is based on a finite-difference operator splitting algorithm with an explicit Euler step for the time advance of the reaction kinetics, and a Crank-Nicholson semi-implicit time step for the transport operator. The anomalous diffusion operators are discretized using an upwind, flux-conserving, Grunwald-Letnikov finite-difference scheme applied to the regularized fractional derivatives. With fractional diffusion of order $\alpha$, fronts exhibit exponential acceleration, $a_L(t) \sim e^{\gamma t/\alpha}$, and develop algebraic decaying tails, $\phi \sim 1/x^{\alpha}$. In the case of tempered fractional diffusion, this phenomenology prevails in the intermediate asymptotic regime $\left(\chi t \right)^{1/\alpha} \ll x \ll 1/\lambda$, where $1/\lambda$ is the scale of the tempering. Outside this regime, i.e. for $x > 1/\lambda$, the tail exhibits the tempered decay $\phi \sim e^{-\lambda x}/x^{\alpha+1}$, and the front velocity approaches the terminal speed $v_*= \left(\gamma-\lambda^\alpha \chi\right)/ \lambda$. Of particular interest is the study of the interplay of regular and fractional diffusion. It is shown that the main role of regular diffusion is to delay the onset of front acceleration. In particular, the crossover time, $t_c$, to transition to the accelerated fractional regime exhibits a logarithmic scaling of the form $t_c \sim \log \left(\chi_d/\chi_f\right)$ where $\chi_d$ and $\chi_f$ are the regular and fractional diffusivities

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“…This paper summarizes briefly a research programme, comprised of five elements: (i) standard deviation analysis and diffusion entropy analysis of solar neutrino data [1,2]; (ii) Mathai's entropic pathway model [3,4]; (iii) fractional reaction and extended thermonuclear functions [5,6]; (iv) fractional reaction and diffusion [7,8]; and (v) fractional reaction-diffusion [9][10][11][12]. Boltzmann translated Clausius' second law of thermodynamics "The entropy of the Universe tends to a maximum" into a crucial quantity that links equilibrium and non-equilibrium (time dependent) properties of physical systems and related entropy to probability, S = k log W, which, later, Einstein called Boltzmann's principle [13].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Solar Neutrino Data from Super-Kamiokande I and II

Haubold

Mathai

Saxena

2014

Entropy

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Abstract:We are going back to the roots of the original solar neutrino problem: the analysis of data from solar neutrino experiments. The application of standard deviation analysis (SDA) and diffusion entropy analysis (DEA) to the Super-Kamiokande I and II data reveals that they represent a non-Gaussian signal. The Hurst exponent is different from the scaling exponent of the probability density function, and both the Hurst exponent and scaling exponent of the probability density function of the Super-Kamiokande data deviate considerably from the value of 0.5, which indicates that the statistics of the underlying phenomenon is anomalous. To develop a road to the possible interpretation of this finding, we utilize Mathai's pathway model and consider fractional reaction and fractional diffusion as possible explanations of the non-Gaussian content of the Super-Kamiokande data.

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Fractional Reaction-Diffusion Equations

Cited by 84 publications

References 31 publications

On the solutions of time-fractional reaction–diffusion equations

On the solutions of time-fractional reaction–diffusion equations

Front propagation in reaction-diffusion systems with anomalous diffusion

Analysis of Solar Neutrino Data from Super-Kamiokande I and II

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