Stationary Fronts in an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>B</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:math>Reaction under Subdiffusion

Froemberg, D.; Sokolov, Igor M.

doi:10.1103/physrevlett.100.108304

Cited by 43 publications

(36 citation statements)

References 21 publications

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“…We have considered three different CTRW models with reactions which have been discussed in the literature. We have derived new non-linear Master equations (14) and (26) for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We have applied the theory to the problem of front propagation in reaction-transport systems with KPP-kinetics and non-Markovian diffusion.…”

Section: Discussionmentioning

confidence: 99%

“…We expand the expression in the brackets (14) for small z and truncate the Taylor series at the second moment. We obtain…”

Section: Subdiffusive Transportmentioning

confidence: 99%

“…The density of other species participating in the reactions are held constant. It has been shown recently that for non-Markovian transport we cannot just add the term r (ρ) ρ to the right hand side of the evolution equation (1) [12][13][14][15][16]. It is convenient to represent the non-linear reaction rate r (ρ) as the difference between the birth rate r + (ρ) and the death rate r − (ρ) r (ρ) = r + (ρ) − r − (ρ) .…”

Section: Introductionmentioning

confidence: 99%

“…In particular, there are many efforts to incorporate the chemical reactions into subdiffusive transport. Different models lead to various fractional reaction-diffusion equations corresponding to kinetic regimes [12][13][14][15][16][17][18][19][20] and subdiffusion-limited reactions [21].…”

Section: Introductionmentioning

confidence: 99%

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Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts

Fedotov

2010

Phys. Rev. E

100

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The main aim of the paper is to incorporate the non-linear kinetic term into non-Markovian transport equations described by a continuous time random walk (CTRW) with non-exponential waiting time distributions. We consider three different CTRW models with reactions. We derive new non-linear Master equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems with Kolmogorov-Petrovskii-Piskunov (KPP) kinetics and anomalous diffusion. We have found an explicit expression for the speed of a propagating front in the case of subdiffusive transport.

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Section: Discussionmentioning

confidence: 99%

“…We expand the expression in the brackets (14) for small z and truncate the Taylor series at the second moment. We obtain…”

Section: Subdiffusive Transportmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts

Fedotov

2010

Phys. Rev. E

100

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“…The predictions of those models are quite different from the predictions of models described in Sec. 2 (see, e.g., [38]). In some limits, integro-differential equations can be reduced to PDEs.…”

Section: Linear Reaction Ratesmentioning

confidence: 99%

Mathematical Modelling of Subdiffusion-reaction Systems

Nepomnyashchy

2015

Math. Model. Nat. Phenom.

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Abstract. A review of recent developments in the theoretical description and mathematical modelling of subdiffusion-reaction systems is presented. A number of subdiffusion-reaction models are discussed, including models of subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators. Specific features of the instability development and pattern formation are discussed. Some applications of subdiffusion-reaction models to specific natural problems are mentioned.

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Stochastic modeling analysis of sequential first-order degradation reactions and non-Fickian transport in steady state plumes

2014

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Stochastic analyses were performed to examine sequential first-order monomolecular reactions at the microscopic scale and both Fickian and non-Fickian plume reactive transport at the macroscopic scale. An analytical solution was derived for the chemical master equation (CME) for a closed system of irreversible first-order monomolecular reactions. Taking a Lagrangian reference frame of particles migrating from a source, analyses show that the relative concentration of each species in the deterministic analytical solution for 1-D steady state plug flow with first-order sequential degradation is mathematically equivalent to the mean of a multinomial distribution of plume particles moving at constant velocity with sequential transformations described by transition probabilities of a discrete state, continuous-time Markov chain. In order to examine the coupling of reaction and transport terms in subdiffusive-reactive transport equations, a closed-form multispecies analytical solution also was derived for steady state advection, dispersion, and sequential first-order reaction. Using a 1-D continuous-time random walk (CTRW) embedded in Markov chains, computationally efficient Monte Carlo simulations of particle movement were performed to more fully examine effects of subdiffusive-reactive transport with an application to steady state, sequentially degrading multispecies plumes at a site in Palm, Bay, FL. The simulation results indicated that non-Fickian steady state plumes can resemble Fickian plumes because linear reactions truncate the waiting time between particle jumps, which removes lower velocity particles from the broad spectrum of velocities in highly heterogeneous media. Results show that fitting of Fickian models to plume concentration data can lead to inaccurate estimates of rate constants because of the wide distribution of travel times in highly heterogeneous media.

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Stationary Fronts in anA+B→0Reaction under Subdiffusion

Cited by 43 publications

References 21 publications

Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts

Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts

Mathematical Modelling of Subdiffusion-reaction Systems

Stochastic modeling analysis of sequential first-order degradation reactions and non-Fickian transport in steady state plumes

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