Subdiffusion-Limited<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">A</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="italic">A</mml:mi></mml:math>Reactions

Yuste, S. B.; Lindenberg, Katja

doi:10.1103/physrevlett.87.118301

Cited by 135 publications

(57 citation statements)

References 25 publications

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“…with initial condition: (1 − ) 2 2 (1 − ) 2 , which can be verified by substituting directly into (35). Table 1 shows the maximum error for the numerical solution for the example with α = 1 5 at time = 1.…”

Section: Numerical Examplementioning

confidence: 93%

“…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%

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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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Section: Numerical Examplementioning

confidence: 93%

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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“…Indeed, a recent detailed discussion on ways to extract accurate parameters and exponents from such experiments concludes that at least the experiments presented in that work, carried out in a gel, reflect subdiffusive rather than diffusive motion [41]. We recently solved the A + A reaction-subdiffusion problem in one dimension [42]. To solve this problem, we generalized methods first applied to the reaction-diffusion A+A problem.…”

Section: Introductionmentioning

confidence: 99%

“…The distribution of intervals evolves linearly, and therefore one can find an exact solution. In the reactiondiffusion problem the description involves a diffusion equation, while the reaction-subdiffusion problem involves a subdiffusion equation [42]; both can be solved exactly. For the A + A → C problem one uses instead the odd/even parity method [7,8], whereby one keeps track of the parity of the number of particles in an interval.…”

Section: Introductionmentioning

confidence: 99%

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Reaction front in anA+B→Creaction-subdiffusion process

2004

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We study the reaction front for the process A + B → C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular its unusual behavior at the center of the reaction zone.

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Subdiffusion‐Limited Reactions

Yuste¹,

Lindenberg²,

Ruiz-Lorenzo³

2008

Anomalous Transport

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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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Subdiffusion-LimitedA+AReactions

Cited by 135 publications

References 25 publications

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

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

Reaction front in anA+B→Creaction-subdiffusion process

Subdiffusion‐Limited Reactions

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