Reaction front 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:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>→</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:mrow><mml:mi>C</mml:mi></mml:math>reaction-subdiffusion process

Yuste, S. B.; Acedo, L.; Lindenberg, Katja

doi:10.1103/physreve.69.036126

Cited by 315 publications

(212 citation statements)

References 95 publications

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“…The case m = n = 1 has been studied by Yuste et al [20]. In the case of normal diffusion, it is known that the width of the reaction zone grows with time as W r ∼ t 1/6 , while the width of the depletion zone, where the concentrations of components A and B are small, grows as W d ∼ t 1/2 .…”

Section: Models For Subdiffusion-limited Reactionsmentioning

confidence: 98%

“…1, the models of subdiffusion-reaction systems can be divided into two groups. The first group of models [19], [20], which are appropriate for subdiffusion-limited reactions, have the structure…”

Section: Models For Subdiffusion-limited Reactionsmentioning

confidence: 99%

“…Models of this kind were derived from the CTRW model for the recombination kinetics [19] and instantaneous creation and annihilation processes in subdiffusive media [17]. It was also suggested for the description of results of Monte-Carlo simulations in [20].…”

Section: Models For Subdiffusion-limited Reactionsmentioning

confidence: 99%

“…Generally, there are two main groups of macroscopic models. In the first group of models, appropriate for diffusion-limited kinetics, the diffusion and reaction terms appear in the equations in an additive way, but the time evolution is described by fractional order derivatives, i.e., the memory kernel is the same for the diffusion and reaction terms [18] - [20]. The other group of models appropriate for activation-limited reactions, is based on the assumption that the reaction constants are unaffected by the diffusion, but the change of the chemical composition due to the reaction affects the diffusion process.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Models For Subdiffusion-limited Reactionsmentioning

confidence: 98%

Section: Models For Subdiffusion-limited Reactionsmentioning

confidence: 99%

Section: Models For Subdiffusion-limited Reactionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mathematical Modelling of Subdiffusion-reaction Systems

Nepomnyashchy

2015

Math. Model. Nat. Phenom.

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“…In the last decade, fractional models, because of their ability to model anomalous transport phenomena, have attracted considerable interest and have played a very important role in various fields of science and engineering. Recently, a growing number of works by many authors from various fields, such as system biology (see [2]), physics (see [4]), chemistry and biochemistry (see [3]), finance (see [5]), hydrology (see [6]) and thermodynamics (see [8]), deal with dynamical systems described by fractional differential equations. Fractional-order models provide an excellent instrument for describing the memory and hereditary properties of various processes.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Liu¹,

Chen²,

Turner³

et al. 2013

Open Physics

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Abstract:Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.PACS (2008)

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Subdiffusion‐Limited Reactions

Yuste¹,

Lindenberg²,

Ruiz-Lorenzo³

2008

Anomalous Transport

Self Cite

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IntroductionAnomalous diffusion processes are ubiquitous in nature [1][2][3][4][5][6]. Their occurrence is usually associated with complex systems that induce spatial and/or temporal correlations in the diffusion process. The signature of normal diffusion is the linear asymptotic dependence of the mean-square displacement of the diffusing entity (hereafter called "the particle") on time, r 2 ∼ t, t → ∞. The signature of anomalous diffusion is a nonlinear dependence on time. In particular, if the growth with time is sublinear, so that(13.1) the particle is said to be subdiffusive. (The process is superdiffusive when the limit goes to infinity.) In this chapter, we focus on the important class of subdiffusive processes for whichand where the (anomalous) diffusion exponent γ satisfies 0 < γ < 1. An interesting class of diffusive processes are so-called diffusion-limited reactions. These are processes in which diffusion is the dominant mixing mechanism and, furthermore, where the time for reactants to find one another is much longer than the time it takes for a reaction to occur following such an encounter. Therefore, in these systems diffusion is the key factor that determines the spatial distribution of reactants and the resultant reaction rate. Since diffusion is not a particularly effective mixing mechanism, diffusionlimited reactions often present extremely interesting spatial as well as temporal characteristics. In this context, it is especially appropriate to point to the pioneering work of Turing on pattern formation in reaction-diffusion systems [7]. Diffusion-limited reactions show up in a vast number of applications including not only chemical (see, e.g., [8]) but also biological (e.g., [9]), ecological (e.g., [10]) and economic processes (e.g., [11]) that have been studied over many decades.

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Reaction front in anA+B→Creaction-subdiffusion process

Cited by 315 publications

References 95 publications

Mathematical Modelling of Subdiffusion-reaction Systems

Mathematical Modelling of Subdiffusion-reaction Systems

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Subdiffusion‐Limited Reactions

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